Nonuniqueness of Generalised Weak Solutions to the Primitive and Prandtl Equations

Abstract

We develop a convex integration scheme for constructing nonunique weak solutions to the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics) in both two and three dimensions. We also develop such a scheme for the construction of nonunique weak solutions to the three-dimensional viscous primitive equations, as well as the two-dimensional Prandtl equations. While in Boutros et al. (Calc Var Partial Differ Equ 62(8):219, 2023) the classical notion of weak solution to the hydrostatic Euler equations was generalised, we introduce here a further generalisation. For such generalised weak solutions, we show the existence and nonuniqueness for a large class of initial data. Moreover, we construct infinitely many examples of generalised weak solutions which do not conserve energy. The barotropic and baroclinic modes of solutions to the hydrostatic Euler equations (which are the average and the fluctuation of the horizontal velocity in the z-coordinate, respectively) that are constructed have different regularities.

1 Introduction

1.1 Problems Considered in this Paper and Context

In this work, we consider the following general equation (with $(d+1)$-dimensional spatial domain for $d=1,2$)

$$\begin{aligned} \partial _t u - \nu _h^* \Delta _h u - \nu _v^* \partial _{zz} u + u\cdot \nabla _h u + w \partial _z u + \nabla _h p = 0, \end{aligned}$$

(1.1)

$$\begin{aligned} \partial _z p = 0, \end{aligned}$$

(1.2)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w = 0, \end{aligned}$$

(1.3)

where the horizontal velocity field $u: {\mathbb {T}}^{d+1} \times (0,T) \rightarrow {\mathbb {R}}^d$, the vertical velocity field $w: {\mathbb {T}}^{d+1} \times (0,T) \rightarrow {\mathbb {R}}$, and the pressure $p: {\mathbb {T}}^{d+1} \rightarrow {\mathbb {R}}$ are unknown, and the horizontal and vertical viscosity parameters $\nu _h^*, \nu _v^* \ge 0$ are given constants. The d-dimensional horizontal gradient is denoted by $\nabla _h$ and the d-dimensional horizontal Laplacian by $\Delta _h$. Since the pressure p is only determined up to an additive constant, we may require that p is mean-free.

In this paper, we are interested in the following cases:

• Taking $d = 2$ and $\nu _h^* = \nu _v^* = 0$ gives the three-dimensional hydrostatic Euler equations of an incompressible fluid (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). In this paper, the terms inviscid primitive equations and hydrostatic Euler equations will be used interchangeably.

• Taking $d = 2$ and $\nu _h^*, \nu _v^* > 0$ leads to the three-dimensional viscous primitive equations. We remark that the cases with anisotropic viscosities ($\nu _h^* > 0$ and $\nu _v^* = 0$, or $\nu _h^* = 0$ and $\nu _v^* > 0$) have also been studied.

• Taking $d = 1$ and $\nu _h^* = \nu _v^* = 0$ yields the two-dimensional inviscid primitive equations (or hydrostatic Euler equations).

• Taking $d = 1$, $\nu _h^* = 0$ and $\nu _v^* > 0$ yields the two-dimensional Prandtl equations.

In this paper, we will develop a convex integration scheme for system (1.1)–(1.3) for the cases mentioned above. In particular, we will work with a generalised notion of weak solution. While classical weak solutions have sufficient Lebesgue integrability for the nonlinearity to make sense as an $L^1 ({\mathbb {T}}^{d+1} \times (0,T))$ function, another notion of weak solution was introduced in Boutros et al. (2023) where the nonlinearity is interpreted as a paraproduct. The generalised weak solutions introduced in this paper treat the nonlinearity in an even more general way, see Sect. 1.3.3.

In all the cases of (1.1)–(1.3) that we are interested in, we will show the existence of such generalised weak solutions (for a dense set of initial data in the relevant spaces). In addition, we will show that such weak solutions are nonunique.

If $\nu _h^* = \nu _v^* = 0$, we recall that classical spatially analytic solutions of (1.1)–(1.3) (see Ghoul et al. 2022; Kukavica et al. 2010, 2011) conserve the energy, i.e. the spatial $L^2 ({\mathbb {T}}^{d+1})$ norm of u. In Boutros et al. (2023), an analogue of Onsager’s conjecture was studied for the three-dimensional hydrostatic Euler equations and it was found that there exist several sufficient regularity criteria for weak solutions which guarantee the conservation of energy. In particular, there exist several notions of weak solutions for these equations, each of which has their own version of the analogue of the Onsager conjecture.

In this work, we will construct generalised weak solutions to these equations, which do not conserve energy and do not satisfy the regularity criteria mentioned above. In other words, in this paper we prove a first result towards the aim of resolving the dissipation part of the analogue of the Onsager conjecture for the inviscid primitive equations (hydrostatic Euler), while the conservation part of the analogue of the Onsager conjecture has been studied in Boutros et al. (2023), as was mentioned before.

1.2 Literature Overview

In this section, we will provide an overview of some of the literature that is related to this work. As both the primitive and Prandtl equations as well as the Onsager conjecture have been the subject matter of many works in recent years, this overview is by no means comprehensive and is by necessity incomplete in reviewing all the relevant work.

Onsager’s conjecture was originally posed in Onsager (1949) for the incompressible Euler equations. The conjecture states that if a weak solution lies in $L^3 ((0,T); C^{0,\alpha } ({\mathbb {T}}^3))$ for $\alpha > \frac{1}{3}$, it must conserve energy. If $\alpha < \frac{1}{3}$, energy might not be conserved.

In Eyink (1994), a proof of a slightly weaker result than the first half of the conjecture was given. A full proof of the first half was then given in Constantin et al. (1994). In Duchon and Robert (2000), a different proof was presented, which relied on an equation of local energy balance and a defect measure. In Bardos and Titi (2018); Bardos et al. (2019) (see also Bardos et al. (2023)), the problem was considered in the presence of physical boundaries and the first half of the conjecture was proved in this case.

The existence of non-energy conserving solutions of the Euler equations of an incompressible fluid was first shown in Scheffer 1993; Shnirelman 2000. To prove the existence of dissipative weak solutions of the Euler equations (and to prove the second half of Onsager’s conjecture), techniques from convex integration were used. They were introduced for the first time in the context of incompressible fluid mechanics in De Lellis and Székelyhidi (2009, 2010).

The second half of the conjecture was then proven in Isett (2018), after gradual success in the papers (De Lellis and Székelyhidi 2013; Buckmaster et al. 2015) (and see references therein). The proof in Isett (2018) relied on the Mikado flows that were developed in Daneri and Székelyhidi (2017). In the work Buckmaster et al. (2019), dissipative Hölder continuous solutions of the Euler equations up to $\frac{1}{3}$ were constructed.

Subsequently, an intermittent version of convex integration was developed. This was first used in Buckmaster and Vicol (2019) to prove the nonuniqueness of very weak (not Leray–Hopf) solutions to the Navier–Stokes equations. In Buckmaster et al. (2021), this result was extended to show the existence of nonunique weak solutions with a bound on the singular set. In Buckmaster et al. (2023) and Novack and Vicol (2023), an intermittent scheme was constructed to prove the existence of non-energy-conserving weak solutions of the Euler equations with Sobolev regularity. In Luo and Titi (2020), the method of Buckmaster and Vicol (2019) was generalised to the hyperviscous Navier–Stokes equations to show the sharpness of the Lions exponent.

After the works (Modena and Székelyhidi 2018, 2019; Modena and Sattig 2020) where a spatially intermittent convex integration scheme was developed for the transport equation, temporal intermittency was introduced to the scheme in Cheskidov and Luo (2021, 2023) to prove the nonuniqueness of weak solutions to the transport equation. This scheme was then adapted to the Navier–Stokes equations in Cheskidov and Luo (2022) to prove the sharpness of one of the Prodi–Serrin criteria and in Cheskidov and Luo (2023) to show that $L^2$ is the critical space for uniqueness for the 2D Navier–Stokes equations.

The primitive equations of oceanic and atmospheric dynamics were introduced in Richardson (1922). They were studied mathematically for the first time in Lions et al. (1992a, 1992b, 1995), in which the global existence of weak solutions was proved. The short time existence of strong solutions was then obtained in Guillén-González et al. (2001). The global well-posedness of the viscous primitive equations was proved in Cao and Titi (2007), see also Kobelkov (2006). In Kukavica and Ziane (2007b, 2007a), different boundary conditions were considered, and in Hieber and Kashiwabara (2016) global well-posedness was established using a semigroup method.

Subsequently, the cases with only horizontal viscosity were studied in Cao et al. (2016, 2017, 2020a). The case with only vertical diffusivity and full viscosity was looked at in Cao et al. (2014b); Cao and Titi (2012). The case with only horizontal diffusivity and full viscosity was investigated in Cao et al. (2014a). The small aspect ratio was rigorously justified in a weak sense in Azérad and Guillén (2001) (see also Bresch et al. 2003). It was subsequently proven in a strong sense with full viscosity in Li and Titi (2019) and with only horizontal viscosity in Li et al. (2022) with error estimates in terms of the small aspect ratio.

The case with only vertical viscosity was studied in Renardy (2009), in which linear ill-posedness was proved. The ill-posedness can be counteracted by adding a linear damping term, see Cao et al. (2020b) for more details. By considering the case of initial data with Gevrey regularity with certain convexity conditions, in Gérard-Varet et al. (2020) local well-posedness was established. By considering small data which are analytic in the horizontal variables, the paper (Paicu et al. 2020) established global well-posedness for the case without rotation and Dirichlet boundary conditions. Finally, Lin et al. (2022) considered the case with rotation and impermeable and stress-free boundary conditions.

The linear and nonlinear ill-posedness of the inviscid primitive equations in all Sobolev spaces was proved in Renardy (2009); Han-Kwan and Nguyen (2016). The ill-posedness results in Sobolev spaces suggest that the natural space for showing local well-posedness of the inviscid primitive equations is the space of analytic functions, which was proved in Ghoul et al. (2022), Kukavica et al. (2010) and Kukavica et al. (2011). In Ghoul et al. (2022), the role of fast rotation in prolonging the life span of solutions was investigated.

In Cao et al. (2015), it was shown that smooth solutions of the inviscid primitive equations can form a singularity in finite time, see also Wong (2015). In Chiodaroli and Michálek (2017), the existence and nonuniqueness of weak solutions with $L^\infty$ data was proved. In Boutros et al. (2023), several sufficient criteria for energy conservation were proved. In the inviscid setting, there have also been works studying the case of initial data with a monotonicity assumption, see Brenier (1999), Kukavica et al. (2014) and Masmoudi and Wong (2012).

In addition to Chiodaroli and Michálek (2017), there have been several papers in which convex integration schemes are developed for geophysical models. In particular, there has been a sequence of works (Tao and Zhang 2018b, 2017, 2018a; Luo et al. 2020) in which nonunique weak solutions for the Boussinesq equations were constructed in a variety of settings. To be precise, in Tao and Zhang (2018a) the effects of vertical viscosity were included, while in Luo et al. (2020) the Boussinesq system with full diffusion for the temperature was studied. Moreover, in Novack (2020) the nonuniqueness of weak solutions was established for the quasi-geostrophic equations.

The equations for the boundary layer were derived in Prandtl (1904). In Oleinik (1963, 1966), the local well-posedness of the equations was shown under a monotonicity assumption. In Sammartino and Caflisch (1998), the local well-posedness for analytic data was proved, while in E and Engquist (1997) the blow-up of solutions for certain classes of $C^\infty$ data was proved. Further local well-posedness results were proved in Kukavica and Vicol (2013), Kukavica et al. (2014), Paicu et al. (2020), Ignatova and Vicol (2016) and Paicu and Zhang (2021). In Xin and Zhang (2004), Xin et al. (2022) and Liu et al. (2016), the global existence of weak solutions was established under the assumption that the pressure is favourable. In Grenier (2000), it was shown that the equations are nonlinearly unstable.

The linear ill-posedness of the Prandtl equations in all Sobolev spaces was shown in Gérard-Varet and Dormy (2010) (for further work see Liu and Yang 2017 and references therein). In the three-dimensional case, a convex integration scheme was developed in Luo and Xin (2018). The analytic local well-posedness has been improved to Gevrey function spaces, see Li et al. (2022) and references therein.

1.3 Definitions and Main Results

1.3.1 Baroclinic and Barotropic Modes

Now we introduce the notion of barotropic and baroclinic modes, which is an important decomposition of the solutions which has been explored extensively in the investigation of the primitive equations. In the construction of the convex integration scheme for the primitive equations, we will not use this decomposition explicitly. However, it is an important idea underlying the scheme.

We will illustrate this concept for the equations in the inviscid case; the viscous case is similar and can be found in Cao and Titi (2007). The 3D inviscid primitive equations are given by

$$\begin{aligned} \partial _t u + u\cdot \nabla _h u + w \partial _z u + \nabla _h p = 0, \end{aligned}$$

(1.4)

$$\begin{aligned} \partial _z p = 0, \end{aligned}$$

(1.5)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w = 0, \end{aligned}$$

(1.6)

where $u: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}^2$ is the horizontal velocity field, $w: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}$ the vertical velocity field and $p: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}$ the pressure.

The barotropic mode ${\overline{u}}$ of a velocity field u is defined as follows

$$\begin{aligned} \begin{aligned} {\overline{u}} (x_1,x_2,t) \,{:=} \int _{{\mathbb {T}}} u(x_1,x_2,z,t) \,\text {d} z. \end{aligned} \end{aligned}$$

(1.7)

The baroclinic mode ${\widetilde{u}}$ is defined as the fluctuation

$$\begin{aligned} \begin{aligned} {\widetilde{u}} \,{:=}\, u - {\overline{u}}. \end{aligned} \end{aligned}$$

(1.8)

The primitive equations (1.4)–(1.6) can then be written formally as a coupled system of evolution equations for the barotropic and baroclinic modes ${\overline{u}}$ and ${\widetilde{u}}$, which are

$$\begin{aligned}&\partial _t {\overline{u}} + ({\overline{u}} \cdot \nabla _h) {\overline{u}} + \overline{\big [ ({\widetilde{u}} \cdot \nabla _h) {\widetilde{u}} + (\nabla _h \cdot {\widetilde{u}}) {\widetilde{u}} \big ]} + \nabla _h p = 0, \end{aligned}$$

(1.9)

$$\begin{aligned}&\partial _t {\widetilde{u}} + ({\widetilde{u}} \cdot \nabla _h ) {\widetilde{u}} + w \partial _z {\widetilde{u}} + ({\widetilde{u}} \cdot \nabla _h) {\overline{u}} + ({\overline{u}} \cdot \nabla _h) {\widetilde{u}} - \overline{\big [ ({\widetilde{u}} \cdot \nabla _h) {\widetilde{u}} + (\nabla _h \cdot {\widetilde{u}}) {\widetilde{u}} \big ]} = 0. \end{aligned}$$

(1.10)

Moreover, we have the following incompressibility conditions

$$\begin{aligned} \nabla _h \cdot {\overline{u}} = \nabla _h \cdot {\widetilde{u}} + \partial _z w = 0, \end{aligned}$$

(1.11)

which formally follow from Eq. (1.6) and the periodicity of the functions.

In the convex integration scheme, we will add separate barotropic and baroclinic perturbations. This leads to different regularities of the barotropic and baroclinic modes of the solution and allows us to control different parts of the error.

The following estimates on the baroclinic and barotropic modes are standard

$$\begin{aligned} \Vert {\overline{u}} \Vert _{L^p} \lesssim \Vert u \Vert _{L^p}, \qquad \Vert {\widetilde{u}} \Vert _{L^p} \lesssim \Vert u \Vert _{L^p}. \end{aligned}$$

1.3.2 Notation

Throughout the paper, we will use the following notation.

• The components of the spatial variable are given by $x=(x_1,z)$ if $d=1$, and $x=(x_1,x_2,z)$ if $d=2$. For $d=1$, $x_1$ represents the horizontal direction; for $d=2$ the horizontal position is given by $(x_1,x_2)$. In both cases, z is the vertical direction.

• The horizontal velocity field is called u; the vertical velocity is denoted by w and the full velocity by ${\textbf{u}}=(u,w)$. They are d-, 1- and $(d+1)$-dimensional, respectively.

• We use the symbol $\nabla _h$ for the horizontal gradient (which equals $\partial _{x_1}$ if $d=1$) and $\nabla$ for the full ($(d+1)$-dimensional) gradient.

• For an integrability parameter $1\le p\le \infty$, the Hölder conjugate is denoted by $p'$, i.e. $\frac{1}{p} + \frac{1}{p'} = 1$.

• Let $1< p\le \infty$. In Sect. 1, $p-$ denotes any parameter $1\le p- <p$. In the other sections, we have to be a bit more precise. In particular, there is a need to quantify the ‘−’ in $p-$. More precisely, there will be a $\delta >0$ and we set $p-:= \frac{1}{\frac{1}{p}+\delta }$. Here we tacitly assume that $\delta$ is sufficiently small, such that $p-\ge 1$.

• For $1\le p,q\le \infty$ and $s\in {\mathbb {R}}$, the Besov space $B_{p,q}^s({\mathbb {T}}^3)$ is defined in Appendix A.1. Let us emphasise here that $B_{2,2}^s({\mathbb {T}}^3)=H^s({\mathbb {T}}^3)$, see Remark A.2.

• Throughout this paper, we will omit the domain of a space-time norm if it is ${\mathbb {T}}^{d+1}\times [0,T]$, e.g. we write $\Vert \cdot \Vert _{L^p(H^s)} = \Vert \cdot \Vert _{L^p((0,T);H^s({\mathbb {T}}^{d+1}))}$.

• In view of Sect. 1.3.1, we define the barotropic and baroclinic part of any quantity $a=a(x)$ by

  $$\begin{aligned} {\overline{a}} = \int _{{\mathbb {T}}} a(x) \,\textrm{d} z, \qquad {\widetilde{a}} = a - {\overline{a}}. \end{aligned}$$

1.3.3 Generalised Weak Solutions

In Boutros et al. (2023), two new types of weak solutions to the hydrostatic Euler equations (1.4)–(1.6) were introduced. In the present paper, we will consider a slightly different notion of weak solution, which we will refer to as a generalised weak solution. This notion of solution is inspired by the notion of a type III weak solution, as introduced in Boutros et al. (2023).

Before we state the results for the different cases of the system (1.1)–(1.3), we will be more specific regarding the notion of weak solution used in this paper. The weak solutions of (1.1)–(1.3) we consider are defined as follows: We assume that $u \in L^2 ({\mathbb {T}}^3 \times (0,T))$, $w \in {\mathcal {D}}' ( {\mathbb {T}}^3 \times (0,T) )$ and $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ for some suitably large $s \in {\mathbb {R}}$. System (1.1)–(1.3) must then be satisfied in the sense of distributions, where the vertical advection term

$$\begin{aligned} \int _0^T \langle u w, \partial _z \phi \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t, \end{aligned}$$

is interpreted as a duality bracket between the term uw and the test function $\phi \in {\mathcal {D}} ( {\mathbb {T}}^3 \times (0,T))$.

If u and w happen to have sufficient regularity, for example when $u \in L^2 ((0,T); H^{s + \delta } ({\mathbb {T}}^3))$ and $w \in L^2 ((0,T); H^{-s} ({\mathbb {T}}^3))$ (for some small $\delta > 0$), then by applying the paradifferential calculus (see Appendix A) we know that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$. This is a stronger notion of solution compared to the notion of a generalised weak solution that we introduced above, as u is required to have (positive) Sobolev regularity and w has to possess some regularity (i.e. it is more than just a distribution).

The reason we introduce these generalised weak solutions to the system (1.1)–(1.3) is that the velocity field does not have isotropic regularity. In particular, the vertical part of the advection term can be formally written as follows

$$\begin{aligned} \partial _z (w u) = - \partial _z \bigg [ \bigg ( \int _0^z \nabla _h \cdot u dz' \bigg ) u \bigg ]. \end{aligned}$$

(1.12)

The only a priori regularity bound for u in the inviscid case is that $u \in L^\infty ((0,T); L^2 ({\mathbb {T}}^3))$ (i.e. u lies in the energy space). Therefore, by relation (1.12) it is natural to consider weak solutions for the system (1.1)–(1.3) such that w lies in a negative Sobolev space with respect to the spatial variables.

As was already mentioned, solutions of such form have been introduced in Boutros et al. (2023) (by applying paradifferential calculus), but their existence was left open. One of the aims of the present work is to prove the existence of solutions of this type for the primitive and Prandtl equations. Generally speaking, this approach provides a general way of defining weak solutions for systems with a loss of derivative (for the system (1.1)–(1.3) the loss of derivative is in the horizontal directions).

In the next few subsections, we will give precise definitions of the notion of weak solution we will use, and we will state the theorems we will prove for the different cases of the system (1.1)–(1.3). But generally speaking, we will split the nonlinearity uw into the barotropic-vertical and baroclinic-vertical interactions, i.e. the terms ${\overline{u}} w$ and ${\widetilde{u}} w$.

The baroclinic mode ${\overline{u}}$ of the constructed solutions will have sufficient regularity such that ${\overline{u}} w$ can be interpreted as a paraproduct. The terms ${\widetilde{u}}$ and w do not have sufficient regularity to apply the paradifferential calculus. However, as part of the convex integration scheme we will obtain separate estimates on ${\widetilde{u}} w$ in order to show that it lies in $L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ for some suitable s. Therefore, the weak solutions we obtain are partly ‘generalised’ (as for the baroclinic-vertical part of the nonlinearity) and partly ‘paradifferential’ (for the barotropic-vertical part of the nonlinearity).

1.3.4 Results for the 3D Inviscid Primitive Equations

We first introduce the notion of weak solution for the 3D inviscid primitive equations (1.4)–(1.6).

Definition 1.1

A triple (u, w, p) is called a weak solution of the hydrostatic Euler equations (1.4)–(1.6) if $u \in L^2 ({\mathbb {T}}^3 \times (0,T))$, $w \in {\mathcal {D}}' ({\mathbb {T}}^3 \times (0,T) )$ and $p \in L^1 ({\mathbb {T}}^3 \times (0,T))$ such that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ (where $s>0$ is referred to as the regularity parameter) and the equations are satisfied in the sense of distributions, i.e.

$$\begin{aligned} \int _0^T \int _{{\mathbb {T}}^3} u \cdot \partial _t \phi _1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} u \otimes u : \nabla _h \phi _1 \,\textrm{d} x\,\textrm{d} t\qquad \quad&\nonumber \\ + \int _0^T \langle uw , \partial _z \phi _1 \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} p \nabla _h \cdot \phi _1 \,\textrm{d} x\,\textrm{d} t&= 0, \end{aligned}$$

(1.13)

$$\begin{aligned} \int _0^T \int _{{\mathbb {T}}^3} p \partial _z \phi _2 \,\textrm{d} x\,\textrm{d} t&= 0, \end{aligned}$$

(1.14)

$$\begin{aligned} \int _0^T \langle {{\textbf {u}}} , \nabla \phi _3 \rangle \,\textrm{d} t&= 0, \end{aligned}$$

(1.15)

for all test functions $\phi _1, \phi _2$ and $\phi _3$ in ${\mathcal {D}} ({\mathbb {T}}^3 \times (0,T))$.

Remark 1.2

We emphasise that this definition of weak solutions to (1.4)–(1.6) is more general than the notion of weak solution introduced in Boutros et al. (2023). While in Boutros et al. (2023) the velocity field of a weak solution has sufficient regularity to automatically guarantee that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ (by using the paradifferential calculus), in Definition 1.1 we do not have sufficient (separate) regularity requirements on u and w such that the product uw is well-defined. Hence, $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ is a separate independent requirement of Definition 1.1.

Remark 1.3

It should be noted that for a general $u \in L^2 ({\mathbb {T}}^3 \times (0,T))$ and $w \in {\mathcal {D}}' ({\mathbb {T}}^3 \times (0,T) )$ the product uw need not be well-defined. Hence, the meaning of the requirement from Definition 1.1 that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ could be unclear in such cases. Therefore, we specify what we mean with uw in this general setting.

Firstly, both u and w can be expressed as a Fourier series. Therefore, one can formally define uw as a Fourier series with coefficients which are the convolution of the Fourier coefficients of u and w (in general the convolutions need not to converge). Therefore, the requirement $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ can be understood as the condition that the convolutions of the Fourier coefficients of u and w converge and also that subsequently the Fourier series is bounded in this Besov norm.

In this paper, we will prove the following result.

Theorem 1.4

Let $T>0$ and suppose there exist smooth solutions of the hydrostatic Euler equations (1.4)–(1.6) $(u_1,w_1,p_1)$ on [0, T/2] and $(u_2,w_2,p_2)$ on [T/2, T]. Moreover, let $1 \le q_1, q_2, q_3 \le \infty$ and¹$0 < s_1, s_3$ be parameters satisfying

$$\begin{aligned} q_2&> 2, \quad q_3 \le q_1, \quad s_1> s_3, \quad \frac{2}{q_1} > s_1 + 1. \end{aligned}$$

(1.16)

Then there exists a weak solution (u, w, p) in the sense of Definition 1.1 with regularity parameter $s=1$ and with the following properties:

• 1. The solution satisfies that

  $$\begin{aligned} (u,w,p)(\cdot ,t) = \left\{ \begin{array}{ll} (u_1,w_1,p_1)(\cdot ,t) &{} \text { if } t\in [0,T/4), \\ (u_2,w_2,p_2)(\cdot ,t) &{} \text { if } t\in (3T/4,T]. \end{array} \right. \end{aligned}$$

  (1.17)

• 2. We have that

  $$\begin{aligned} {\overline{u}}&\in L^2({\mathbb {T}}^3\times (0,T)) \cap L^{q_1} ((0,T); H^{s_1} ({\mathbb {T}}^3)), \\ {\widetilde{u}}&\in L^{q_2-} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3-} ((0,T); H^{s_3} ({\mathbb {T}}^3)) , \\ w&\in L^{q_2'} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3'} ((0,T) ; H^{-s_3} ({\mathbb {T}}^3)), \end{aligned}$$

  where ${\overline{u}}$ and ${\widetilde{u}}$ denote the barotropic and baroclinic modes of u, respectively.

Remark 1.5

Alternatively one can construct a weak solution with the properties stated in Theorem 1.4 where the only difference is that the endpoint time integrability is attained for ${\widetilde{u}}$ rather than w. In other words,

$$\begin{aligned} {\widetilde{u}}&\in L^{q_2} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3} ((0,T); H^{s_3} ({\mathbb {T}}^3)) , \\ w&\in L^{q_2'-} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3'-} ((0,T) ; H^{-s_3} ({\mathbb {T}}^3)), \end{aligned}$$

see Remarks 2.6 and 5.6. To this end however, we have to require that $q_3< q_1$ (strictly) in (1.16).

Remark 1.6

By proceeding as in Sect. 7, we can achieve in addition that ${\overline{u}},{\widetilde{u}}\in L^1((0,T);W^{1,1}({\mathbb {T}}^3))$. To this end, however, we have to require the constraints (1.27) rather than (1.16), see also Theorem 1.13 and Remark 1.16.

Remark 1.7

Again we would like to remark that the solutions constructed in Theorem 1.4 are partially ‘generalised’ (see Sect. 1.3.3) and partially ‘paradifferential’ as in Boutros et al. (2023). In particular, they have been inspired by the type III weak solutions that were introduced in Boutros et al. (2023).

More precisely, from the regularities of ${\overline{u}}$ and w stated in Theorem 1.4 it follows that ${\overline{u}} w \in L^1 ((0,T); B^{- 1}_{1,\infty } ({\mathbb {T}}^3))$ (see the proof of Theorem 1.4 in Sects. 4–6 for details). The term ${\widetilde{u}} w$ is estimated directly in $L^1 ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$ as part of the convex integration scheme, as one cannot obtain the regularity of the product ${\widetilde{u}} w$ simply from the regularities of ${\widetilde{u}}$ and w (as they are insufficient to apply the paradifferential calculus directly).

The specific form of the perturbations allows for a direct estimate, as was done for example in Cheskidov and Luo (2023). Therefore, the interpretation of the term ${\overline{u}} w$ can be seen as ‘paradifferential’, while the interpretation of the term ${\widetilde{u}} w$ is in the sense of a ‘generalised weak solution’ (as in Definition 1.1).

Remark 1.8

In addition, we would like to emphasise that in the presence of physical boundaries the primitive equations are often studied with no-normal flow boundary conditions on the top and bottom of the channel, i.e. $w |_{z=0,1} = 0$. However, in the convex integration scheme developed in this paper we will work on the three-dimensional torus rather than the channel. Note that solutions in the torus can be understood as solutions in the channel with an in-flow out-flow boundary condition, i.e.

$$\begin{aligned} w ( x_1, x_2, 0, t) = w (x_1,x_2, 1, t) = w_B (x_1,x_2,t), \end{aligned}$$

for a flow $w_B$. In our case, $w_B$ will be constructed as part of the convex integration scheme. In other words, we will not solve the boundary value problem for given $w_B$ and in particular, not for the case of the impermeability boundary condition $w_B = 0$.

We also remark that the constructed flow $w_B$ belongs to the space $L^{q_2'} ((0,T); L^2 ({\mathbb {T}}^2)) \cap L^{q_3'} ((0,T); H^{-s_3} ({\mathbb {T}}^2))$, where the parameters $q_2', q_3'$ and $s_3$ are the same as in Theorem 1.4.

Remark 1.9

In this paper, we will not consider the role of density variations in the Boussinesq approximation model of the primitive equations. If one takes density effects into account and applies the Boussinesq approximation, the full inviscid primitive equations are given by

$$\begin{aligned} \partial _t u + u\cdot \nabla _h u + w \partial _z u + \nabla _h p = 0, \end{aligned}$$

(1.18)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w = 0, \end{aligned}$$

(1.19)

$$\begin{aligned} \partial _z p + g \rho = 0, \end{aligned}$$

(1.20)

$$\begin{aligned} \rho = \rho (T,S,p), \end{aligned}$$

(1.21)

$$\begin{aligned} \partial _t T + u \cdot \nabla _h T + w \partial _z T = 0, \end{aligned}$$

(1.22)

$$\begin{aligned} \partial _t S + u \cdot \nabla _h S + w \partial _z S = 0, \end{aligned}$$

(1.23)

where $\rho$ is the density, g is the gravitational constant, T is the temperature and S is the salinity. Below we will ignore the salinity effects.

We note that for system (1.18)–(1.23) to be fully determined, an equation of state $\rho (T,p)$ needs to be provided. The well-posedness results for the viscous case in Cao and Titi (2007) and Kukavica and Ziane (2007a) consider the case of a linear equation of state, i.e. the assumption

$$\begin{aligned} \rho = \rho _0 - \alpha (T - T_0), \end{aligned}$$

where $\rho _0$ and $T_0$ are the mean density and temperature, respectively. The thermal expansion coefficient is denoted by $\alpha$. In Korn (2021), the global well-posedness of the viscous primitive equations with a nonlinear equation of state was established.

If one includes density variations as part of the equations, the nature of the pressure changes. Namely, the pressure can now be split into a surface pressure $p_{\textrm{s}}$ (which is independent of z and in our case can be taken to be equal to the pressure at $z = 0$) and the hydrostatic pressure $p_{\textrm{hyd}}$ (which is the integral with respect to the vertical coordinate of the density)

$$\begin{aligned} p (x_1,x_2,z,t)= & {} p_{\textrm{s}} (x_1,x_2,t) + p_{\textrm{hyd}} (x_1,x_2,z,t), \\ p_{\textrm{hyd}} (x_1,x_2,z,t)= & {} \int _0^z \rho (x_1,x_2,z',t) \,\textrm{d} z'. \end{aligned}$$

The surface pressure solves an elliptic problem, and it has played an important role in the development of numerical schemes to solve the primitive equations, see, for example, Samelson et al. (2003), Pinardi et al. (1995), Smith et al. (1992) and Dukowicz et al. (1993).

In the convex integration scheme developed in this paper, introducing density variations as part of the dynamics leads to an additional linear error in the construction (which is coming from the hydrostatic pressure) which with the current approach can probably not be controlled. We leave the consideration of this very important additional effect to future work.

Theorem 1.4 allows to show the nonuniqueness and existence of solutions which do not conserve energy:

Corollary 1.10

For any analytic initial data, there exist infinitely many global-in-time weak solutions (u, w, p) of the hydrostatic Euler equations (1.4)–(1.6) (in the sense of Definition 1.1 which satisfy the regularity properties of Theorem 1.4) and they do not conserve energy.

Proof

We take the smooth local-in-time solution for the given choice of analytic data (whose existence can be proven using the methods from Kukavica et al. 2011; Ghoul et al. 2022) as the first solution $(u_1, w_1, p_1)$ on [0, T/2], and the zero solution on [T/2, T] as the second solution $(u_2, w_2, p_2)$. Then Theorem 1.4 yields a weak solution, which we may extend by zero for $t>T$. If the initial data are nonzero, we can conclude that the energy is not conserved as it is positive on [0, T/4) and zero on $(3T/4,\infty )$.

Another global-in-time weak solution can be constructed similarly with replacing T by T/4. This solution has positive energy on [0, T/16) while the energy is zero on $(3T/16,\infty )$. Consequently, the two solutions cannot coincide. Repeating this argument leads to infinitely many global-in-time weak solutions with the same initial data, which are smooth and unique for a small initial interval of time, but which do not conserve energy.

For zero initial data, we observe that Theorem 1.4 allows one to ‘connect’ any analytic initial data with any analytic data in finite time. Hence, we may connect the zero initial data to arbitrary analytic data with positive energy at $t={\widetilde{T}}$. On the time interval $[{\widetilde{T}},\infty )$, we then proceed as above. $\square$

Remark 1.11

Finally, we should emphasise that we do not claim that the solutions constructed in Theorem 1.4 belong to $u \in L^\infty ((0,T); L^2 ({\mathbb {T}}^3))$. Therefore, it is not possible to directly compare the results in this paper with the Onsager-type conjectures which were formulated in Boutros et al. (2023), and there is no admissibility criterion for the constructed weak solutions at this point. We leave the study of the sharpness of the conjectures from Boutros et al. (2023) to future work.

1.3.5 Results for the 3D Viscous Primitive Equations

We now consider the viscous primitive equations, which are given by

$$\begin{aligned} \partial _t u - \nu _h^* \Delta _h u - \nu _v^* \partial _{zz} u + u\cdot \nabla _h u + w\partial _z u + \nabla _h p&= 0, \end{aligned}$$

(1.24)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(1.25)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w&= 0, \end{aligned}$$

(1.26)

where $\nu _h^*$ and $\nu _v^*$ are the horizontal and vertical viscosities. As before, $u: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}^2$ is the horizontal velocity field, $w: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}$ the vertical velocity and $p: {\mathbb {T}}^3 \times (0,T) \rightarrow {\mathbb {R}}$ the pressure. We have the following notion of weak solution for these equations.

Definition 1.12

A triple (u, w, p) is called a weak solution of the viscous primitive equations (1.24)–(1.26) if $u \in L^2 ({\mathbb {T}}^3 \times (0,T)) \cap L^1 ((0,T); W^{1,1} ({\mathbb {T}}^3))$, $w \in {\mathcal {D}}' ({\mathbb {T}}^3 \times (0,T))$ and $p \in L^1 ( {\mathbb {T}}^3 \times (0,T))$ such that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^3))$ (where $s > 0$ is referred to as the regularity parameter) and the equations are satisfied in the sense of distributions, i.e.

$$\begin{aligned}&\int _0^T \int _{{\mathbb {T}}^3} u \cdot \partial _t \phi _1 \,\textrm{d} x\,\textrm{d} t- \nu _h^* \int _0^T \int _{{\mathbb {T}}^3}\nabla _h u : \nabla _h \phi _1 \,\textrm{d} x\,\textrm{d} t- \nu _v^* \int _0^T \int _{{\mathbb {T}}^3} \partial _z u \cdot \partial _z \phi _1 \,\textrm{d} x\,\textrm{d} t\\&\qquad + \int _0^T \int _{{\mathbb {T}}^3} u\otimes u : \nabla _h \phi _1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \langle uw, \partial _z \phi _1 \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t\\&\qquad + \int _0^T \int _{{\mathbb {T}}^3} p \nabla _h \cdot \phi _1 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \int _{{\mathbb {T}}^3} p \partial _z \phi _2 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \langle \textbf{u}, \nabla \phi _3 \rangle \,\textrm{d} t= 0, \end{aligned}$$

for all test functions $\phi _1$, $\phi _2$ and $\phi _3$ in ${\mathcal {D}} ({\mathbb {T}}^3 \times (0,T))$.

In this paper, we will prove the following result.

Theorem 1.13

Let $T>0$ and suppose there exist smooth solutions of the viscous primitive equations (1.24)–(1.26) $(u_1,w_1,p_1)$ on [0, T/2] and $(u_2,w_2,p_2)$ on [T/2, T]. Moreover, let $1 \le q_1, q_2, q_3 \le \infty$ and $0 < s_1, s_3$ be parameters satisfying the following relations

$$\begin{aligned} q_2&> 2, \quad q_3 < q_1 , \quad s_1> s_3, \quad \frac{2}{q_1}> s_1 + 1, \quad s_3> \frac{1}{2\left( 1-\frac{1}{q_2}\right) } \left( \frac{1}{q_3} - \frac{1}{q_2}\right) . \end{aligned}$$

(1.27)

Then there exists a weak solution (u, w, p) in the sense of Definition 1.12 with regularity parameter $s=1$ and with the following properties:

• 1. The solution satisfies that

  $$\begin{aligned} (u,w,p)(\cdot ,t) = \left\{ \begin{array}{ll} (u_1,w_1,p_1)(\cdot ,t) &{} \text { if } t\in [0,T/4), \\ (u_2,w_2,p_2)(\cdot ,t) &{} \text { if } t\in (3T/4,T]. \end{array} \right. \end{aligned}$$

• 2. We have that

  $$\begin{aligned} {\overline{u}}&\in L^2 ({\mathbb {T}}^3 \times (0,T)) \cap L^{q_1} ((0,T); H^{s_1} ({\mathbb {T}}^3)) \cap L^1 ((0,T); W^{1,1} ({\mathbb {T}}^3)), \\ {\widetilde{u}}&\in L^{q_2-} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3-} ((0,T); H^{s_3} ({\mathbb {T}}^3)) \cap L^1 ((0,T); W^{1,1} ({\mathbb {T}}^3)) ,\\ w&\in L^{q_2'-} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3'-} ((0,T); H^{-s_3} ({\mathbb {T}}^3)). \end{aligned}$$

Remark 1.14

Similar to Theorem 1.4, one can even obtain endpoint time integrability for w. With the modification described in Remarks 1.5, 2.6 and 5.6, one can alternatively establish endpoint time integrability for ${\widetilde{u}}$.

Remark 1.15

The reader should notice that there exist parameters $1 \le q_1, q_2, q_3 \le \infty$ and $0 < s_1, s_3$ satisfying (1.27). Indeed for every $q_3<3/2$, we have

$$\begin{aligned} \frac{1}{q_3} > \frac{1}{4q_3} + \frac{1}{2}. \end{aligned}$$

Hence, there exists $q_1$ with

$$\begin{aligned} \frac{1}{q_3}> \frac{1}{q_1} > \frac{1}{4q_3} + \frac{1}{2}. \end{aligned}$$

Thus, $q_3<q_1$ and for $q_2>2$ sufficiently large, the estimate

$$\begin{aligned} \frac{2}{q_1} -1 > \frac{1}{2\left( 1-\frac{1}{q_2}\right) } \left( \frac{1}{q_3} - \frac{1}{q_2}\right) \end{aligned}$$

holds since the right-hand side converges to $\frac{1}{2q_3}$ for $q_2\rightarrow \infty$. This allows to choose $s_1$ and $s_3$ such that

$$\begin{aligned} \frac{2}{q_1} -1> s_1> s_3 > \frac{1}{2\left( 1-\frac{1}{q_2}\right) } \left( \frac{1}{q_3} - \frac{1}{q_2}\right) , \end{aligned}$$

so all constraints in (1.27) are satisfied.

Remark 1.16

We would like to emphasise that Theorem 1.13 holds for any choice of viscosities $\nu _h^*,\nu _v^*\in {\mathbb {R}}$, in particular even for the inviscid case $\nu _h^*=\nu _v^*=0$.

Remark 1.17

In the case of the 2D and 3D viscous primitive equations, it has been shown in Medjo (2010), Ju (2017) and Petcu (2007) that the so-called z-weak solutions of the viscous primitive equations are unique. Such weak solutions possess the regularity

$$\begin{aligned} u, \partial _z u \in L^\infty ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^2 ( (0,T); H^1 ({\mathbb {T}}^3)). \end{aligned}$$

The solutions constructed in Theorem 1.13 possess the regularity $u, \partial _z u \in L^1 ((0,T); W^{1,1} ({\mathbb {T}}^3)) \cap L^1 ((0,T); H^{1-} ({\mathbb {T}}^3))$. However, the convex integration scheme used to prove Theorem 1.13 is currently unable to obtain solutions such that $u, \partial _z u \in L^{2-} ((0,T); H^1 ({\mathbb {T}}^3))$. We leave the study of the sharpness of the z-weak solution regularity class with respect to the (non)uniqueness problem to future work.

Remark 1.18

Finally, we emphasise that the solutions constructed in Theorem 1.13 are not of Leray–Hopf type, as they do not have a finite rate of mean energy dissipation (i.e. the horizontal velocity field does not belong to the space $L^2 ((0,T);H^1({\mathbb {T}}^3))$).

We now obtain the global existence of weak solutions as a corollary.

Corollary 1.19

For $\nu _h^*,\nu _v^*>0$ and any initial data $u_0 \in H^{1} ({\mathbb {T}}^3)$, there exist infinitely many global-in-time weak solutions (u, w, p) of the viscous primitive equations (1.24)–(1.26) (in the sense of Definition 1.12) which satisfy the regularity properties of Theorem 1.13.

Proof

The proof works exactly as the proof of Corollary 1.10 where the corresponding local (even global) well-posedness result can be achieved by using the methods from Cao and Titi (2007). $\square$

Remark 1.20

The proof of nonuniqueness of global weak solutions works equally well in the three cases of full, horizontal or vertical viscosity, which were studied in the works (Cao et al. 2016, 2017, 2020a). Moreover, in the case of full viscosity the result can also be adapted to classes of initial data belonging to different function spaces, by relying on the well-posedness results from Giga et al. (2020).

1.3.6 Results for the 2D Hydrostatic Euler Equations

It is also possible to develop a convex integration scheme for the two-dimensional hydrostatic Euler equations. They are given by

$$\begin{aligned} \partial _t u + u \partial _{x_1} u + w \partial _z u + \partial _{x_1} p&= 0, \end{aligned}$$

(1.28)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(1.29)

$$\begin{aligned} \partial _{x_1} u + \partial _z w&=0, \end{aligned}$$

(1.30)

where $u: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ is the horizontal velocity, $w: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ is the vertical velocity and $p: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ is the pressure. We first state the definition of weak solution to these equations.

Definition 1.21

A triple (u, w, p) is called a weak solution of the two-dimensional hydrostatic Euler equations (1.28)–(1.30) if $u \in L^2 ({\mathbb {T}}^2 \times (0,T))$, $w \in {\mathcal {D}}' ({\mathbb {T}}^2 \times (0,T))$ and $p \in L^1 ({\mathbb {T}}^2 \times (0,T))$ such that $u w \in L^1 ( (0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^2))$ (where $s > 0$ is the regularity parameter) and the equations are satisfied in the sense of distributions, i.e.

$$\begin{aligned}&\int _0^T \int _{{\mathbb {T}}^2} u \partial _t \phi _1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^2} u^2 \partial _{x_1} \phi _1 \,\textrm{d} x\,\textrm{d} t\\&\qquad +\int _0^T \langle u w , \partial _z \phi _1 \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^2} p \partial _{x_1} \phi _1 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \int _{{\mathbb {T}}^2} p \partial _z \phi _2 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \langle \textbf{u}, \nabla \phi _3 \rangle \,\textrm{d} t= 0, \end{aligned}$$

for all test functions $\phi _1$, $\phi _2$ and $\phi _3$ in ${\mathcal {D}}({\mathbb {T}}^2 \times (0,T))$.

In particular, we will prove the following theorem.

Theorem 1.22

Let $T>0$ and suppose there exist smooth solutions of the two-dimensional hydrostatic Euler equations (1.28)–(1.30) $(u_1,w_1,p_1)$ on [0, T/2] and $(u_2,w_2,p_2)$ on [T/2, T]. Moreover, let $1 \le q_2, q_3 \le \infty$ and $0 < s_3$ be parameters satisfying²

$$\begin{aligned} \frac{3}{2} q_2 \left( \frac{1}{q_3} - \frac{1}{q_2}\right)> s_3> \frac{1}{1-\frac{2}{q_2}} \left( \frac{1}{q_3} - \frac{1}{q_2}\right) >0, \quad 1\ge s_3. \end{aligned}$$

(1.31)

Then there exists a weak solution (u, w, p) in the sense of Definition 1.21 with regularity parameter $s=1$ and with the following properties:

• 1. The solution satisfies that

  $$\begin{aligned} (u,w,p)(\cdot ,t) = \left\{ \begin{array}{ll} (u_1,w_1,p_1)(\cdot ,t) &{} \text { if } t\in [0,T/4), \\ (u_2,w_2,p_2)(\cdot ,t) &{} \text { if } t\in (3T/4,T]. \end{array} \right. \end{aligned}$$

• 2. We have that

  $$\begin{aligned} u&\in L^{q_2-} ((0,T); L^2 ({\mathbb {T}}^2)) \cap L^{q_3-} ((0,T); H^{s_3} ({\mathbb {T}}^2)), \\ w&\in L^{q_2'-} ((0,T); L^2 ({\mathbb {T}}^2)) \cap L^{q_3'-} ((0,T); H^{-s_3} ({\mathbb {T}}^2)). \end{aligned}$$

Remark 1.23

It might seem slightly odd to label the parameters by $q_2$, $q_3$ and $s_3$ (rather than $q_1$ etc.). The reason we chose to do so is because it will allow for easy comparisons with the three-dimensional scheme from Theorem 1.4. We emphasise that there are no equivalent parameters to $q_1$ and $s_1$ in the two-dimensional version of the scheme.

Remark 1.24

Remark 1.14 is also true in the context of the two-dimensional hydrostatic Euler equations (1.28)–(1.30), see Remark 8.6.

Remark 1.25

By proceeding as in Sect. 9, we can achieve in addition that ${\overline{u}},{\widetilde{u}}\in L^1((0,T);W^{1,1}({\mathbb {T}}^3))$. In contrast with the three-dimensional case (cf. Remark 1.6), in two dimensions there is no need to require stronger constraints for the parameters, see also Theorem 1.29 and Remark 1.31.

We observe that it is possible to establish a two-dimensional analogue of Corollary 1.10 using the local well-posedness result from Kukavica et al. (2010) and Kukavica et al. (2011) for analytic data in the channel. This yields existence of infinitely many global weak solutions for suitable initial data.

Remark 1.26

We observe that a suitable modification (essentially by changing the scaling of the Mikado flows and densities in Proposition 3.9) of the proof of Theorem 1.22 yields the existence of nonunique weak solutions of the two-dimensional hydrostatic Euler equations in the energy space, i.e. solutions such that $u \in L^\infty ((0,T); L^2 ({\mathbb {T}}^2))$. To the knowledge of the authors, this is the first example of a temporally intermittent convex integration scheme that is able to construct weak solutions lying in the energy space (which in essence is due to the structure of the primitive equations). Moreover, we expect that by using some of the ideas in for example (Cheskidov and Luo 2023) one can achieve control over the energy profile (but this lies outside the scope of this work).

Remark 1.27

By taking $q_3 = 3$, $s_3 > \frac{2}{3}$ and $q_2 > \frac{13}{3}$ in Theorem 1.22, we observe that the solutions constructed have the regularity $u \in L^3 ((0,T); B^{1/3+}_{3,\infty } ({\mathbb {T}}^2))$. This regularity is sufficient (by an argument similar to Boutros et al. 2023) to conclude that the horizontal advection term is not responsible for any dissipation or increase in energy. In this setting, the mechanism resulting in non-conservation of energy is ‘purely hydrostatic’, i.e. completely due to the irregularity of the vertical velocity w.

In fact, by taking $q_3 = 4$, $s_3 > \frac{3}{4}$ and $q_2 > 6$ and using a more refined argument one can show that $u \in L^4 ((0,T); B^{1/2+}_{4,\infty } ({\mathbb {T}}^2))$ and that $w \in L^{6/5-} ((0,T); L^2 ({\mathbb {T}}^2))$, by using the estimates from Proposition 8.5 and the specific form of the perturbation. We recall from Boutros et al. (2023) that if $u \in L^4 ((0,T); B^{1/2+}_{4,\infty } ({\mathbb {T}}^2))$ and $w \in L^2 ({\mathbb {T}}^2 \times (0,T))$, then the energy is conserved. Therefore, the result from Theorem 1.22 would be sharp if we could show that $w \in L^{2-} ((0,T); L^2 ({\mathbb {T}}^2))$.

1.3.7 Results for the 2D Prandtl Equations

Now we turn to studying the two-dimensional Prandtl equations, which are given by

$$\begin{aligned} \partial _t u - \nu _v^* \partial _{zz} u + u \partial _{x_1} u + w \partial _z u + \partial _{x_1} p&= 0, \end{aligned}$$

(1.32)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(1.33)

$$\begin{aligned} \partial _{x_1} u + \partial _z w&=0, \end{aligned}$$

(1.34)

where $u: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ is the horizontal velocity, $w: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ is the vertical velocity and $p: {\mathbb {T}}^2 \times (0,T) \rightarrow {\mathbb {R}}$ the pressure.

We observe that these equations differ from the two-dimensional hydrostatic Euler equations (1.28)–(1.30) by the vertical viscosity term $\nu _v^* \partial _{zz} u$. We introduce the following notion of weak solution to the Prandtl equations (1.32)–(1.34).

Definition 1.28

A triple (u, w, p) is called a weak solution of the two-dimensional Prandtl equations (1.32)–(1.34) if $u \in L^2 ({\mathbb {T}}^2 \times (0,T))\cap L^1((0,T);W^{1,1}({\mathbb {T}}^2))$, $w \in {\mathcal {D}}' ( {\mathbb {T}}^2 \times (0,T))$ and $p \in L^1 ({\mathbb {T}}^2 \times (0,T))$ such that $u w \in L^1 ((0,T); B^{-s}_{1,\infty } ({\mathbb {T}}^2))$ (where $s > 0$ is the regularity parameter) and the equations are satisfied in the sense of distributions, i.e.

$$\begin{aligned}&\int _0^T \int _{{\mathbb {T}}^2} u \partial _t \phi _1 \,\textrm{d} x\,\textrm{d} t- \nu _v^* \int _0^T \int _{{\mathbb {T}}^2} \partial _z u \partial _{z} \phi _1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^2} u^2 \partial _{x_1} \phi _1 \,\textrm{d} x\,\textrm{d} t\\&\qquad + \int _0^T \langle u w , \partial _z \phi _1 \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^2} p \partial _{x_1} \phi _1 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \int _{{\mathbb {T}}^2} p \partial _z \phi _2 \,\textrm{d} x\,\textrm{d} t= 0, \\&\int _0^T \langle {{\textbf {u}}} , \nabla \phi _3 \rangle \,\textrm{d} t= 0, \end{aligned}$$

for all test functions $\phi _1$, $\phi _2$ and $\phi _3$ in ${\mathcal {D}} ( {\mathbb {T}}^2 \times (0,T))$.

We will prove the following result.

Theorem 1.29

Let $T>0$ and suppose there exist smooth solutions of the two-dimensional Prandtl equations (1.32)–(1.34) $(u_1,w_1,p_1)$ on [0, T/2] and $(u_2,w_2,p_2)$ on [T/2, T]. Moreover, let $1 \le q_2, q_3 \le \infty$ and $0 < s_3$ be parameters satisfying

$$\begin{aligned} \frac{3}{2} q_2 \left( \frac{1}{q_3} - \frac{1}{q_2}\right)> s_3> \frac{1}{1-\frac{2}{q_2}} \left( \frac{1}{q_3} - \frac{1}{q_2}\right) >0, \quad 1\ge s_3. \end{aligned}$$

(1.35)

Then there exists a weak solution (u, w, p) in the sense of Definition 1.28 with regularity parameter $s=1$ and with the following properties:

• 1. The solution satisfies that

  $$\begin{aligned} (u,w,p)(\cdot ,t) = \left\{ \begin{array}{ll} (u_1,w_1,p_1)(\cdot ,t) &{} \text { if } t\in [0,T/4), \\ (u_2,w_2,p_2)(\cdot ,t) &{} \text { if } t\in (3T/4,T]. \end{array} \right. \end{aligned}$$

• 2. We have that

  $$\begin{aligned} u&\in L^{q_2-} ((0,T); L^2 ({\mathbb {T}}^2)) \cap L^{q_3-} ((0,T); H^{s_3} ({\mathbb {T}}^2)) \cap L^1((0,T);W^{1,1}({\mathbb {T}}^2)), \\ w&\in L^{q_2'-} ((0,T); L^2 ({\mathbb {T}}^2)) \cap L^{q_3'-} ((0,T); H^{-s_3} ({\mathbb {T}}^2)) . \end{aligned}$$

Remark 1.30

Remark 1.14 is also true in the context of the Prandtl equations (1.32)–(1.34).

Remark 1.31

Similar to Remark 1.16, Theorem 1.29 holds for any $\nu _v^* \in {\mathbb {R}}$, in particular even for the inviscid case $\nu _v^* = 0$.

We note that it is possible to establish an analogue of Corollary 1.10 where one has to use the local well-posedness result from Paicu et al. (2020, p. 6) (see also Wang et al. 2021, p. 7186) for analytic data in the strip/channel. A straightforward adaption of the proof of Corollary 1.10 yields the existence of infinitely many global weak solutions for suitable initial data.

Remark 1.32

We note that Theorem 1.29 also applies to the 2D viscous primitive equations, with the same constraints (1.35). This is because the $L^1 ((0,T); W^{1,1} ({\mathbb {T}}^2))$ regularity needed to control the horizontal and vertical viscosities has already been obtained in the proof of Theorem 1.29 (and Proposition 9.1).

1.4 Further Remarks and Outline of the Paper

Now that we have presented the results for the four cases of system (1.1)–(1.3) that we consider in this paper, we would like to make some further remarks on these results. Some conclusions that can be drawn are:

• 1. There exist weak solutions of the inviscid primitive equations (1.4)–(1.6) that do not conserve energy. Compared to the solutions constructed in Chiodaroli and Michálek (2017), the solutions that we construct in this paper have Sobolev regularity. Moreover, they are related to the notion of type III weak solutions, as introduced in Boutros et al. (2023), as the barotropic-vertical part of the nonlinearity is interpreted as a paraproduct.

• 2. In addition, the scheme is able to construct solutions where the baroclinic and barotropic modes have different regularities. This is expected, as the loss of derivative in the advective term only occurs in the baroclinic equation. The barotropic mode must have higher Sobolev regularity than the baroclinic mode in the scheme as otherwise the paraproduct between the vertical velocity and the barotropic mode will not make sense.

• 3. As far as we can tell, this is the first proof of nonuniqueness of weak solutions for the viscous primitive equations. It shows that although the system is globally well-posed (as shown in Cao and Titi 2007), at low regularity the system has nonunique weak solutions. This is true even if one has sufficiently regular Sobolev data for which global well-posedness holds in the class of strong solutions.

• 4. To the best of our knowledge, this is the first convex integration scheme for the two-dimensional Prandtl equations (in the three-dimensional case there is the work Luo and Xin 2018), as well as the two-dimensional hydrostatic Euler equations. Indeed, the fact that in this paper we consider a new class of weak solutions to the Prandtl equations (as discussed in Sect. 1.3.3) makes it possible to address several of the issues raised in Luo and Xin (2018). In particular, we are able to use Mikado flows (while in Luo and Xin 2018 linear plane waves are used for the perturbation) and (temporal and spatial) intermittency as part of the construction for the Prandtl equations. It was left open in Luo and Xin (2018) whether these tools can be applied to the Prandtl case.

There are a few new features of the scheme that we wish to highlight:

• We have introduced a splitting of the Reynolds stress tensor into a barotropic and baroclinic part. We add perturbations to separately deal with both these parts of the error. We then ensure that the interactions between the two perturbations are controlled.

• The splitting of the Reynolds stress tensor requires us to construct and use horizontal and vertical inverse divergence operators, as the barotropic part depends only on the horizontal variables, while the baroclinic part is mean-free with respect to the z-variable.

• Having two parts of the perturbation allows us to use different scalings of the temporal intermittency functions for the barotropic and baroclinic parts of the perturbation. This makes it possible to ensure that the different perturbations have different regularities, such that the interactions between the different parts can be controlled. In particular, this is crucial to control the terms ${\overline{u}}_p \otimes {\widetilde{u}}_p$ and $w_p {\overline{u}}_p$ (the barotropic-baroclinic and vertical-baroclinic parts of the nonlinearity).

Now we present an outline of the paper. In Sects. 2–6, we will develop the convex integration scheme for the 3D inviscid primitive equations, in order to prove Theorem 1.4. In Sect. 2, we state the core inductive proposition of the convex integration scheme and prove Theorem 1.4 using this proposition. In Sect. 3, we discuss several preliminaries. In particular, we introduce the inverse divergence operators, the spatial building blocks for the convex integration, as well as the temporal intermittency functions. In addition, we will discuss the choice of the frequency parameters.

In Sect. 4, we introduce the perturbation that will be used in each iteration of the convex integration scheme, and compute the new Reynolds stress tensor after adding the perturbation. We will prove the estimates on the perturbation required for Proposition 2.4 in Sect. 5. The estimates on the Reynolds stress tensor are proved in Sect. 6.

In Sects. 7–9, we will develop convex integration schemes to study the other cases of Eqs. (1.1)–(1.3) that we are interested in this paper. These schemes differ from the scheme presented in Sects. 3–6 in some aspects, while other parts are similar. Therefore, for the sake of brevity, in Sects. 7–9 we will focus on the parts that differ from the convex integration scheme for the 3D inviscid primitive equations.

In Sect. 7, we provide an extension of the convex integration scheme to the viscous primitive equations with full viscosity and prove Theorem 1.13. The cases with anisotropic viscosities can be studied in a similar manner. In Sect. 8, we investigate the two-dimensional hydrostatic Euler equations and prove Theorem 1.22. Finally, in Sect. 9 we consider the (two-dimensional) Prandtl equations and provide the proof for Theorem 1.29.

In Appendix A, we give a short introduction to Littlewood–Paley theory, Besov spaces and paradifferential calculus, in order to make the paper self-contained. In Appendix B, we state the improved Hölder inequality, which was introduced in Modena and Székelyhidi (2018, Lemma 2.1), and we prove an oscillatory paraproduct estimate based on this inequality. Moreover, we provide another proof of Lemma 5.3 as an alternative to the proof given in Sect. 5.1.2. This lemma states a new inequality needed to control the interaction between the vertical velocity and the baroclinic mode, which turns out to be a critical part of the scheme.

2 The Inductive Proposition

The following underdetermined system of equations is called the hydrostatic Euler–Reynolds system

$$\begin{aligned} \partial _t u + u \cdot \nabla _h u + w \partial _z u + \nabla _h p&= \nabla _h \cdot R_h + \partial _z R_v, \end{aligned}$$

(2.1)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(2.2)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w&=0 , \end{aligned}$$

(2.3)

where $u, w, p, R_h$ and $R_v$ are the unknowns. Here the horizontal Reynolds stress tensor³$R_h: {\mathbb {T}}^2 \times [0,T] \rightarrow {\mathcal {S}}^{2 \times 2}$ is a function of $(x_1,x_2,t)$, while the vertical Reynolds stress tensor $R_v: {\mathbb {T}}^3 \times [0,T] \rightarrow {\mathbb {R}}^2$ is a function of $(x_1,x_2,z,t)$, and which is mean-free with respect to z, i.e. $\int _0^1 R_v \,\textrm{d} z=0$. We will only work with smooth solutions to this system.

Remark 2.1

Notice that $R_h$ is independent of z. Hence, we have $\overline{R_h}=R_h$, see Sect. 1.3.2. Moreover, by definition $R_v$ is mean-free with respect to z and thus $\widetilde{R_v}=R_v$.

The following definition is inspired by Cheskidov and Luo (2022, Definition 2.1).

Definition 2.2

We say that a smooth solution $(u,w,p,R_h,R_v)$ of the hydrostatic Euler–Reynolds system (2.1)–(2.3) is well-prepared if there exists a time interval $I \subseteq [0,T]$ and parameter $\tau > 0$ such that $R_h (x,t) = 0$, $R_v (x,t) = 0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau$.

Remark 2.3

In the definition of well-preparedness, the trivial case $I = [0,T]$ (i.e. without restrictions on the support of $R_h$ and $R_v$) has not been excluded. In this case, the perturbations considered in the inductive proposition will be supported on the whole time interval, but the estimates stated in Proposition 2.4 also hold when $I = [0,T]$. Including the trivial case in Definition 2.2 therefore allows us to phrase Proposition 2.4 in a more general way.

The core of the proof of Theorem 1.4 will revolve around proving the following inductive proposition.

Proposition 2.4

Suppose $(u, w, p, R_h,R_v)$ is a smooth solution of the hydrostatic Euler–Reynolds system (2.1)–(2.3) which is well-prepared with associated time interval I and parameter $\tau >0$. Moreover, consider parameters $1 \le q_1, q_2, q_3 \le \infty$ and $0<s_1, s_3$ which satisfy the following constraints⁴

$$\begin{aligned} q_2&> 2, \quad \frac{2}{q_1}> s_1 + 1, \quad \frac{2}{q_3} > s_3 + \frac{2}{q_2}. \end{aligned}$$

(2.4)

Finally, let $\delta , \epsilon > 0$ be arbitrary. Then there exists another smooth solution $(u + {\overline{u}}_p + {\widetilde{u}}_p, w + w_p, p + P, R_{h,1}, R_{v,1})$ of the hydrostatic Euler–Reynolds system (2.1)–(2.3) which is well-prepared with respect to the same time interval I and parameter $\tau /2$ and has the following properties:

• 1. $({\overline{u}}_p,{\widetilde{u}}_p,w_p)(x,t)=(0,0,0)$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

• 2. The following estimates are satisfied⁵⁶

  $$\begin{aligned} \Vert R_{h,1} \Vert _{L^1 (L^1)}&\le \epsilon , \end{aligned}$$

  (2.5)
  $$\begin{aligned} \Vert R_{v,1} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \epsilon , \end{aligned}$$

  (2.6)
  $$\begin{aligned} \Vert {\overline{u}}_p \Vert _{L^{q_1} (H^{s_1})}&\le \epsilon , \end{aligned}$$

  (2.7)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_2-} (L^2)}&\le \epsilon , \end{aligned}$$

  (2.8)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_3-} (H^{s_3})}&\le \epsilon , \end{aligned}$$

  (2.9)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_2'-} (L^2)}&\le \epsilon , \end{aligned}$$

  (2.10)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_3'-} (H^{-s_3})}&\le \epsilon . \end{aligned}$$

  (2.11)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_2'} (L^2)}&\lesssim \Vert R_h \Vert _{L^1(L^1)}, \end{aligned}$$

  (2.12)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_3'} (H^{-s_3})}&\lesssim \Vert R_h \Vert _{L^1(L^1)}. \end{aligned}$$

  (2.13)

• 3. Moreover, we have the following bounds

  $$\begin{aligned} \Vert {\overline{u}}_p \Vert _{L^{2} (L^2)}&\lesssim \Vert R_h \Vert _{L^1 (L^1)}^{1/2}, \end{aligned}$$

  (2.14)
  $$\begin{aligned} \Vert w_p {\widetilde{u}}_p + w {\widetilde{u}}_p + w_p u \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

  (2.15)

Remark 2.5

Note that when writing ${\overline{u}}_p$, ${\widetilde{u}}_p$, we implicitly require ${\overline{u}}_p$ to be independent of z and ${\widetilde{u}}_p$ mean-free with respect to z, see Sect. 1.3.1.

Remark 2.6

Another version of Proposition 2.4 where (2.12) and (2.13) are replaced by

$$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_2} (L^2)}&\lesssim \Vert R_h \Vert _{L^1(L^1)}, \end{aligned}$$

(2.16)

$$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_3} (H^{s_3})}&\lesssim \Vert R_h \Vert _{L^1(L^1)}, \end{aligned}$$

(2.17)

is true as well, see Remark 5.6. This way we obtain the endpoint time integrability for ${\widetilde{u}}$ rather than w in Theorem 1.4, see Remark 1.5.

Next we prove Theorem 1.4 using Proposition 2.4.

Proof of Theorem 1.4

We first take $\chi _1$ and $\chi _2$ to be a $C^\infty$ partition of unity of [0, T] such that $\chi _1 \equiv 1$ on [0, 3T/8] and $\chi _2 \equiv 1$ on [5T/8, T]. Then we define $(u_0, w_0, p_0)$ as follows

$$\begin{aligned} \begin{aligned} (u_0, w_0, p_0) \,{:=}\, \chi _1 (u_1, w_1, p_1) + \chi _2 (u_2, w_2, p_2). \end{aligned} \end{aligned}$$

(2.18)

For a suitable choice of $\chi _1$ and $\chi _2$, $(u_0, w_0, p_0)$ is no longer a solution of the hydrostatic Euler equations, but with a proper definition⁷ of $R_{h,0},R_{v,0}$ it solves the hydrostatic Euler–Reynolds system (2.1)–(2.3). Moreover, $(u_0,w_0,p_0,R_{h,0},R_{v,0})$ is well-prepared for the time interval $I \,{:=}\, [T/4,3T/4] \subset [0,T]$ and parameter $\tau _0 {:=} T/16$.

Taking the sequence $\epsilon _n = 2^{-n}$, $\delta _n=\delta$ with a suitable choice of $\delta >0$ (see below) and applying Proposition 2.4 inductively, we find a sequence of well-prepared solutions

$$\begin{aligned} \bigg (u_0 + \sum _{k=1}^n \big ( {\overline{u}}_k + {\widetilde{u}}_k \big ), w_0 + \sum _{k=1}^n w_{k}, p_0 + \sum _{k=1}^n P_{k}, R_{h,n}, R_{v,n}\bigg ), \end{aligned}$$

(2.19)

of the hydrostatic Euler–Reynolds system with a sequence of well-preparedness parameters $\{ \tau _n \}$ (and the same time interval I). Note that $\tau _n \rightarrow 0$.

Estimates (2.5), (2.7)–(2.14) imply that the sequence $\Big \{ {\overline{u}}_0 + \sum _{k=1}^n {\overline{u}}_k \Big \}$ is a Cauchy sequence in the space $L^2 ({\mathbb {T}}^3 \times (0,T)) \cap L^{q_1} ((0,T); H^{s_1} ({\mathbb {T}}^3))$, the sequence $\Big \{ {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big \}$ is Cauchy in $L^{q_2-} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3-} ((0,T); H^{s_3} ({\mathbb {T}}^3))$ and the sequence $\Big \{ w_0 + \sum _{k=1}^n w_{k} \Big \}$ is Cauchy in $L^{q_2'} ((0,T); L^2 ({\mathbb {T}}^3)) \cap L^{q_3'} ((0,T); H^{-s_3} ({\mathbb {T}}^3))$. In particular, by choosing $\delta$ appropriately we can identify the limits ${\overline{u}}, {\widetilde{u}}$ and w, as $n \rightarrow \infty$, lying in the spaces stated in Theorem 1.4.

Now we define the pressure p by

$$\begin{aligned} \begin{aligned} p \,{:=} - (\Delta _h)^{-1} \big ( \nabla _h \cdot ( \nabla _h \cdot (\overline{u\otimes u})) \big ), \end{aligned} \end{aligned}$$

(2.20)

where $u={\overline{u}}+{\widetilde{u}}$.

Next, we check that the triple $({\overline{u}} + {\widetilde{u}}, w,p)$ is a weak solution in the sense of Definition 1.1. We first show that $uw\in L^1((0,T);B_{1,\infty }^{-1}({\mathbb {T}}^3))$. According to Lemma A.6, $\Big ( {\overline{u}}_0 + \sum _{k=1}^n {\overline{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big ) \xrightarrow []{n \rightarrow \infty } {\overline{u}} w$ in $L^1 ((0,T); B_{1,\infty }^{-1} ({\mathbb {T}}^3))$. Here we have also used that $1\ge s_1>s_3$ (which follows from (1.16)) as well as Lemma A.3, and the fact that $\frac{1}{q_1} + \frac{1}{q_3'}\le 1$ (which follows from $q_3\le q_1$, see (1.16)) in order to obtain $L^1$ integrability in time.

Next, we need to check that $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big ) \xrightarrow []{n \rightarrow \infty } {\widetilde{u}} w$ in $L^1 ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$. By estimates (2.6) and (2.15), it follows that $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big ) \xrightarrow []{n \rightarrow \infty } {\widetilde{U}} W$ in $L^1 ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$ for some element ${\widetilde{U}} W \in L^1 ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$. We need to show that ${\widetilde{U}} W = {\widetilde{u}} w$ (in a suitable Besov space).

By using estimates (2.5), (2.9) and (2.13), we find that $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big ) \xrightarrow []{n \rightarrow \infty } {\widetilde{u}} w$ in $L^{1-} ((0,T); B^{-s_3}_{1,\infty } ({\mathbb {T}}^3))$, where we have also applied Lemmas A.4 and A.5 (as well as the completeness of $L^p$ spaces for $0< p < 1$, see, for example, Komornik 2016,  Proposition 10.5). Therefore, because of the fact that $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big )$ converges both to ${\widetilde{U}} W$ and ${\widetilde{u}} w$ in $L^{1-} ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$ (which is a quasi-normed space), it follows that ${\widetilde{U}} W = {\widetilde{u}} w$ in $L^{1-} ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$.

We note that the identification can also be made in a different way. The fact that $\Big \{ {\overline{u}}_0 + \sum _{k=1}^n {\overline{u}}_k \Big \}$ and $\Big \{ w_0 + \sum _{k=1}^n w_{k} \Big \}$ are Cauchy sequences in $L^1 ({\mathbb {T}}^3 \times (0,T))$ means that subsequences converge almost everywhere in ${\mathbb {T}}^3\times (0,T)$ to $\widetilde{u}$ and w, respectively. Hence, $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big )$ converges to ${\tilde{u}} w$ almost everywhere in ${\mathbb {T}}^3 \times (0,T)$.

One can also show that $\Big ( {\widetilde{u}}_0 + \sum _{k=1}^n {\widetilde{u}}_k \Big ) \Big ( w_0 + \sum _{k=1}^n w_{k} \Big )$ converges in $L^{1-} ((0,T); L^1({\mathbb {T}}^3))$ to $\tilde{U} W$ (which follows from slightly adapting the proof of Lemma 5.5 and related estimates). Therefore, by Proposition 10.5 in Komornik (2016) there is another subsequence that converges to $\tilde{U} W$ almost everywhere in ${\mathbb {T}}^3 \times (0,T)$. Therefore, we are able to conclude that $\tilde{U} W = {\tilde{u}} w$ almost everywhere in ${\mathbb {T}}^3 \times (0,T)$.

Furthermore, we observe that (1.14) immediately follows from the definition of p. Moreover since for any $n\in {\mathbb {N}}$ the quintuple (2.19) satisfies (2.3), we find that (u, w, p) complies with (1.15). In order to show (1.13), we define the abbreviations

$$\begin{aligned} \begin{aligned} u_n&\,{:=}\, u_0 + \sum _{k=1}^n \big ( {\overline{u}}_k + {\widetilde{u}}_k \big ), \\ w_n&\,{:=}\, w_0 + \sum _{k=1}^n w_{k} ,\\ p_n&\,{:=}\, p_0 + \sum _{k=1}^n P_{k} . \end{aligned} \end{aligned}$$

Since (2.19) satisfies (2.1), we observe that

$$\begin{aligned}&\int _0^T \int _{{\mathbb {T}}^3} u_n\cdot \partial _t\varphi \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} \big ( u_n\otimes u_n \big ) : \nabla _h \varphi \,\textrm{d} x\,\textrm{d} t\nonumber \\&\qquad + \int _0^T \langle u_n w_n, \partial _z \varphi \rangle _{B^{-1}_{1,\infty } \times B^1_{\infty ,1}} \,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} p_n \nabla _h\cdot \varphi \,\textrm{d} x\,\textrm{d} t\nonumber \\&\quad = \int _0^T \int _{{\mathbb {T}}^3} R_{h,n} : \nabla _h \varphi \,\textrm{d} x\,\textrm{d} t+ \int _0^T\langle R_{v,n} , \partial _z \varphi \rangle _{B^{-1}_{1,\infty } \times B^1_{\infty ,1}} \,\textrm{d} t, \end{aligned}$$

(2.21)

for any $n\in {\mathbb {N}}$ and any test function $\varphi \in {\mathcal {D}} ({\mathbb {T}}^3 \times (0,T))$. Note that (2.5) and (2.6) imply $R_{h,n} \xrightarrow []{n \rightarrow \infty } 0$ in $L^1 ((0,T); L^1 ({\mathbb {T}}^3))$ and $R_{v,n} \xrightarrow []{n \rightarrow \infty } 0$ in $L^1 ((0,T); B^{-1}_{1,\infty } ({\mathbb {T}}^3))$, respectively. Hence, by taking the limit we deduce from (2.21) that

$$\begin{aligned} \int _0^T \!\int _{{\mathbb {T}}^3} u\cdot \partial _t\varphi \,\textrm{d} x\,\textrm{d} t\!+\! \int _0^T \!\int _{{\mathbb {T}}^3} \big ( u\otimes u \big ): \nabla _h \varphi \,\textrm{d} x\,\textrm{d} t\! +\! \int _0^T \langle uw, \partial _z \varphi \rangle _{B^{-1}_{1,\infty } \times B^1_{\infty ,1}} \,\textrm{d} t= 0,\nonumber \\ \end{aligned}$$

(2.22)

for any test function $\varphi \in {\mathcal {D}} ({\mathbb {T}}^3 \times (0,T))$ which is either mean-free with respect to z, or independent of z with $\nabla _h\cdot \varphi =0$. Here we have used that for any $n\in {\mathbb {N}}$ and $\varphi$ mean-free with respect to z,

$$\begin{aligned} \int _0^T \int _{{\mathbb {T}}^3} p_n \nabla _h \cdot \varphi \,\textrm{d} x\,\textrm{d} t= \int _0^T \int _{{\mathbb {T}}^2} \bigg [ p_n \nabla _h \cdot \bigg ( \int _{{\mathbb {T}}} \varphi \,\textrm{d} z\bigg ) \bigg ] \,\textrm{d} x_1\,\textrm{d} x_2\,\textrm{d} t= 0, \end{aligned}$$

according to (2.2), and, furthermore, that for any $n\in {\mathbb {N}}$ and $\varphi$ independent of z with $\nabla _h\cdot \varphi =0$

$$\begin{aligned} \int _0^T \int _{{\mathbb {T}}^3} p_n \nabla _h \cdot \varphi \,\textrm{d} x\,\textrm{d} t= 0. \end{aligned}$$

Now we are ready to prove (1.13). We may split the test function $\phi _1 = {\overline{\phi }}_1 + {\widetilde{\phi }}_1$ into the barotropic and baroclinic parts and use the Helmholtz decomposition to find test functions $\varphi ,\psi$, which are independent of z, and such that ${\overline{\phi }}_1= \varphi + \nabla _h \psi$ and $\nabla _h\cdot \varphi = 0$. Then by (2.22) we have

$$\begin{aligned}&\int _0^T \int _{{\mathbb {T}}^3} u \cdot \partial _t \phi _1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} \big ( u \otimes u \big ) : \nabla _h \phi _1 \,\textrm{d} x\,\textrm{d} t\\&\qquad + \int _0^T \langle uw , \partial _z \phi _1 \rangle _{B^{-s}_{1,\infty } \times B^s_{\infty ,1}} \,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} p \nabla _h \cdot \phi _1 \,\textrm{d} x\,\textrm{d} t\\&\quad = \int _0^T \int _{{\mathbb {T}}^3} p \nabla _h\cdot {\widetilde{\phi }}_1 \,\textrm{d} x\,\textrm{d} t+ \int _0^T \int _{{\mathbb {T}}^3} u \cdot \partial _t \nabla _h \psi \,\textrm{d} x\,\textrm{d} t\\&\qquad + \int _0^T \int _{{\mathbb {T}}^3} \Big ( (u\otimes u ):\nabla _h (\nabla _h \psi ) + p \Delta _h\psi \Big ) \,\textrm{d} x\,\textrm{d} t\\&\quad = 0, \end{aligned}$$

where the latter equality follows from the fact that p is independent of z, (1.15) and the definition of p in equation (2.20).

Finally, we observe that equation (1.17) follows from Proposition 2.4 because the time interval I of well-preparedness stays the same for the sequence (2.19). In particular, all the perturbations have support in the time interval I. Therefore, since $(u_0, w_0, p_0)$ agrees with $(u_1, w_1, p_1)$ on [0, T/4) and with $(u_2, w_2, p_2)$ on (3T/4, T], the constructed solution (u, w, p) will have the same properties, as no perturbations with support in $[0,T/4) \cup (3T/4, T]$ have been added. $\square$

After recalling some preliminaries in Sect. 3, we prove Proposition 2.4 in Sects. 4–6.

3 Preliminaries

3.1 Outline

In this paper, we are going to use Mikado flows as building blocks. These have been introduced in Daneri and Székelyhidi (2017) and are built upon a geometric lemma which goes back to Nash (1954), see also (Székelyhidi 2012, Lemma 3.3). Later on, concentrated Mikado flows have been introduced in Modena and Székelyhidi (2018) in order to construct solutions with Sobolev regularity. In this paper, the term Mikado flows will always refer to such Mikado flows with concentration.

In the proof of Proposition 2.4, we will handle the two error terms $R_h$ and $R_v$ separately. To treat $R_v$, we use two-dimensional Mikado flows and Mikado densities in two directions, whereas for $R_h$ we use two-dimensional Mikado flows in several directions which are given by the above-mentioned geometric lemma. We use the version of the Mikado flows which was introduced in Cheskidov and Luo (2022). We recall these flows in Sect. 3.4. We call the perturbation which reduces the error $R_v$ vertical and the one which reduces $R_h$ horizontal.

The above-mentioned concentration is represented by the spatial concentration parameters $\mu _h$ and $\mu _v$ which are used in the horizontal and vertical perturbation, respectively. Moreover, the perturbations will be highly oscillating flows, and this oscillation is represented by the spatial oscillation parameters $\sigma _h$ and $\sigma _v$, which are again used in the horizontal and vertical perturbation, respectively.

Finally, we will use intermittent flows. To this end we introduce temporal intermittency functions in Sect. 3.5. These are time-dependent functions which contain the temporal concentration parameters $\kappa _h$, $\kappa _v$ and temporal oscillation parameters $\nu _h$, $\nu _v$.

3.2 The Parameters

As mentioned in Sect. 3.1, we have two sets of four parameters, namely $\{\mu _h,\sigma _h,\kappa _h,\nu _h\}$ and $\{\mu _v,\sigma _v,\kappa _v,\nu _v\}$, so eight parameters in total. In addition to that, we work with two “master parameters” $\lambda _h$ and $\lambda _v$, which the other parameters depend on via

$$\begin{aligned} \begin{aligned} \nu _i&= \lambda ^{a_i}_i, \quad \sigma _i = \lambda ^{b_i}_i, \\ \kappa _i&= \lambda ^{c_i}_i, \quad \mu _i = \lambda _i \end{aligned} \end{aligned}$$

(3.1)

for $i=h,v$ and fixed exponents $a_i,b_i,c_i>0$. These exponents are determined in the following lemma. We will later fix $\lambda _h,\lambda _v$. These parameters will be very large and such that $\sigma _h,\sigma _v\in {\mathbb {N}}$, as well as $\kappa _h, \kappa _v > 1$.

Lemma 3.1

Let $1 \le q_1, q_2, q_3 \le \infty$ and $0<s_1, s_3$ satisfy the following conditions⁸

$$\begin{aligned} \frac{2}{q_1}> s_1 + 1, \quad \frac{2}{q_3} > s_3 + \frac{2}{q_2}. \end{aligned}$$

(3.2)

Then we can choose $a_i,b_i,c_i>0$ for $i=h,v$ in (3.1) with the property that there exist $\gamma _h, \gamma _v > 0$ such that

$$\begin{aligned} \kappa _h^{1/2-1/q_1} (\sigma _h \mu _h)^{s_1}&\le \lambda _h^{-\gamma _h}, \end{aligned}$$

(3.3)

$$\begin{aligned} \sigma _h^{-1} \nu _h \kappa _h^{1/2} \mu _h^{-1}&\le \lambda _h^{-\gamma _h}, \end{aligned}$$

(3.4)

$$\begin{aligned} \kappa _v^{1/q_2 - 1/q_3} (\sigma _v \mu _v)^{s_3}&= 1, \end{aligned}$$

(3.5)

$$\begin{aligned} \sigma _v^{-1} \nu _v \kappa _v^{1/2} \mu _v^{-1}&\le \lambda _v^{-\gamma _v}, \end{aligned}$$

(3.6)

$$\begin{aligned} \kappa _v^{-\delta }&\le \lambda _v^{-\gamma _v} , \end{aligned}$$

(3.7)

and in addition $\mu _i,\sigma _i,\kappa _i,\nu _i\ge \lambda _i^{\gamma _i}$ for $i=h,v$.

Proof

We choose $0< a_h, a_v < 1$ and set

$$\begin{aligned} \begin{aligned} b_h&\,{:=}\, \frac{2s_1}{\frac{2}{q_1} -s_1 - 1}, \\ b_v&\,{:=}\, \frac{s_3}{\frac{2}{q_3}-s_3 - \frac{2}{q_2}}, \\ c_h&\,{:=}\, 2b_h, \\ c_v&\,{:=}\, 2b_v. \end{aligned} \end{aligned}$$

Notice that (3.2) ensures that $b_h,b_v>0$. Consequently $c_h,c_v>0$.

By taking logarithms, inequalities (3.3)–(3.7) are equivalent to

$$\begin{aligned} -\bigg ( \frac{1}{2} - \frac{1}{q_1} \bigg ) c_h - s_1 (b_h+1)&\ge \gamma _h, \end{aligned}$$

(3.8)

$$\begin{aligned} b_h - a_h - \frac{1}{2} c_h + 1&\ge \gamma _h, \end{aligned}$$

(3.9)

$$\begin{aligned} -\bigg ( \frac{1}{q_2} - \frac{1}{q_3} \bigg ) c_v - s_3 (b_v+1)&=0 , \end{aligned}$$

(3.10)

$$\begin{aligned} b_v - a_v - \frac{1}{2} c_v + 1&\ge \gamma _v, \end{aligned}$$

(3.11)

$$\begin{aligned} \delta c_v&\ge \gamma _v. \end{aligned}$$

(3.12)

Using the definition of $b_v$ and $c_v$, we immediately conclude that (3.10) is valid.

The required additional estimates $\mu _i,\sigma _i,\kappa _i,\nu _i\ge \lambda _i^{\gamma _i}$ for $i=h,v$ translate into the bounds $1,a_i,b_i,c_i\ge \gamma _i$. Since these upper bounds for $\gamma _i$ are positive, it remains to show that the upper bounds given by the left-hand sides of (3.8), (3.9), (3.11) and (3.12) are positive as well.

It is obvious that $\delta c_v >0$. Furthermore, from our choice of $a_i,b_i,c_i$ we obtain

$$\begin{aligned} b_i - a_i -\frac{1}{2}c_i + 1 = 1 -a_i >0, \end{aligned}$$

and

$$\begin{aligned} -\bigg ( \frac{1}{2} - \frac{1}{q_1} \bigg ) c_h - s_1 (b_h+1) = b_h \bigg ( \frac{2}{q_1} -s_1 - 1 \bigg ) -s_1 = s_1 > 0. \end{aligned}$$

$\square$

Remark 3.2

When proving Proposition 2.4, inequality (3.3) ensures that $\Vert {\overline{u}}_p\Vert _{L^{q_1}(H^{s_1})}$ can be made small (see Sect. 5.1.1), while inequality (3.5) guarantees that both $\Vert {\widetilde{u}}_p\Vert _{L^{q_3-}(H^{s_3})}$ and $\Vert w_p\Vert _{L^{q_3'-}(H^{-s_3})}$ can be made small (see Sect. 5.1.2). Moreover, (3.7) will be used at several points during the proof. Finally, inequalities (3.4) and (3.6) make sure that the temporal parts of the linear error are controlled, see Sect. 6.3.1.

3.3 Inverse Divergence Operators

Like in most of the convex integration schemes in the context of fluid dynamics in the literature, we will need inverse divergence operators in order to define the new Reynolds stress tensors $R_{h,1}$ and $R_{v,1}$. In this context, the first inverse divergence operator goes back to De Lellis and Székelyhidi (2013). In this paper, we will work with three inverse divergence operators. The horizontal inverse divergence ${\mathcal {R}}_h$ and its bilinear version ${\mathcal {B}}$ will be used to define the new horizontal Reynolds stress tensor $R_{h,1}$. Those operators are treated in Sects. 3.3.1 and 3.3.2, respectively. In order to determine the new vertical Reynolds stress tensor $R_{v,1}$, we need a “vertical inverse divergence” which is just an integral in z. It is introduced in Sect. 3.3.3.

3.3.1 Horizontal Inverse Divergence

Our horizontal inverse divergence coincides with the two-dimensional inverse divergence from Cheskidov and Luo (2022). It is based upon the inverse divergence introduced in De Lellis and Székelyhidi (2013) and is defined as follows.

Definition 3.3

We define the map⁹${\mathcal {R}}_h: C^\infty ({\mathbb {T}}^2; {\mathbb {R}}^2) \rightarrow C^\infty ({\mathbb {T}}^2; {\mathcal {S}}_0^{2 \times 2} )$ by¹⁰

$$\begin{aligned} \begin{aligned} ({\mathcal {R}}_h v)_{ij} \,{:=}\, {\mathcal {R}}_{ijk,h} v_k, \end{aligned} \end{aligned}$$

(3.13)

where

$$\begin{aligned} {\mathcal {R}}_{ijk,h} := - \Delta _h^{-1} \partial _k \delta _{ij} + \Delta _h^{-1} \partial _i \delta _{jk} + \Delta _h^{-1} \partial _j \delta _{ik} \end{aligned}$$

(3.14)

for $i,j,k\in \{1,2\}$.

The following lemma, which can also be found in Cheskidov and Luo (2022, Appendix B), summarises some properties of the map ${\mathcal {R}}_h$.

Lemma 3.4

• 1. The following identities hold

  $$\begin{aligned} \nabla _h \cdot ({\mathcal {R}}_h v)&= v - \int _{{\mathbb {T}}^2} v \,\textrm{d} x, \qquad \text { for all } v\in C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^2), \end{aligned}$$

  (3.15)
  $$\begin{aligned} {\mathcal {R}}_h \Delta _h v&= \nabla _h v + \nabla _h v^T, \qquad \text { for all divergence-free } v\in C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^2). \end{aligned}$$

  (3.16)

• 2. For $1 \le p \le \infty$, the operator ${\mathcal {R}}_h$ is bounded, i.e. for all $f \in C^\infty ({\mathbb {T}}^2; {\mathbb {R}}^2)$ we have that

  $$\begin{aligned} \Vert {\mathcal {R}}_h f \Vert _{L^p} \lesssim \Vert f \Vert _{L^p}. \end{aligned}$$

  (3.17)

  If f is mean-free, i.e. $\int _{{\mathbb {T}}^2} f \,\textrm{d} x= 0$, then

  $$\begin{aligned} \Vert {\mathcal {R}}_h f (\sigma \cdot ) \Vert _{L^p } \lesssim \sigma ^{-1} \Vert f \Vert _{L^p}, \qquad \text {for any } \sigma \in {\mathbb {N}}. \end{aligned}$$

  (3.18)

• 3. The operator ${\mathcal {R}}_h\nabla _h: C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^{2\times 2}) \rightarrow C^\infty ({\mathbb {T}}^2;{\mathcal {S}}_0^{2\times 2})$ is a Calderón-Zygmund operator, in particular for any $1<p<\infty$ and all $A\in C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^{2\times 2})$ we have

  $$\begin{aligned} \Vert {\mathcal {R}}_h\nabla _h \cdot A \Vert _{L^p} \lesssim \Vert A\Vert _{L^p}. \end{aligned}$$

  (3.19)

For the proof, we refer to Cheskidov and Luo (2022, Appendix B).

3.3.2 Horizontal Bilinear Inverse Divergence

Next we recall the bilinear inverse divergence operator from Cheskidov and Luo (2022). For our purposes, we call it horizontal bilinear inverse divergence.

Definition 3.5

We define ${\mathcal {B}}: C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^2) \times C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^{2\times 2}) \rightarrow C^\infty ({\mathbb {T}}^2;{\mathcal {S}}_0^{2\times 2})$ by

$$\begin{aligned} ( {\mathcal {B}} (b, A) )_{ij} = b_l {\mathcal {R}}_{ijk,h} A_{lk} - {\mathcal {R}}_h (\partial _i b_l {\mathcal {R}}_{ijk,h} A_{lk} ), \end{aligned}$$

(3.20)

or written without components (where we have abused notation)

$$\begin{aligned} {\mathcal {B}} (b,A) = b {\mathcal {R}}_h A - {\mathcal {R}}_h ( \nabla _h b {\mathcal {R}}_h A). \end{aligned}$$

(3.21)

We will also use the following lemma from Cheskidov and Luo (2022).

Lemma 3.6

For $1 \le p \le \infty$, $b \in C^\infty ({\mathbb {T}}^2; {\mathbb {R}}^2)$ and $A \in C^\infty ({\mathbb {T}}^2; {\mathbb {R}}^{2 \times 2})$ with $\int _{{\mathbb {T}}^2} A \,\textrm{d} x= 0$, it holds that

$$\begin{aligned} \nabla _h \cdot ( {\mathcal {B}} (b, A)) = b A - \int _{{\mathbb {T}}^2} b A \,\textrm{d} x. \end{aligned}$$

(3.22)

Moreover, we have the following estimate

$$\begin{aligned} \Vert {\mathcal {B}} (b, A) \Vert _{L^p} \lesssim \Vert b \Vert _{C^1} \Vert {\mathcal {R}}_h A \Vert _{L^p}. \end{aligned}$$

(3.23)

The proof of Lemma 3.6 can be found in Cheskidov and Luo (2022, Appendix B).

3.3.3 Vertical Inverse Divergence

Finally, we introduce the vertical inverse divergence as follows.

Definition 3.7

We define the map¹¹${\mathcal {R}}_v: C^\infty _{0,z}({\mathbb {T}}^3;{\mathbb {R}}^2) \rightarrow C^\infty ({\mathbb {T}}^3;{\mathbb {R}}^2)$ by

$$\begin{aligned} ({\mathcal {R}}_v v) (x_1,x_2,z) := \int _0^z v (x_1,x_2,z') \,\textrm{d} z' - \int _0^1 \int _0^{z'} v (x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z'. \end{aligned}$$

(3.24)

The vertical inverse divergence operator has the following properties.

Lemma 3.8

• 1. The following identities hold for any $v\in C^\infty _{0,z}({\mathbb {T}}^3;{\mathbb {R}}^2)$

  $$\begin{aligned} \int _{{\mathbb {T}}} {\mathcal {R}}_v v \,\textrm{d} z&= 0, \end{aligned}$$

  (3.25)
  $$\begin{aligned} \partial _z {\mathcal {R}}_v v&= v , \end{aligned}$$

  (3.26)
  $$\begin{aligned} {\mathcal {R}}_v (\partial _{zz} v)&= \partial _z v. \end{aligned}$$

  (3.27)

• 2. For $1 \le p,q \le \infty$ and $s\in {\mathbb {R}}$, the operator ${\mathcal {R}}_v$ satisfies the following estimates

  $$\begin{aligned} \Vert {\mathcal {R}}_v f \Vert _{L^p}&\lesssim \Vert f \Vert _{L^p}, \end{aligned}$$

  (3.28)
  $$\begin{aligned} \Vert {\mathcal {R}}_v f \Vert _{B^{s}_{p,q}}&\lesssim \Vert f \Vert _{B^{s}_{p,q}}. \end{aligned}$$

  (3.29)

  Moreover,

  $$\begin{aligned} \Vert {\mathcal {R}}_v f (\sigma \cdot ) \Vert _{L^p } \lesssim \sigma ^{-1} \Vert f \Vert _{L^p} \qquad \text { for any }\sigma \in {\mathbb {N}}. \end{aligned}$$

  (3.30)

• 3. For any $1\le p\le \infty$ and all $v\in C^\infty ({\mathbb {T}}^3;{\mathbb {R}}^2)$, we have

  $$\begin{aligned} \Vert {\mathcal {R}}_v\partial _z v \Vert _{L^p} \lesssim \Vert v\Vert _{L^p}. \end{aligned}$$

  (3.31)

Proof

The identities (3.25), (3.26) are just a simple consequence of the definition of ${\mathcal {R}}_v$. We also observe that

$$\begin{aligned} {\mathcal {R}}_v (\partial _{zz} v) (x_1,x_2,z)&= \int _0^z \partial _{zz} v (x_1,x_2,z') \,\textrm{d} z' - \int _0^1 \int _0^{z'} \partial _{zz} v (x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \\&= \partial _z v (x_1,x_2,z) - \int _0^1 \partial _z v (x_1,x_2,z') \,\textrm{d} z' = \partial _z v (x_1,x_2,z), \end{aligned}$$

i.e. (3.27).

Estimate (3.28) is established simply by moving the $L^p$ norm inside the integral. In order to prove estimate (3.29), we first observe that

$$\begin{aligned} {\mathcal {R}}_v \Delta _j f = \Delta _j {\mathcal {R}}_v f, \end{aligned}$$

(3.32)

which can be verified by a direct computation. Alternatively, thanks to equation (3.25) we have $(\widehat{{\mathcal {R}}_v f})_k = \frac{{\widehat{f}}_k}{2\pi i k_3}$ (where $k_3 \ne 0$), and then equation (3.32) follows by using the definition of the Littlewood–Paley blocks. From (3.28) and (3.32), we obtain

$$\begin{aligned} \Vert \Delta _j {\mathcal {R}}_v f \Vert _{L^p} = \Vert {\mathcal {R}}_v \Delta _j f \Vert _{L^p} \lesssim \Vert \Delta _j f \Vert _{L^p}, \end{aligned}$$

which implies (3.29).

To prove (3.30), we set $f_{\sigma } (x_1,x_2,z) \,{:=}\, f (x_1,x_2,\sigma z)$ and compute

$$\begin{aligned} {\mathcal {R}}_v f_{\sigma }(x_1,x_2,z)&= \int _0^z f_{\sigma }(x_1,x_2,z') \,\textrm{d} z' - \int _0^1 \int _0^{z'} f_{\sigma }(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \\&= \sigma ^{-1} \int _0^{\sigma z} f(x_1,x_2,z') \,\textrm{d} z' - \sigma ^{-1} \int _0^1 \int _0^{\sigma z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \\&= \sigma ^{-1} \int _0^{\sigma z} f(x_1,x_2,z') \,\textrm{d} z' - \sigma ^{-2} \int _0^{\sigma } \int _0^{z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \\&= \sigma ^{-1} \int _0^{\sigma z} f(x_1,x_2,z') \,\textrm{d} z' - \sigma ^{-1} \int _0^{1} \int _0^{z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z'. \end{aligned}$$

Hence,

$$\begin{aligned}&\Vert {\mathcal {R}}_v f_{\sigma }(x_1,x_2,\cdot ) \Vert _{L^p({\mathbb {T}})} \\&\quad \le \sigma ^{-1} \bigg ( \int _{{\mathbb {T}}} \bigg |\int _0^{\sigma z} f(x_1,x_2,z') \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p}\\&\qquad + \sigma ^{-1} \bigg ( \int _{{\mathbb {T}}} \bigg |\int _0^{1} \int _0^{z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p} \\&\quad \le \sigma ^{-1} \bigg ( \sigma ^{-1} \int _{\sigma {\mathbb {T}}} \bigg |\int _0^{z} f(x_1,x_2,z') \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p} \\&\qquad + \sigma ^{-1} \bigg ( \int _{{\mathbb {T}}} \bigg |\int _0^{1} \int _0^{z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p} \\&\quad \le \sigma ^{-1} \bigg ( \int _{{\mathbb {T}}} \bigg |\int _0^{z} f(x_1,x_2,z') \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p} \\&\qquad + \sigma ^{-1} \bigg ( \int _{{\mathbb {T}}} \bigg |\int _0^{1} \int _0^{z'} f(x_1,x_2,z'') \,\textrm{d} z'' \,\textrm{d} z' \bigg |^p \,\textrm{d} z\bigg )^{1/p} \\&\quad \lesssim \sigma ^{-1} \Vert f(x_1,x_2,\cdot ) \Vert _{L^p({\mathbb {T}})}. \end{aligned}$$

This implies

$$\begin{aligned} \Vert {\mathcal {R}}_v f(\sigma \cdot ) \Vert _{L^p({\mathbb {T}}^3)}&= \bigg (\int _{{\mathbb {T}}^2} \Vert {\mathcal {R}}_v f(\sigma x_1, \sigma x_2, \sigma \cdot ) \Vert ^p_{L^p({\mathbb {T}})} \,\textrm{d} x_1 \,\textrm{d} x_2 \bigg )^{1/p} \\&\lesssim \sigma ^{-1} \bigg (\int _{{\mathbb {T}}^2} \Vert f(\sigma x_1,\sigma x_2,\cdot ) \Vert ^p_{L^p({\mathbb {T}})} \,\textrm{d} x_1 \,\textrm{d} x_2 \bigg )^{1/p} \\&= \sigma ^{-1} \Vert f \Vert _{L^p({\mathbb {T}}^3)} , \end{aligned}$$

i.e. (3.30). The case $p = \infty$ follows in a similar fashion.

Finally, observe that

$$\begin{aligned} {\mathcal {R}}_v \partial _z v&= \int _0^z \partial _z v \,\textrm{d} z' - \int _0^1 \int _0^{z'} \partial _z v \,\textrm{d} z'' \,\textrm{d} z' \\&= v - \int _0^1 v \,\textrm{d} z' , \end{aligned}$$

which immediately yields (3.31). $\square$

3.4 Building Blocks for the Perturbation

Next we recall the building blocks. We begin with the Mikado flows and Mikado densities which we use to handle $R_v$. We state their existence together with their most important properties in the following proposition. The construction of the Mikado flows and densities is nowadays standard and goes back to Daneri and Székelyhidi (2017). For the proof of the following proposition, we refer to Cheskidov and Luo (Cheskidov and Luo 2022, Section 4.1).

Proposition 3.9

For each $k\in \{1,2\}$, there exist functions $W_k\in C^\infty ({\mathbb {T}}^2; {\mathbb {R}}^2)$ and $\phi _k\in C^\infty ({\mathbb {T}}^2;{\mathbb {R}})$ (referred to as the Mikado flows and Mikado densities, respectively) depending on a parameter $\mu _v$, with the following properties:

• 1. The functions $W_k, \phi _k$ have zero mean for all $k\in \{1,2\}$. Moreover,

  $$\begin{aligned} \int _{{\mathbb {T}}^2} W_k \phi _k \,\textrm{d} x= {\textbf{e}}_k \quad \text { for all }k\in \{1,2\}, \end{aligned}$$

  (3.33)

  where ${\textbf{e}}_k$ denotes the k-th standard basis vector in ${\mathbb {R}}^2$, and by construction $W_k=\phi _k {\textbf{e}}_k$.

• 2. For any $k\in \{1,2\}$ there exists¹²$\Omega _k\in C^\infty ({\mathbb {T}}^2;{\mathcal {A}}^{2\times 2})$ with zero mean such that $W_k=\nabla _h \cdot \Omega _k$. In particular, $\nabla _h\cdot W_k=0$. Moreover $\nabla _h\cdot (W_k \phi _k)= W_k \cdot \nabla _h \phi _k = 0$.

• 3. For all $s\ge 0$, $1\le p\le \infty$ and $k,k'\in \{1,2\}$ with $k\ne k'$ the following estimates hold:

  $$\begin{aligned} \Vert \phi _k\Vert _{W^{s,p}({\mathbb {T}}^2)}&\lesssim \mu _v^{\frac{1}{2}-\frac{1}{p} + s} ; \end{aligned}$$

  (3.34)
  $$\begin{aligned} \Vert W_k\Vert _{W^{s,p}({\mathbb {T}}^2)}&\lesssim \mu _v^{\frac{1}{2}-\frac{1}{p} + s}; \end{aligned}$$

  (3.35)
  $$\begin{aligned} \Vert \Omega _k\Vert _{W^{s,p}({\mathbb {T}}^2)}&\lesssim \mu _v^{-\frac{1}{2}-\frac{1}{p}+s}; \end{aligned}$$

  (3.36)
  $$\begin{aligned} \Vert W_k\otimes W_{k'}\Vert _{L^p({\mathbb {T}}^2)}&\lesssim \mu _v^{1-\frac{2}{p}}. \end{aligned}$$

  (3.37)

  Here the implicit constant may depend on s, p, but it does not depend on $\mu _v$.

Let us now recall the Mikado flows which we will use to treat $R_h$. In the following proposition, $B_{1/2}({\mathbb {I}})$ denotes the closed ball in ${\mathcal {S}}^{2\times 2}$ around the identity matrix ${\mathbb {I}}$ with radius 1/2. For the proof we refer to Cheskidov and Luo (2022, Lemma 4.2, Theorem 4.3).

Proposition 3.10

There exists $N\in {\mathbb {N}}$, $N\ge 3$ and for each $k\in \Lambda \,{:=}\,\{3,\ldots ,N\}$ there exists a flow $W_k\in C^\infty ({\mathbb {T}}^2;{\mathbb {R}}^2)$ (called Mikado flows) depending on a parameter $\mu _h$ and a function $\Gamma _k\in C^\infty (B_{1/2}({\mathbb {I}});{\mathbb {R}})$, with the following properties:

• 1. The flows $W_k$ have zero mean, i.e. $\int _{{\mathbb {T}}^2} W_k \,\textrm{d} x= 0$, for all $k\in \Lambda$. Moreover

  $$\begin{aligned} \sum _{k\in \Lambda } \Gamma _k^2(R) \int _{{\mathbb {T}}^2} W_k\otimes W_k \,\textrm{d} x= R \quad \text { for all }R\in B_{1/2}({\mathbb {I}}). \end{aligned}$$

  (3.38)

• 2. For any $k\in \Lambda$ there exists $\Omega _k\in C^\infty ({\mathbb {T}}^2;{\mathcal {A}}^{2\times 2})$ with zero mean such that $W_k=\nabla _h \cdot \Omega _k$. In particular, $\nabla _h\cdot W_k=0$. Moreover, $\nabla _h\cdot (W_k\otimes W_k)= W_k \cdot \nabla _h W_k = 0$.

• 3. For all $s\ge 0$, $1\le p\le \infty$ and $k,k'\in \Lambda$ with $k\ne k'$ the following estimates hold:

  $$\begin{aligned} \Vert W_k\Vert _{W^{s,p}({\mathbb {T}}^2)}&\lesssim \mu _h^{\frac{1}{2}-\frac{1}{p}+s}; \end{aligned}$$

  (3.39)
  $$\begin{aligned} \Vert \Omega _k\Vert _{W^{s,p}({\mathbb {T}}^2)}&\lesssim \mu _h^{-\frac{1}{2}-\frac{1}{p}+s}; \end{aligned}$$

  (3.40)
  $$\begin{aligned} \Vert W_k\otimes W_{k'}\Vert _{L^p({\mathbb {T}}^2)}&\lesssim \mu _h^{1-\frac{2}{p}}. \end{aligned}$$

  (3.41)

  Here the implicit constant may depend on s, p, but it does not depend on $\mu _h$.

The following lemma is a simple corollary of (3.34)–(3.36), (3.39) and (3.40).

Lemma 3.11

Let $\sigma \in {\mathbb {N}}$. Then we have the following bounds for all $s \ge 0$, $1 \le p \le \infty$ and $k\in \{1,2\}$, $k'\in \Lambda$:

$$\begin{aligned} \Vert \phi _k (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _v)^s \mu _v^{\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(3.42)

$$\begin{aligned} \Vert W_k (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _v)^s \mu _v^{\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(3.43)

$$\begin{aligned} \Vert W_{k'} (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _h)^s \mu _h^{\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(3.44)

$$\begin{aligned} \Vert \Omega _{k} (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _v)^s \mu _v^{-\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(3.45)

$$\begin{aligned} \Vert \Omega _{k'} (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _h)^s \mu _h^{-\frac{1}{2} - \frac{1}{p}}. \end{aligned}$$

(3.46)

Proof

For any $s\in {\mathbb {N}}_0$, the estimates simply follow from taking derivatives and (3.34)–(3.36), (3.39) and (3.40). Then by interpolation we obtain the desired estimates for any $s\ge 0$. $\square$

3.5 Intermittency

As was done in Cheskidov and Luo (2021, 2022, 2023), we now introduce the temporal intermittency functions, which differ for the horizontal and vertical perturbations. We first fix a non-negative function $G \in C^\infty _c ((0,1/2))$ with

$$\begin{aligned} \int _{[0,1]} G^2 (t) \,\textrm{d} t= 1. \end{aligned}$$

(3.47)

3.5.1 Horizontal Temporal Intermittency Functions

We set

$$\begin{aligned} \begin{aligned} g_h (t) \,{:=}\, \kappa _h^{1/2} G(\kappa _h t); \end{aligned} \end{aligned}$$

more precisely, $g_h$ is the 1-periodic extension of the right-hand side (where we require that $\kappa _h > 1$). Note that from (3.47) we obtain the normalisation identity

$$\begin{aligned} \int _{[0,1]} g_h^2 \,\textrm{d} t= 1, \end{aligned}$$

(3.48)

and furthermore, it is straightforward to verify that

$$\begin{aligned} \Vert g_h \Vert _{L^p([0,1])} \lesssim \kappa _h^{1/2 - 1/p}, \end{aligned}$$

(3.49)

for any $p\in [1,\infty ]$. Subsequently, we introduce the temporal correction function

$$\begin{aligned} h_h(t) := \int _0^t \big (g_h^2(\tau ) - 1\big ) \,\textrm{d} \tau . \end{aligned}$$

Due to (3.48), $h_h$ is 1-periodic and we have

$$\begin{aligned} \Vert h_h \Vert _{L^\infty ([0,1])} \le 1. \end{aligned}$$

(3.50)

3.5.2 Vertical Temporal Intermittency Functions

The vertical temporal oscillation functions are given by the 1-periodic extension (assuming that $\kappa _v > 1$) of

$$\begin{aligned} g_{v,1}^-(t) := \kappa _v^{1/q_2} G(\kappa _v t), \quad g_{v,1}^+(t) := \kappa _v^{1-1/q_2} G(\kappa _v t). \end{aligned}$$

The corresponding temporal correction function is defined by

$$\begin{aligned} h_{v,1}(t) := \int _0^t \big (g_{v,1}^-(\tau )g_{v,1}^+(\tau ) - 1\big ) \,\textrm{d} \tau . \end{aligned}$$

In addition to that, we need temporal oscillation functions where the argument of G is shifted. Those are defined as the 1-periodic extension of

$$\begin{aligned} g_{v,2}^-(t) := \kappa _v^{1/q_2} G(\kappa _v (t-1/2)), \quad g_{v,2}^+(t) := \kappa _v^{1-1/q_2} G(\kappa _v (t-1/2)) \end{aligned}$$

with corresponding correction function

$$\begin{aligned} h_{v,2}(t) := \int _0^t \big (g_{v,2}^-(\tau )g_{v,2}^+(\tau ) - 1\big ) \,\textrm{d} \tau . \end{aligned}$$

Since G has compact support in (0, 1/2) and $\kappa _v > 1$, the functions $g_{v,1}^\pm$ and $g_{v,2}^\pm$ have disjoint supports.

Note that due to the fact that $q_2>2$, we have $1/q_2<1-1/q_2$ which justifies the notation $g_{v,k}^-$, $g_{v,k}^+$ for $k=1,2$.

Similar to the horizontal temporal functions which we introduced in Sect. 3.5.1, we have the following estimates for any $p\in [1,\infty ]$ and $k\in \{1,2\}$

$$\begin{aligned} \Vert g_{v,k}^- \Vert _{L^p([0,1])}&\lesssim \kappa _v^{1/q_2 - 1/p}, \end{aligned}$$

(3.51)

$$\begin{aligned} \Vert g_{v,k}^+ \Vert _{L^p([0,1])}&\lesssim \kappa _v^{1-1/q_2 - 1/p}, \end{aligned}$$

(3.52)

$$\begin{aligned} \Vert h_{v,k} \Vert _{L^\infty ([0,1])}&\le 1. \end{aligned}$$

(3.53)

Finally, in a similar manner one can show that

$$\begin{aligned} \Vert g_{v,k}^- \Vert _{W^{n,p}([0,1])} \lesssim \kappa _v^{1/q_2+n-1/p} \end{aligned}$$

(3.54)

for any $n\in {\mathbb {N}}_0$ and $p\in [1,\infty ]$.

4 Velocity Perturbation and New Reynolds Stress Tensor

In Sects. 4, 5 and 6, we prove Proposition 2.4; hence, we suppose that the assumptions of Proposition 2.4 hold.

The perturbation will be written as

$$\begin{aligned} {\overline{u}}_p&= u_{p,h} + u_{c,h} + u_{t,h} , \end{aligned}$$

(4.1)

$$\begin{aligned} {\widetilde{u}}_p&= u_{p,v} + u_{c,v} + u_{t,v} , \end{aligned}$$

(4.2)

$$\begin{aligned} w_p&= w_{p,v} + w_{t,v}, \end{aligned}$$

(4.3)

where $u_{p,h}$ and $u_{p,v}$ are referred to as the horizontal and vertical principal parts of the perturbation, while $u_{c,h}$, $u_{c,v}$, $u_{t,h}$ and $u_{t,v}$ are referred to as the horizontal and vertical spatial and temporal correctors.

Remark 4.1

We would like to remark that $u_{p,h}, u_{c,h}$ and $u_{t,h}$ do no depend on z, while $u_{p,v}, u_{c,v}$ and $u_{t,v}$ are mean-free with respect to z. Therefore, the first three are indeed a barotropic perturbation, while the latter three are a baroclinic perturbation. This is already hidden in (4.1) and (4.2).

In Sects. 4.1 and 4.3, we define the horizontal perturbation ${\overline{u}}_p$ and the vertical perturbation ${\widetilde{u}}_p$, respectively. The pressure perturbation P is determined in Sect. 4.2. Finally, we define the new Reynolds stress tensors $R_{h,1}$ and $R_{v,1}$ in Sect. 4.4.

4.1 The Horizontal Perturbation

We begin by constructing the horizontal perturbation which consists (see above) of a principal part $u_{p,h}$, a spatial corrector $u_{c,h}$ and a temporal corrector $u_{t,h}$.

In order to construct $u_{p,h}$, we introduce a cutoff function $\chi$. First we choose ${\widetilde{\chi }}\in C^\infty ([0,\infty ))$ to be increasing and satisfying

$$\begin{aligned} {\widetilde{\chi }} (\sigma ) = \left\{ \begin{array}{ll} 4 \Vert R_h \Vert _{L^1 (L^1)} &{} \text { if } 0 \le \sigma \le \Vert R_h \Vert _{L^1 (L^1 )}, \\ 4 \sigma &{} \text { if } \sigma \ge 2 \Vert R_h \Vert _{L^1 (L^1)}. \end{array} \right. \end{aligned}$$

Next we define the function

$$\begin{aligned} \chi (x,t) := {\widetilde{\chi }} \big (|R_h(x,t)| \big ). \end{aligned}$$

It is straightforward to check that ${\mathbb {I}} - \frac{R_h}{\chi } \in B_{1/2} ({\mathbb {I}})$ for all $(x,t) \in {\mathbb {T}}^2 \times [0,T]$. This means in particular that we can evaluate the functions $\Gamma _k$ (see Proposition 3.10) at ${\mathbb {I}} - \frac{R_h}{\chi }$.

We now introduce a temporal smooth cutoff function $\theta \in C^\infty ([0,T];[0,1])$ which satisfies

$$\begin{aligned} \theta (t) = {\left\{ \begin{array}{ll} 1 \quad \text {if } {{\,\textrm{dist}\,}}(t, I^c) \ge \tau , \\ 0 \quad \text {if } {{\,\textrm{dist}\,}}(t, I^c) \le \frac{1}{2}\tau , \end{array}\right. } \end{aligned}$$

(4.4)

in order to achieve the desired property of the supports of the perturbations ${\overline{u}}_p, {\widetilde{u}}_p, w_p$. The horizontal principal perturbation is then defined by

$$\begin{aligned} u_{p,h} (x,t) := \sum _{k \in \Lambda } a_k (x,t) W_k (\sigma _h x), \end{aligned}$$

(4.5)

where the $W_k$ are given by Proposition 3.10 and the amplitude functions are

$$\begin{aligned} \begin{aligned} a_k (x,t)\, {:=}\, \theta (t) g_h (\nu _h t) \chi ^{1/2}(x,t) \Gamma _k \bigg ( {\mathbb {I}} - \frac{R_h(x,t)}{\chi (x,t)} \bigg ). \end{aligned} \end{aligned}$$

(4.6)

Notice that $u_{p,h}$ does not need to be divergence free. To overcome this, we define the corrector $u_{c,h}$ as

$$\begin{aligned} \begin{aligned} u_{c,h}\, {:=}\, \sigma _h^{-1} \sum _{k \in \Lambda } \nabla _h a_k \cdot \Omega _k (\sigma _h x). \end{aligned} \end{aligned}$$

(4.7)

Hence,

$$\begin{aligned} u_{p,h} + u_{c,h} = \sigma _h^{-1} \sum _{k \in \Lambda } \nabla _h \cdot (a_k (x,t) \Omega _k (\sigma _h x)), \end{aligned}$$

(4.8)

which implies

$$\begin{aligned} \nabla _h \cdot (u_{p,h} + u_{c,h}) = \sigma _h^{-1} \sum _{k \in \Lambda } (\nabla _h \otimes \nabla _h): (a_k (x,t) \Omega _k (\sigma _h x)) = 0, \end{aligned}$$

(4.9)

as $\Omega _k$ is skew-symmetric. Moreover, using the definition of $\theta$ in (4.4), we have $u_{p,h}=u_{c,h}=0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

Next, we define the horizontal temporal corrector to be

$$\begin{aligned} u_{t,h} := \nu _h^{-1} h_h (\nu _h t) (\nabla _h \cdot R_h - \nabla _h \Delta _h^{-1} [(\nabla _h \otimes \nabla _h): R_h] ). \end{aligned}$$

(4.10)

It is straightforward to check that $(\nabla _h \otimes \nabla _h): R_h$ is mean-free (so that the inverse Laplacian $\Delta _h^{-1}$ can be applied to this expression), $\nabla _h\cdot u_{t,h}=0$, and that $u_{t,h}=0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

Finally, notice that $u_{p,h}$, $u_{c,h}$ and $u_{t,h}$ are indeed independent of z.

4.2 The Pressure Perturbation

The pressure perturbation is defined as follows

$$\begin{aligned} P := - \theta ^2 g_h^2 (\nu _h t) \chi + \nu _h^{-1} \Delta _h^{-1} (\nabla _h \otimes \nabla _h): \partial _t (h_h (\nu _h t) R_h). \end{aligned}$$

(4.11)

Note that $\partial _z P=0$, since $R_h$, and hence also $\chi$, are independent of z.

4.3 The Vertical Perturbation

We denote the components of the vertical Reynolds stress tensor as $R_{v,k}$, $k=1,2$, i.e.

$$\begin{aligned} R_v = \begin{pmatrix} R_{v,1} \\ R_{v,2} \end{pmatrix}. \end{aligned}$$

This allows us to define the vertical principal perturbation as

$$\begin{aligned} u_{p,v}(x,t)&\,{:=} -\sum _{k=1}^2 \frac{g_{v,k}^- (\nu _v t) \theta (t) R_{v,k}(x,t) W_k (\sigma _v x)}{\Vert R_h \Vert _{L^1 (L^1 )}} , \end{aligned}$$

(4.12)

$$\begin{aligned} w_{p,v}(x,t)&\,{:=} \sum _{k=1}^2 g_{v,k}^+ (\nu _v t) \theta (t) \phi _k (\sigma _v x) \Vert R_h \Vert _{L^1 (L^1 )}, \end{aligned}$$

(4.13)

where $W_k$ and $\phi _k$ are given by Proposition 3.9.

We now introduce the vertical spatial corrector in order to make the perturbation divergence free. We set

$$\begin{aligned} u_{c,v} \,{:=} - \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k=1}^2 \nabla _h \big (g_{v,k}^- (\nu _v t) \theta R_{v,k}\big ) \Omega _k (\sigma _v x). \end{aligned}$$

(4.14)

Observe that

$$\begin{aligned} u_{p,v} + u_{c,v} = -\frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \nabla _h \cdot \big (g_{v,k}^- (\nu _v t) \theta R_{v,k} \Omega _k (\sigma _v x)\big ), \end{aligned}$$

(4.15)

and hence

$$\begin{aligned} \nabla _h\cdot (u_{p,v} + u_{c,v}) = -\frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 (\nabla _h\otimes \nabla _h): \big (g_{v,k}^- (\nu _v t) \theta R_{v,k} \Omega _k (\sigma _v x)\big ) = 0, \end{aligned}$$

since $\Omega _k$ is skew-symmetric. Notice that $w_{p,v}$ is independent of z. Again according to the definition of $\theta$ in (4.4), $u_{p,v}=u_{c,v}=w_{p,v}=0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

Moreover, we introduce the vertical temporal corrector to be

$$\begin{aligned} u_{t,v} := \nu _v^{-1} \sum _{k=1}^2 h_{v,k} (\nu _v t) \partial _z R_{v,k} {\textbf{e}}_k. \end{aligned}$$

(4.16)

Since $u_{t,v}$ does not need to be divergence free, we introduce the corrector

$$\begin{aligned} w_{t,v} := - \nu _v^{-1} \sum _{k=1}^2 h_{v,k} (\nu _v t) \partial _{k} R_{v,k}. \end{aligned}$$

(4.17)

It is then simple to check that $\nabla _h\cdot u_{t,v} + \partial _z w_{t,v} = 0$. Similar to $u_{t,h}$, see above, we have $u_{t,v}=0$ and $w_{t,v}=0$ whenever ${{\,\textrm{dist}\,}}(t,I^c)\le \tau /2$.

Finally, notice that $u_{p,v}$, $u_{c,v}$ and $u_{t,v}$ are mean-free with respect to z because $R_v$ is mean-free with respect to z.

4.4 New Reynolds Stress Tensors

The goal of this section is to define the new Reynolds stress tensors $R_h$ and $R_v$. These will consist of several pieces.

4.4.1 Horizontal Oscillation Error

Let us first define

$$\begin{aligned} R_{\textrm{far}}&\,{:=} \sum _{k,k' \in \Lambda , k \ne k'} a_{k} a_{k'} W_k (\sigma _h x) \otimes W_{k'} (\sigma _h x), \end{aligned}$$

(4.18)

$$\begin{aligned} R_{\textrm{osc},x,h}&\,{:=} \sum _{k \in \Lambda } {\mathcal {B}} \bigg ( \nabla _h (a_k^2), W_k (\sigma _h x) \otimes W_k (\sigma _h x) - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg ), \end{aligned}$$

(4.19)

$$\begin{aligned} R_{\textrm{osc},t,h}&\,{:=}\, \nu _h^{-1} h_h (\nu _h t) \partial _t R_h. \end{aligned}$$

(4.20)

where ${\mathcal {B}}$ is the bilinear inverse divergence operator from Sect. 3.3.2. Moreover, we set

$$\begin{aligned} R_{\textrm{osc},h} = R_{\textrm{osc},x,h} + R_{\textrm{osc},t,h} + R_{\textrm{far}}. \end{aligned}$$

(4.21)

Lemma 4.2

We have

$$\begin{aligned} \partial _t u_{t,h} + \nabla _h \cdot (u_{p,h} \otimes u_{p,h} + R_h) + \nabla _h P = \nabla _h \cdot R_{\textrm{osc},h}. \end{aligned}$$

(4.22)

Proof

Let us first look at the term $\nabla _h \cdot ( u_{p,h} \otimes u_{p,h} + R_h )$. We may write

$$\begin{aligned} \nabla _h \cdot ( u_{p,h} \otimes u_{p,h} + R_h ) = \nabla _h \cdot \bigg ( \sum _{k \in \Lambda } a_k^2 W_k (\sigma _h x) \otimes W_k (\sigma _h x) + R_h \bigg ) + \nabla _h \cdot R_{\textrm{far}}. \end{aligned}$$

Using the definition of the $a_k$, items 1 and 2 of Proposition 3.10 and Lemma 3.6 we find

$$\begin{aligned}&\nabla _h \cdot \bigg ( \sum _{k \in \Lambda } a_k^2 W_k (\sigma _h x) \otimes W_k (\sigma _h x) + R_h \bigg ) \\&\quad = \nabla _h \cdot \bigg [ \sum _{k \in \Lambda } a_k^2 \bigg ( W_k (\sigma _h x) \otimes W_k (\sigma _h x) - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x+ \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg ) + R_h \bigg ] \\&\quad = \sum _{k \in \Lambda } \nabla _h (a_k^2) \cdot \bigg ( W_k (\sigma _h x) \otimes W_k (\sigma _h x) - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg ) \\&\qquad + \theta ^2 g_h^2 (\nu _h t) \nabla _h \chi + (1 - \theta ^2 g_h^2(\nu _h t)) \nabla _h \cdot R_h \\&\quad = \nabla _h \cdot R_{\textrm{osc},x,h} + \theta ^2 g_h^2 (\nu _h t) \nabla _h \chi + (1 - \theta ^2 g_h^2(\nu _h t)) \nabla _h \cdot R_h . \end{aligned}$$

Next we compute

$$\begin{aligned} \partial _t u_{t,h}&= \nu _h^{-1} \partial _t (h_h (\nu _h t)) \nabla _h \cdot R_h + \nu _h^{-1} h_h (\nu _h t) \nabla _h \cdot \partial _t R_h \\&\qquad - \nu _h^{-1} \nabla _h \Delta _h^{-1} (\nabla _h \otimes \nabla _h) : \partial _t (h_h (\nu _h t) R_h) ) \\&= (g_h^2 (\nu _h t) - 1) \nabla _h \cdot R_h + \nabla _h \cdot R_{\textrm{osc},t,h} \\&\qquad - \nu _h^{-1} \nabla _h \Delta _h^{-1} (\nabla _h \otimes \nabla _h) : \partial _t (h_h (\nu _h t) R_h) . \end{aligned}$$

Hence, we have shown

$$\begin{aligned}{} & {} \partial _t u_{t,h} + \nabla _h \cdot (u_{p,h} \otimes u_{p,h} + R_h) + \nabla _h P \\{} & {} \quad = \nabla _h \cdot (R_{\textrm{osc},x,h} + R_{\textrm{osc},t,h} + R_{\textrm{far}}) + g_h^2 (\nu _h t) (1 - \theta ^2) \nabla _h \cdot R_h. \end{aligned}$$

If ${{\,\textrm{dist}\,}}(t, I^c)\le \tau$, then $R_h=0$ by well-preparedness. If ${{\,\textrm{dist}\,}}(t, I^c)\ge \tau$, then $\theta (t)=1$ and hence $1 - \theta ^2=0$. This completes the proof of (4.22). $\square$

4.4.2 Vertical Oscillation Error

We define

$$\begin{aligned} R_{\textrm{osc},x,v}&\,{:=}\, - \sum _{k=1}^2 g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) \theta ^2 R_{v,k} \bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ), \end{aligned}$$

(4.23)

$$\begin{aligned} R_{\textrm{osc},t,v}&\,{:=} \,\nu _v^{-1} \sum _{k=1}^2 h_{v,k} (\nu _v t) \partial _t R_{v,k} {\textbf{e}}_k . \end{aligned}$$

(4.24)

and set

$$\begin{aligned} R_{\textrm{osc},v} = R_{\textrm{osc},x,v} + R_{\textrm{osc},t,v}. \end{aligned}$$

(4.25)

Lemma 4.3

We have

$$\begin{aligned} \partial _t u_{t,v} + \partial _z ( w_{p,v} u_{p,v} + R_v) = \partial _z R_{\textrm{osc},v}. \end{aligned}$$

(4.26)

Proof

First we observe that

$$\begin{aligned} w_{p,v} u_{p,v} = - \sum _{k=1}^2 g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) \theta ^2 R_{v,k} \phi _{k} (\sigma _v x) W_{k} (\sigma _v x), \end{aligned}$$

since $g_{v,1}^\pm g_{v,2}^\pm = 0$, see Sect. 3.5.2. Hence, we obtain using Proposition 3.9

$$\begin{aligned}&\partial _z (w_{p,v} u_{p,v} + R_v)\\&\quad = - \sum _{k=1}^2 g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) \theta ^2 \partial _z R_{v,k} \bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \\&\qquad - \sum _{k=1}^2 \big ( g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) - 1 \big ) \partial _z R_{v,k} {\textbf{e}}_k + \sum _{k=1}^2 g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) (1 - \theta ^2) \partial _z R_{v,k} {\textbf{e}}_k \\&\quad = \partial _z R_{\textrm{osc},x,v} - \sum _{k=1}^2 \big ( g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) - 1 \big ) \partial _z R_{v,k} {\textbf{e}}_k . \end{aligned}$$

Here we used that $(1 - \theta ^2) \partial _z R_{v,k}=0$, see the proof of Lemma 4.2. Moreover, a straightforward computation shows

$$\begin{aligned} \partial _t u_{t,v} - \sum _{k=1}^2 \big ( g_{v,k}^- (\nu _v t) g_{v,k}^+ (\nu _v t) - 1 \big ) \partial _z R_{v,k} {\textbf{e}}_k&= \nu _v^{-1} \sum _{k=1}^2 h_{v,k}(\nu _v t) \partial _t \partial _z R_{v,k} {\textbf{e}}_k \\&= \partial _z R_{\textrm{osc},t,v}, \end{aligned}$$

which finishes the proof of Lemma 4.3. $\square$

4.4.3 Linear Errors

Next we define the horizontal and vertical linear errors by

$$\begin{aligned} R_{\textrm{lin},h} \,{:=}\,&{\mathcal {R}}_h\bigg [ \partial _t \big (u_{p,h}+u_{c,h}\big ) + \nabla _h \cdot \bigg ( {\overline{u}}\otimes \big (u_{p,h}+u_{c,h}+u_{t,h}\big ) + \overline{u\otimes \big (u_{p,v}+u_{c,v}+u_{t,v}\big )} \\&+ \big (u_{p,h}+u_{c,h}+u_{t,h}\big )\otimes {\overline{u}} + \overline{\big (u_{p,v}+u_{c,v}+u_{t,v}\big )\otimes u} \bigg ) \bigg ], \end{aligned}$$

and

Note that the arguments of the operators ${\mathcal {R}}_h$ and ${\mathcal {R}}_v$ satisfy the required properties, i.e. they are independent of z and mean-free with respect to z, respectively.

For convenience, let us write

$$\begin{aligned} \begin{aligned} R_{\text {lin},t,h}&\,{:=}\, {\mathcal {R}}_h \partial _t \big (u_{p,h}+u_{c,h}\big ), \\ R_{\text {lin},t,v}&\,{:=}\, {\mathcal {R}}_v \partial _t \big (u_{p,v}+u_{c,v}\big ), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} R_{\text {lin},x,h}&\,{:=}\, R_{\text {lin},h} - R_{\text {lin},t,h}, \\ R_{\text {lin},x,v}&\,{:=}\, R_{\text {lin},v} - R_{\text {lin},t,v}. \end{aligned} \end{aligned}$$

4.4.4 Corrector Errors

Finally, we define the horizontal and vertical corrector errors by

$$\begin{aligned} R_{\textrm{cor},h}&\,{:=}\, {\mathcal {R}}_h\bigg [ \nabla _h \cdot \bigg ( \big (u_{c,h}+u_{t,h}\big ) \otimes \big (u_{c,h}+u_{t,h}\big ) + u_{p,h} \otimes \big (u_{c,h}+u_{t,h}\big )\\&\qquad \, + \big (u_{c,h}+u_{t,h}\big ) \otimes u_{p,h} \\&\qquad \,+ \overline{\big (u_{p,v}+u_{c,v}+u_{t,v}\big ) \otimes \big (u_{p,v}+u_{c,v}+u_{t,v}\big )} \bigg )\bigg ] \end{aligned}$$

and

As in Sect. 4.4.3, we remark that the arguments of the operators ${\mathcal {R}}_h$ and ${\mathcal {R}}_v$ satisfy the required properties, i.e. they are independent of z and mean-free with respect to z, respectively.

4.4.5 Conclusion

The new Reynolds stress tensors $R_{h,1}, R_{v,1}$ are then given by

$$\begin{aligned} R_{h,1} \,{:=}\, R_{\textrm{osc},h} + R_{\textrm{lin},h} + R_{\textrm{cor},h},\qquad R_{v,1} \,{:=}\, R_{\textrm{osc},v} + R_{\textrm{lin},v} + R_{\textrm{cor},v}. \end{aligned}$$

First, we note that by definition $R_{\textrm{osc},h}$, $R_{\textrm{lin},h}$ and $R_{\textrm{cor},h}$ (and consequently also $R_{h,1}$) are independent of z. Moreover, by definition $R_{\textrm{osc},v}$ is mean-free with respect to z, and so are $R_{\textrm{lin},v}$ and $R_{\textrm{cor},v}$ according to Lemma 3.8 (consequently $R_{v,1}$ has the same property).

Next we remark that $R_{h,1} (x,t) = R_{v,1} (x,t) = 0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \frac{\tau }{2}$. Indeed, for the oscillation errors $R_{\textrm{osc},h}$, $R_{\textrm{osc},v}$ this follows from the fact that $R_h=R_v=0$ whenever ${{\,\textrm{dist}\,}}(t,I^c)\le \tau$, and the definition of $\theta$, see (4.4). The fact that $R_{\textrm{lin},h}=0$, $R_{\textrm{lin},v}=0$, $R_{\textrm{cor},h}=0$, and $R_{\textrm{cor},v}=0$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \frac{\tau }{2}$ immediately follows from item 1 of Proposition 2.4, which we have proved in Sects. 4.1 and 4.3.

Finally, with the help of

• the fact that $\nabla _h\cdot (u+{\overline{u}}_p+{\widetilde{u}}_p) + \partial _z (w+w_p)=0$,

• Lemmas 4.2 and 4.3,

• Lemmas 3.4 and 3.8,

• the fact that $\int _{{\mathbb {T}}^2} \big (u_{p,h} + u_{c,h}\big ) \,\textrm{d} x=0$, see (4.8),

• and $\partial _z u_{p,h} = \partial _z u_{c,h} = \partial _z u_{t,h} = \partial _z w_{p,v} = 0$, see Sects. 4.1 and 4.3,

a long but straightforward computation shows

$$\begin{aligned}&\partial _t (u+{\overline{u}}_p+{\widetilde{u}}_p) + (u+{\overline{u}}_p+{\widetilde{u}}_p)\cdot \nabla _h (u+{\overline{u}}_p+{\widetilde{u}}_p) \\&\qquad + (w+w_p) \partial _z (u+{\overline{u}}_p+{\widetilde{u}}_p) + \nabla _h (p+P) \\&\quad = \partial _t \big ({\overline{u}}_p+{\widetilde{u}}_p\big ) + \nabla _h \cdot \Big ( u\otimes \big ({\overline{u}}_p+{\widetilde{u}}_p\big ) + \big ({\overline{u}}_p+{\widetilde{u}}_p\big )\otimes u + \big ({\overline{u}}_p+{\widetilde{u}}_p\big )\otimes \big ({\overline{u}}_p+{\widetilde{u}}_p\big ) \Big ) \\&\qquad + \partial _z \Big ( w\big ({\overline{u}}_p+{\widetilde{u}}_p\big ) + w_p u + w_p \big ({\overline{u}}_p+{\widetilde{u}}_p\big )\Big ) + \nabla _h P + \nabla _h \cdot R_h + \partial _z R_v \\&\quad = \nabla _h \cdot R_{h,1} + \partial _z R_{v,1}. \end{aligned}$$

Hence, $(u + {\overline{u}}_p + {\widetilde{u}}_p, w + w_p, p + P, R_{h,1}, R_{v,1})$ solves (2.1).

5 Estimates on the Perturbation

In the remaining sections, we will use the following convention: for quantities $Q_1$ and $Q_2$ we write $Q_1 \lesssim Q_2$ if there exists a constant C such that $Q_1 \le C Q_2$. In general, we require that C does not depend on $(u,w,p,R_h,R_v)$. However, if the right-hand side $Q_2$ only contains powers of the parameters $\mu _i,\sigma _i,\kappa _i,\nu _i,\lambda _i$ ($i=h,v$), then the implicit constant C may depend on $(u,w,p,R_h,R_v)$.

5.1 Principal Perturbation

5.1.1 Horizontal Principal Perturbation

First, we estimate the horizontal part of the principal perturbation. We recall the following lemma from Cheskidov and Luo (2022).

Lemma 5.1

We have the following estimates for all $n,m\in {\mathbb {N}}_0$, $p\in [1,\infty ]$

$$\begin{aligned} \Vert \partial ^n_t \nabla ^m a_k \Vert _{L^p (L^\infty )}&\lesssim (\nu _h \kappa _h)^n \kappa _h^{1/2-1/p}, \end{aligned}$$

(5.1)

$$\begin{aligned} \Vert a_k (t) \Vert _{L^2}&\lesssim \theta (t) g_h (\nu _h t) \bigg ( \int _{{\mathbb {T}}^2} \chi (x,t) \,\textrm{d} x\bigg )^{1/2}, \quad \text { for any }t\in [0,T]. \end{aligned}$$

(5.2)

The implicit constant in (5.1) might depend on u or $R_h$, whereas the implicit constant in (5.2) neither depends on t nor on u or $R_h$.

For the proof, we refer to Cheskidov and Luo (2022, Lemma 5.2).

With Lemma 5.1 at hand, we are ready to prove the estimates on the horizontal principal perturbation.

Lemma 5.2

If $\lambda _h$ is chosen sufficiently large (depending on $R_h$), then the horizontal principal perturbation satisfies the following estimates

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^2 (L^2)}&\lesssim \Vert R_h \Vert _{L^1(L^1)}^{1/2}, \end{aligned}$$

(5.3)

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^{q_1} (H^{s_1})}&\lesssim \lambda _h^{-\gamma _h}. \end{aligned}$$

(5.4)

Proof

By applying the improved Hölder inequality in Lemma B.1 and Lemmas 3.11 and 5.1, we find that

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^2 (L^2)}&\le \sum _{k \in \Lambda } \Vert a_k W_k(\sigma _h\cdot ) \Vert _{L^2(L^2)}\\&\lesssim \sum _{k \in \Lambda } \bigg ( \Vert a_k \Vert _{L^2 (L^2)} \Vert W_k \Vert _{L^2} + \sigma _h^{-1/2} \Vert a_k \Vert _{L^2 (C^1)} \Vert W_k \Vert _{L^2} \bigg ) \\&\lesssim \bigg \Vert g_h(\nu _h \cdot ) \bigg ( \int _{{\mathbb {T}}^2} \chi (\cdot ,x) \,\textrm{d} x\bigg )^{1/2} \bigg \Vert _{L^2} + C_{u,R_h} \sigma ^{-1/2}_h \end{aligned}$$

with a constant $C_{u,R_h}$ depending on $u,R_h$. Since $t\mapsto \big ( \int _{{\mathbb {T}}^2} \chi (\cdot ,x) \,\textrm{d} x\big )^{1/2}$ is smooth, we can apply Lemma B.1 once again to obtain

$$\begin{aligned}&\bigg \Vert g_h(\nu _h \cdot ) \bigg ( \int _{{\mathbb {T}}^2} \chi (\cdot ,x) \,\textrm{d} x\bigg )^{1/2} \bigg \Vert _{L^2} \\&\quad \lesssim \bigg \Vert \bigg ( \int _{{\mathbb {T}}^2} \chi (\cdot ,x) \,\textrm{d} x\bigg )^{1/2} \bigg \Vert _{L^2} \Vert g_h\Vert _{L^2} + \nu _h^{-1/2} \bigg \Vert \bigg ( \int _{{\mathbb {T}}^2} \chi (\cdot ,x) \,\textrm{d} x\bigg )^{1/2} \bigg \Vert _{C^1} \Vert g_h\Vert _{L^2} \\&\quad \lesssim \Vert \chi \Vert _{L^1(L^1)}^{1/2} + C_{u,R_h} \nu _h^{-1/2}, \end{aligned}$$

where we made use of (3.48). Since $\chi (x,t)\le 4|R_h(x,t)| + 4\Vert R_h\Vert _{L^1(L^1)}$, we have

$$\begin{aligned} \Vert \chi \Vert _{L^1(L^1)} \lesssim \Vert R_h\Vert _{L^1(L^1)} + \Vert R_h\Vert _{L^1(L^1)} \lesssim \Vert R_h\Vert _{L^1(L^1)}. \end{aligned}$$

So we have shown that

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^2 (L^2)}&\lesssim \Vert R_h\Vert _{L^1(L^1)}^{1/2} + C_{u,R_h} \nu _h^{-1/2} + C_{u,R_h} \sigma ^{-1/2}_h, \end{aligned}$$

which implies (5.3) by taking $\lambda _h$ sufficiently large (depending on $R_h$).

Furthermore, by applying Lemmas 3.1, 3.11 and 5.1 we find that

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^{q_1} (H^{s_1})}&\le \sum _{k \in \Lambda } \Vert a_k \Vert _{L^{q_1} (W^{1,\infty })} \Vert W_k(\sigma _h\cdot ) \Vert _{H^{s_1}} \lesssim \kappa _h^{1/2 - 1/q_1} (\sigma _h \mu _h)^{s_1} \le \lambda _h^{-\gamma _h} \end{aligned}$$

which proves (5.4). $\square$

5.1.2 Vertical Principal Perturbation

Let us first show the following lemma.

Lemma 5.3

For all $1\le p\le \infty$ and $k\in \{1,2\}$, we have

$$\begin{aligned} \bigg \Vert \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \lesssim \sigma _v^{-1}. \end{aligned}$$

(5.5)

Proof

Using equation (3.15), we find

$$\begin{aligned}&\bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \\&\quad = R_{v,k} \nabla _h \cdot \bigg [ {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg ] \\&\quad = \nabla _h \cdot \bigg [ R_{v,k} {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg ] \\&\qquad - \nabla _h R_{v,k} \cdot {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v x) W_k (\sigma _v x) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) . \end{aligned}$$

Next, we observe that according to Lemma 3.11

$$\begin{aligned}{} & {} \bigg \Vert \phi _k W_k - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{L^1} \lesssim \Vert \phi _k W_k\Vert _{L^1} + 1 \nonumber \\{} & {} \quad \le \Vert \phi _k\Vert _{L^2} \Vert W_k\Vert _{L^2} + 1 \lesssim 1. \end{aligned}$$

(5.6)

Then by Lemmas 3.4 and A.3, and inequality (5.6), we obtain

$$\begin{aligned}&\bigg \Vert \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\quad \lesssim \bigg \Vert \nabla _h \cdot \bigg [ R_{v,k} {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg ] \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\qquad + \bigg \Vert \nabla _h R_{v,k} \cdot {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\quad \lesssim \bigg \Vert R_{v,k} {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^p (L^1)} \\&\qquad + \bigg \Vert \nabla _h R_{v,k} \cdot {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^p (L^1)} \\&\quad \lesssim \Big (\Vert R_{v,k} \Vert _{L^p(L^\infty )} + \Vert \nabla _h R_{v,k} \Vert _{L^p(L^\infty )}\Big ) \bigg \Vert {\mathcal {R}}_h \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^1} \\&\quad \lesssim \sigma _v^{-1} C_{R_v} \bigg \Vert \phi _k W_k - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg \Vert _{L^1} \lesssim \sigma _v^{-1}. \end{aligned}$$

$\square$

Remark 5.4

We present an alternative proof of Lemma 5.3 in the appendix, see Sect. A.3.

Next we estimate the vertical principal perturbation.

Lemma 5.5

If $\lambda _v$ is chosen sufficiently large (depending on $R_v$), then the vertical principal perturbation satisfies the following inequalities

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_3-} (H^{s_3})}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(5.7)

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_2-} (L^2)}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(5.8)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_3'-} (H^{-s_3})}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(5.9)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_2'-} (L^2)}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(5.10)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_3'} (H^{-s_3})}&\lesssim \Vert R_h \Vert _{L^1(L^1)}, \end{aligned}$$

(5.11)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_2'} (L^2)}&\lesssim \Vert R_h \Vert _{L^1(L^1)}, \end{aligned}$$

(5.12)

$$\begin{aligned} \Vert w_{p,v} u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty } )}&\lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

(5.13)

Proof

According to (3.51) and Lemmas 3.1 and 3.11, we obtain

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_3-} (H^{s_3})}&\le \frac{1}{\Vert R_h \Vert _{L^1 (L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_3-}} \Vert R_v \Vert _{L^\infty (W^{1,\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{H^{s_3}} \\&\lesssim \kappa _v^{1/q_2 - 1/q_3 - \delta } (\sigma _v \mu _v)^{s_3} = \kappa _v^{-\delta } \le \lambda _v^{-\gamma _v}. \end{aligned}$$

Similarly,

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_2-} (L^2)}&\le \frac{1}{\Vert R_h \Vert _{L^1 (L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_2-}} \Vert R_v \Vert _{L^\infty (L^{\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{L^2} \nonumber \\&\lesssim \kappa _v^{1/q_2 - 1/q_2 - \delta } = \kappa _v^{-\delta } \le \lambda _v^{-\gamma _v}. \end{aligned}$$

(5.14)

So we have shown (5.7), (5.8).

Next, notice that in accordance with Proposition 3.9 and Lemma 3.11 (keeping in mind that $s_3\le 1$ and hence $-s_3+1\ge 0$)

$$\begin{aligned} \Vert \phi _k (\sigma _v \cdot ) \Vert _{H^{-s_3}}&= \Vert W_k (\sigma _v \cdot ) \Vert _{H^{-s_3}} = \sigma _v^{-1} \big \Vert \nabla _h\cdot \big [\Omega _k (\sigma _v \cdot ) \big ]\big \Vert _{H^{-s_3}} \\&\lesssim \sigma _v^{-1} \Vert \Omega _k (\sigma _v \cdot ) \Vert _{H^{-s_3+1}} \lesssim \sigma _v^{-1} (\sigma _v \mu _v)^{-s_3+1} \mu _v^{-1} = (\sigma _v \mu _v)^{-s_3}. \end{aligned}$$

Together with (3.52) and Lemma 3.1, this yields

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_3'-} (H^{-s_3})}&\le \sum _{k=1}^2 \Vert \phi _k (\sigma _v \cdot ) \Vert _{H^{-s_3}} \Vert R_h \Vert _{L^1(L^1)} \Vert g_{v,k}^+ (\nu _v \cdot ) \Vert _{L^{q_3'-}} \nonumber \\&\lesssim \kappa _v^{1 - 1/q_2 - 1/q_3' -\delta } (\sigma _v \mu _v)^{-s_3} = \kappa _v^{-\delta } \le \lambda _v^{-\gamma _v}. \end{aligned}$$

(5.15)

Similarly, using Lemma 3.11, we obtain

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_2'-} (L^2)}&\le \sum _{k=1}^2 \Vert \phi _k (\sigma _v \cdot ) \Vert _{L^2} \Vert R_h \Vert _{L^1(L^1)} \Vert g_{v,k}^+ (\nu _v \cdot ) \Vert _{L^{q_2'-}} \nonumber \\&\lesssim \kappa _v^{1 - 1/q_2 - 1/q_2' -\delta } = \kappa _v^{-\delta } \le \lambda _v^{-\gamma _v}. \end{aligned}$$

(5.16)

Hence we have proved (5.9), (5.10). Additionally, from (5.15) and (5.16) we see that (5.11) and (5.12) hold.

Finally, we derive estimate (5.13) for the product $u_{p,v} w_{p,v}$. Because $g_{v,1}^\pm g_{v,2}^\pm = 0$, see Sect. 3.5.2, and the improved Hölder inequality (Lemma B.1), we have

$$\begin{aligned} \Vert w_{p,v} u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty } )}&\lesssim \sum _{k=1}^2 \Big \Vert g_{v,k}^- (\nu _v \cdot ) g_{v,k}^+ (\nu _v \cdot ) \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\lesssim \sum _{k=1}^2 \Vert g_{v,k}^- g_{v,k}^+ \Vert _{L^1} \Big \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\qquad + \sum _{k=1}^2 \nu _v^{-1} \Vert g_{v,k}^- g_{v,k}^+ \Vert _{L^1} \Big \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{C^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

First, observe that

$$\begin{aligned} \Vert g_{v,k}^- g_{v,k}^+ \Vert _{L^1} \le \Vert g_{v,k}^- \Vert _{L^2} \Vert g_{v,k}^+ \Vert _{L^2} \lesssim \kappa _v^{1/q_2 - 1/2 +1 -1/q_2 -1/2} = 1, \end{aligned}$$

(5.17)

according to (3.51), (3.52). Next, we estimate

$$\begin{aligned} \Big \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{C^1 (B^{-1}_{1,\infty })}&\lesssim \Big \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{C^1 (L^1)} \\&\lesssim \Vert R_{v,k} \Vert _{C^1(L^\infty )} \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) \Vert _{L^1} \\&\lesssim C_{R_v} \Vert \phi _k (\sigma _v \cdot ) \Vert _{L^2} \Vert W_k (\sigma _v \cdot ) \Vert _{L^2} \\&\lesssim C_{R_v}, \end{aligned}$$

where $C_{R_v}$ is a constant depending on $R_v$ and where we used Lemmas 3.11 and A.3. Moreover, we obtain by Lemma 5.3

$$\begin{aligned}&\Big \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) R_{v,k} \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \le \bigg \Vert \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \bigg \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\qquad + \bigg |\int _{{\mathbb {T}}^2 } \phi _k (x) W_k (x) \,\textrm{d} x\bigg |\Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \sigma _v^{-1} + \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

Hence, we have shown

$$\begin{aligned} \Vert w_{p,v} u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty } )} \lesssim \Vert R_v\Vert _{L^1(B^{-1}_{1,\infty })} + C_{R_v} \Big ( \sigma _v^{-1} + \nu _v^{-1} \Big ) \end{aligned}$$

(5.18)

which implies (5.13) by choosing $\lambda _v$ sufficiently large (depending on $R_v$). $\square$

Remark 5.6

As already mentioned in Remark 2.6, we can establish (2.16), (2.17) instead of (2.12), (2.13). To this end, we have to replace (4.12) and (4.13) by

$$\begin{aligned} \begin{aligned} u_{p,v}(x,t)&\,{:=}\, -\sum _{k=1}^2 g_{v,k}^- (\nu _v t) \theta (t) R_{v,k}(x,t) W_k (\sigma _v x)\frac{\Vert R_h \Vert _{L^1 (L^1 )}}{\Vert R_v \Vert _{L^\infty (W^{1,\infty } )}} , \\ w_{p,v}(x,t)&\,{:=}\, \sum _{k=1}^2 g_{v,k}^+ (\nu _v t) \theta (t) \phi _k (\sigma _v x) \frac{\Vert R_v \Vert _{L^\infty (W^{1,\infty } )}}{\Vert R_h \Vert _{L^1 (L^1 )}}. \end{aligned} \end{aligned}$$

Then (5.11), (5.12) are no longer true. Instead, we find

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_3} (H^{s_3})}&\le \frac{\Vert R_h \Vert _{L^1 (L^1 )}}{\Vert R_v \Vert _{L^\infty (W^{1,\infty } )}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_3}} \Vert R_v \Vert _{L^\infty (W^{1,\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{H^{s_3}} \\&\lesssim \Vert R_h \Vert _{L^1 (L^1 )}\kappa _v^{1/q_2 - 1/q_3} (\sigma _v \mu _v)^{s_3} = \Vert R_h \Vert _{L^1 (L^1 )} , \\ \Vert u_{p,v} \Vert _{L^{q_2} (L^2)}&\le \frac{\Vert R_h \Vert _{L^1 (L^1 )}}{\Vert R_v \Vert _{L^\infty (W^{1,\infty } )}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_2}} \Vert R_v \Vert _{L^\infty (L^{\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{L^2} \\&\lesssim \Vert R_h \Vert _{L^1 (L^1 )}\kappa _v^{1/q_2 - 1/q_2} = \Vert R_h \Vert _{L^1 (L^1 )}. \end{aligned}$$

Further modifications are straightforward.

5.2 Spatial Correctors

Lemma 5.7

The spatial correctors satisfy the following estimates

$$\begin{aligned} \Vert u_{c,h} \Vert _{L^{q_1} (H^{s_1})} + \Vert u_{c,h} \Vert _{L^2(L^\infty )}&\lesssim \lambda _h^{-\gamma _h}, \\ \Vert u_{c,v} \Vert _{L^{q_2} (L^\infty )} + \Vert u_{c,v} \Vert _{L^{q_3} (H^{s_3})}&\lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

Proof

By using Lemmas 3.1, 3.11 and 5.1 as well as estimate (3.51), one gets that

$$\begin{aligned} \Vert u_{c,h} \Vert _{L^{q_1} (H^{s_1})}&\le \sigma _h^{-1} \sum _{k \in \Lambda } \Vert \nabla a_k \Vert _{L^{q_1} (W^{1,\infty })} \Vert \Omega _k (\sigma _h \cdot ) \Vert _{H^{s_1}} \\ {}&\lesssim \sigma _h^{-1} \kappa _h^{1/2 - 1/q_1} (\sigma _h \mu _h)^{s_1} \mu _h^{-1} \lesssim \lambda _h^{-\gamma _h}, \\ \Vert u_{c,h} \Vert _{L^2 (L^\infty )}&\le \sigma _h^{-1} \sum _{k \in \Lambda } \Vert \nabla a_k \Vert _{L^{2} (L^\infty )} \Vert \Omega _k (\sigma _h \cdot ) \Vert _{L^\infty } \lesssim \sigma _h^{-1} \mu _h^{-1/2} \lesssim \lambda _h^{-\gamma _h}, \\ \Vert u_{c,v} \Vert _{L^{q_2} (L^\infty )}&\le \frac{\sigma _v^{-1}}{\Vert R_h\Vert _{L^1(L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_2}} \Vert \nabla _h R_{v,k} \Vert _{L^\infty (L^\infty )} \Vert \Omega _k (\sigma _v \cdot ) \Vert _{L^\infty } \\&\lesssim \sigma _v^{-1} \mu _v^{-1/2} \lesssim \lambda _v^{-\gamma _v}, \\ \Vert u_{c,v} \Vert _{L^{q_3} (H^{s_3})}&\le \frac{\sigma _v^{-1}}{\Vert R_h\Vert _{L^1(L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_3}} \Vert \nabla _h R_{v,k} \Vert _{L^\infty (W^{1,\infty })} \Vert \Omega _k (\sigma _v \cdot ) \Vert _{H^{s_3}} \\&\lesssim \sigma _v^{-1} \kappa _v^{1/q_2 - 1/q_3} (\sigma _v \mu _v)^{s_3} \mu _v^{-1} \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

$\square$

5.3 Temporal Correctors

Lemma 5.8

The temporal correctors satisfy the estimates

$$\begin{aligned} \Vert u_{t,h} \Vert _{L^\infty (W^{n,\infty })}&\lesssim \lambda _h^{-\gamma _h}, \\ \Vert u_{t,v} \Vert _{L^\infty (W^{n,\infty })}&\lesssim \lambda _v^{-\gamma _v}, \\ \Vert w_{t,v} \Vert _{L^\infty (W^{n,\infty })}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

where $n\in {\mathbb {N}}$ is arbitrary, and the implicit constant may depend on n.

Proof

Using (3.50) and (3.53), we obtain

$$\begin{aligned} \Vert u_{t,h} \Vert _{L^\infty (W^{n,\infty })}&\le \nu _h^{-1} \Vert h_h (\nu _h \cdot ) \Vert _{L^\infty } C_{R_h} \lesssim \nu _h^{-1} \le \lambda _h^{-\gamma _h}, \\ \Vert u_{t,v} \Vert _{L^\infty (W^{n,\infty })}&\le \nu _v^{-1} \Vert h_{v,k} (\nu _v \cdot ) \Vert _{L^\infty } C_{R_v} \lesssim \nu _v^{-1} \lesssim \lambda _v^{-\gamma _v}, \\ \Vert w_{t,v} \Vert _{L^\infty (W^{n,\infty })}&\le \nu _v^{-1} \Vert h_{v,k} (\nu _v \cdot ) \Vert _{L^\infty } C_{R_v} \lesssim \nu _v^{-1} \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

$\square$

5.4 Conclusion

We have already shown in Sect. 4 that ${\overline{u}}_p$, ${\widetilde{u}}_p$ and $w_p$ satisfy

$$\begin{aligned} \nabla _h \cdot ({\overline{u}}_p + {\widetilde{u}}_p ) + \partial _z w_p =0, \end{aligned}$$

as well as item 1 of Proposition 2.4. Hence, $(u+{\overline{u}}_p + {\widetilde{u}}_p,w+w_p)$ fulfills (2.3). Additionally, we have shown in Sect. 4 that $\partial _z P=0$ and hence $p+P$ satisfies (2.2). Moreover, we proved that (2.1) holds. Consequently, $(u + {\overline{u}}_p + {\widetilde{u}}_p, w + w_p, p + P, R_{h,1}, R_{v,1})$ is indeed a solution of the hydrostatic Euler–Reynolds system (2.1)–(2.3). We also have shown in Sect. 4 that $(u + {\overline{u}}_p + {\widetilde{u}}_p, w + w_p, p + P, R_{h,1}, R_{v,1})$ is well-prepared for the time interval I and parameter $\tau /2$.

Furthermore, estimates (2.7)–(2.14) of Proposition 2.4 are a simple consequence of Lemmas 5.2, 5.5, 5.7 and 5.8, where one has to choose $\lambda _h$, $\lambda _v$ sufficiently large, depending on $R_h$ and $R_v$, respectively.

In addition, estimate (2.15) can be derived from Lemmas 5.5, 5.7, 5.8 as well. Indeed, Lemma 5.5 already proves $\Vert w_{p,v} u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty } )} \lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}$. Moreover, from Lemmas A.3, 5.5, 5.7 and 5.8 we obtain

$$\begin{aligned} \Vert w_{p,v} (u_{c,v} + u_{t,v}) \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \Vert w_{p,v} (u_{c,v} + u_{t,v}) \Vert _{L^1 (L^1)} \\&\lesssim \Vert w_{p,v} \Vert _{L^{q_2'} (L^2)} ( \Vert u_{c,v} \Vert _{L^{q_2} (L^2)} + \Vert u_{t,v} \Vert _{L^{q_2} (L^2)} ) \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

Similarly (from the proof of Lemma 5.5 we obtain $\Vert u_{p,v}\Vert _{L^{q_2}(L^2)}\lesssim C_{R_h,R_v}$)

$$\begin{aligned} \Vert w_{t,v} {\widetilde{u}}_p \Vert _{L^1 (B^{-1}_{1,\infty })} \lesssim \Vert w_{t,v} {\widetilde{u}}_p \Vert _{L^1 (L^1)} \lesssim \Vert w_{t,v} \Vert _{L^{q_2'} (L^2)} \Vert {\widetilde{u}}_p \Vert _{L^{q_2} (L^2)} \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

Finally, Lemmas 5.5, 5.7 and 5.8 yield

$$\begin{aligned} \Vert w {\widetilde{u}}_p + w_p u \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \Vert w {\widetilde{u}}_p + w_p u \Vert _{L^1 (L^1)} \\&\lesssim \Vert w \Vert _{L^\infty (L^\infty )} \Vert {\widetilde{u}}_p \Vert _{L^1 (L^1)} + \Vert w_p \Vert _{L^1 (L^1)} \Vert u \Vert _{L^\infty (L^\infty )} \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

Hence, if $\lambda _v$ is chosen sufficiently large, depending on $R_v$, we obtain (2.15).

6 Estimates on the Stress Tensor

In order to finish the proof of Proposition 2.4, it remains to show estimates (2.5), (2.6). These two estimates simply follow from Lemmas 6.1, 6.2, 6.3 and 6.4, which we prove in this section, below.

6.1 Oscillation Error

6.1.1 Horizontal Part

Lemma 6.1

If $\lambda _h$ is chosen sufficiently large (depending on $R_h$), then the horizontal oscillation error satisfies

$$\begin{aligned} \Vert R_{\textrm{osc},h} \Vert _{L^1 (L^1)} \le \frac{\epsilon }{3}. \end{aligned}$$

(6.1)

Proof

Using Lemmas 3.1, 3.4, 3.6 and 5.1, we estimate $R_{\textrm{osc},x,h}$ as follows

$$\begin{aligned} \Vert R_{\textrm{osc},x,h} \Vert _{L^1 (L^1)}&= \bigg \Vert \sum _{k \in \Lambda } {\mathcal {B}} \bigg ( \nabla _h (a_k^2), W_k (\sigma _h \cdot ) \otimes W_k (\sigma _h \cdot ) - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^1(L^1)} \\&\le \sum _{k \in \Lambda } \Vert \nabla _h (a_k^2) \Vert _{L^1 (C^1)} \bigg \Vert {\mathcal {R}}_h \bigg ( W_k (\sigma _h \cdot ) \otimes W_k (\sigma _h \cdot ) - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^1} \\&\le \sigma _h^{-1} \sum _{k \in \Lambda } \Vert \nabla _h (a_k^2) \Vert _{L^1 (C^1)} \bigg \Vert W_k \otimes W_k - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg \Vert _{L^1} \\&\lesssim \sigma _h^{-1} \kappa _h^{-1/2} . \end{aligned}$$

Here we have used that (similar to (5.6))

$$\begin{aligned} \bigg \Vert W_k \otimes W_k - \int _{{\mathbb {T}}^2} W_k \otimes W_k \,\textrm{d} x\bigg \Vert _{L^1} \lesssim \Vert W_k\otimes W_k\Vert _{L^1} + 1 \le \Vert W_k\Vert _{L^2}^2 + 1 \lesssim 1,\nonumber \\ \end{aligned}$$

(6.2)

according to Lemma 3.11.

Next, we obtain from (3.50)

$$\begin{aligned} \Vert R_{\textrm{osc},t,h} \Vert _{L^1 (L^1)}&\le \nu _h^{-1} \Vert h_h (\nu _h \cdot ) \Vert _{L^\infty } \Vert \partial _t R_h \Vert _{L^1 (L^1)} \lesssim \nu _h^{-1}. \end{aligned}$$

Finally, using Lemma 5.1 and Proposition 3.10 we get

$$\begin{aligned} \Vert R_{\textrm{far}} \Vert _{L^1 (L^1)}&\le \bigg \Vert \sum _{k,k' \in \Lambda , k \ne k'} a_{k} a_{k'} W_k (\sigma _h \cdot ) \otimes W_{k'} (\sigma _h \cdot ) \bigg \Vert _{L^1(L^1)} \\&\lesssim \sum _{k,k' \in \Lambda , k \ne k'} \Vert a_{k}\Vert _{L^2(L^\infty )} \Vert a_{k'}\Vert _{L^2(L^\infty )} \Big \Vert W_k (\sigma _h \cdot ) \otimes W_{k'} (\sigma _h \cdot ) \Big \Vert _{L^1} \\&\lesssim \sum _{k,k' \in \Lambda , k \ne k'} \Vert W_k \otimes W_{k'} \Vert _{L^1}\lesssim \mu _h^{-1}. \end{aligned}$$

By choosing $\lambda _h$ large enough (depending on $R_h$), we conclude with (6.1). $\square$

6.1.2 Vertical Part

Lemma 6.2

If $\lambda _v$ is chosen sufficiently large (depending on $R_v$), then the vertical oscillation error satisfies

$$\begin{aligned} \Vert R_{\textrm{osc},v} \Vert _{L^1 (B^{-1}_{1,\infty })} \le \frac{\epsilon }{3}. \end{aligned}$$

(6.3)

Proof

Using (5.17) and Lemma 5.3, we find

$$\begin{aligned} \Vert R_{\textrm{osc},x,v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \sum _{k=1}^2 \Big \Vert g_{v,k}^-(\nu _v\cdot )g_{v,k}^+(\nu _v\cdot )\Big \Vert _{L^1}\Vert \theta ^2 \Vert _{L^\infty } \\&\qquad \bigg \Vert R_{v,k} \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k W_k \,\textrm{d} x\bigg ) \bigg \Vert _{L^\infty (B^{-1}_{1,\infty })} \\&\lesssim \,\sigma _v^{-1}. \end{aligned}$$

For the temporal part of the error, we obtain by (3.53)

$$\begin{aligned} \Vert R_{\textrm{osc},t,v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \Vert R_{\textrm{osc},t,v} \Vert _{L^\infty (L^\infty )} \\&=\, \nu _v^{-1} \sum _{k=1}^2 \Vert h_{v,k} (\nu _v \cdot )\Vert _{L^\infty } \Vert \partial _t R_{v,k} \Vert _{L^\infty (L^\infty )} \lesssim \nu _v^{-1}. \end{aligned}$$

Consequently, (6.3) follows by choosing $\lambda _v$ sufficiently large, depending on $R_v$. $\square$

6.2 Corrector Error

Lemma 6.3

If $\lambda _h$ and $\lambda _v$ are sufficiently large (depending on $R_h$ and $R_v$, respectively), then the corrector errors satisfy the estimates

$$\begin{aligned} \Vert R_{\textrm{cor},h} \Vert _{L^1 (L^1)}&\le \frac{\epsilon }{3}, \end{aligned}$$

(6.4)

$$\begin{aligned} \Vert R_{\textrm{cor},v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \frac{\epsilon }{3}. \end{aligned}$$

(6.5)

Proof

First, we estimate $R_{\textrm{cor},h}$. Since estimate (3.19) does not hold for $p=1$, we have to introduce a suitable $r>1$. Let us fix $1<r\le 2$ such that $1-\frac{1}{2}\delta c_v\le \frac{1}{r}$, where $c_v>0$ is given by Lemma 3.1. More precisely, if $1-\frac{1}{2}\delta c_v>0$, we choose $1<r\le \min \left\{ 2,\frac{1}{1-\frac{1}{2}\delta c_v}\right\}$ which is possible due to $1-\frac{1}{2}\delta c_v<1$. On the other hand, if $1-\frac{1}{2}\delta c_v\le 0$, we simply take $1<r\le 2$. Moreover, we set $\frac{1}{{\widetilde{r}}}=\frac{1}{r}-\frac{1}{2}$. Then (similar to (5.14)), we obtain by (3.1), (3.51) and Lemmas 3.1 and 3.11

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_2-} (L^{{\widetilde{r}}})}&\le \frac{1}{\Vert R_h \Vert _{L^1 (L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{q_2-}} \Vert R_v \Vert _{L^\infty (L^{\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{L^{{\widetilde{r}}}} \nonumber \\&\lesssim \kappa _v^{1/q_2 - 1/q_2 - \delta } \mu _v^{1/2-1/{\widetilde{r}}} = \kappa _v^{-\delta } \mu _v^{1-1/r}\lesssim \kappa _v^{-\delta } \mu _v^{\frac{1}{2}\delta c_v} \nonumber \\ {}&= \kappa _v^{-\delta + \frac{1}{2}\delta } \lesssim \lambda _v^{-\frac{1}{2}\gamma _v}. \end{aligned}$$

(6.6)

Now we are ready to estimate $R_{\textrm{cor},h}$. Using Lemmas 3.4, 5.2, 5.5, 5.7 and 5.8, and bound (6.6) we get¹³

$$\begin{aligned}&\Vert R_{\textrm{cor},h} \Vert _{L^1 (L^1)} \lesssim \Vert R_{\textrm{cor},h} \Vert _{L^1 (L^r)} \\&\quad \lesssim \bigg \Vert \big ( u_{c,h} + u_{t,h} \big ) \otimes \big ( u_{c,h} + u_{t,h} \big ) + u_{p,h} \otimes \big (u_{c,h} + u_{t,h} \big ) + \big (u_{c,h} + u_{t,h} \big )\otimes u_{p,h} \\&\qquad + \overline{\big ( u_{p,v} + u_{c,v} + u_{t,v} \big ) \otimes \big ( u_{p,v} + u_{c,v} + u_{t,v} \big )} \bigg \Vert _{L^1 (L^r)} \\&\quad \lesssim \Vert u_{p,h} \Vert _{L^2 (L^2)} \Big ( \Vert u_{c,h} \Vert _{L^2 (L^\infty )} + \Vert u_{t,h} \Vert _{L^2(L^\infty )}\Big ) + \Vert u_{c,h} \Vert _{L^2 (L^\infty )}^2 + \Vert u_{t,h} \Vert _{L^2 (L^\infty )}^2 \\&\qquad + \Vert u_{p,v} \Vert _{L^2 (L^2)} \Vert u_{p,v} \Vert _{L^2 (L^{{\widetilde{r}}})} + \Vert u_{c,v} \Vert _{L^2 (L^\infty )}^2 + \Vert u_{t,v} \Vert _{L^2 (L^\infty )}^2 \\&\quad \lesssim \Vert R_h \Vert _{L^1(L^1)}^{1/2} \lambda _h^{-\gamma _h} + \lambda _h^{-2\gamma _h} + \lambda _v^{-\frac{3}{2}\gamma _v} + \lambda _v^{-2\gamma _v}, \end{aligned}$$

which implies (6.4) as soon as $\lambda _h$ and $\lambda _v$ are suitably large (depending on $R_h$ and $R_v$, respectively).

Finally, according to Lemmas 3.8, 5.2, 5.5, 5.7, 5.8 and A.3

In these estimates, we have used the fact that the time interval [0, T] is finite. Then, (6.5) follows by choosing $\lambda _h$ and $\lambda _v$ large enough (again depending on $R_h$ and $R_v$, respectively). $\square$

6.3 Linear Error

Lemma 6.4

If $\lambda _h$ and $\lambda _v$ are chosen sufficiently large (depending on $R_h$ and $R_v$, respectively), then the linear errors satisfy the estimates

$$\begin{aligned} \Vert R_{\textrm{lin},h} \Vert _{L^1 (L^1)}&\le \frac{\epsilon }{3}, \end{aligned}$$

(6.7)

$$\begin{aligned} \Vert R_{\textrm{lin},v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \frac{\epsilon }{3}. \end{aligned}$$

(6.8)

In order to prove this lemma, we consider the time derivative (see Sect. 6.3.1) and advective terms (see Sect. 6.3.2) separately.

Proof of Lemma 6.4

We simply conclude using Lemmas 6.5 and 6.6 by choosing $\lambda _h$ and $\lambda _v$ large enough. $\square$

6.3.1 Time Derivative

Lemma 6.5

For the time derivative part of the linear error, the following bounds hold

$$\begin{aligned} \Vert R_{\textrm{lin},t,h} \Vert _{L^1 (L^1)}&\lesssim \lambda _h^{-\gamma _h}, \end{aligned}$$

(6.9)

$$\begin{aligned} \Vert R_{\textrm{lin},t,v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

(6.10)

Proof

According to (4.8), we have

$$\begin{aligned} \partial _t (u_{p,h} + u_{c,h}) = \sigma _h^{-1} \sum _{k \in \Lambda } \nabla _h \cdot (\partial _t a_k (x,t) \Omega _k (\sigma _h x)). \end{aligned}$$

Using Lemmas 3.1, 3.4, 3.11 and 5.1, we thus find

$$\begin{aligned} \Vert R_{\textrm{lin},t,h} \Vert _{L^1 (L^1)}&\lesssim \Vert {\mathcal {R}}_h \partial _t (u_{p,h} + u_{c,h}) \Vert _{L^1 (L^2)} \\&\lesssim \sigma _h^{-1} \sum _{k \in \Lambda } \Vert \partial _t a_k \Vert _{L^1 (L^\infty )} \Vert \Omega _k (\sigma _h \cdot ) \Vert _{L^2} \le \sigma _h^{-1} \nu _h \kappa _h^{1/2} \mu _h^{-1} \lesssim \lambda _h^{-\gamma _h}. \end{aligned}$$

Similarly, (4.15) implies

$$\begin{aligned} \partial _t (u_{p,v} + u_{c,v}) = - \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \nabla _h \cdot \partial _t\big (g_{v,k}^- (\nu _v t) \theta R_{v,k} \Omega _k (\sigma _v x)\big ). \end{aligned}$$

Hence, from Lemmas 3.1, 3.8, 3.11 and A.3, the assumption $q_2>2$, and estimate (3.54) we obtain

$$\begin{aligned}&\Vert R_{\textrm{lin},t,v} \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \Big \Vert {\mathcal {R}}_v \nabla _h \cdot \partial _t\big (g_{v,k}^- (\nu _v \cdot ) \theta R_{v,k} \Omega _k (\sigma _v \cdot )\big ) \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \Big \Vert \nabla _h \cdot \partial _t\big (g_{v,k}^- (\nu _v \cdot ) \theta R_{v,k} \Omega _k (\sigma _v \cdot )\big ) \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \Big \Vert \partial _t\big (g_{v,k}^- (\nu _v \cdot ) \theta R_{v,k} \Omega _k (\sigma _v \cdot )\big ) \Big \Vert _{L^1 (L^1)} \\&\quad \lesssim \frac{\sigma _v^{-1}}{\Vert R_h \Vert _{L^1 (L^1 )}} \sum _{k = 1}^2 \Vert g_{v,k}^- (\nu _v \cdot )\Vert _{W^{1,1}} \Vert \theta \Vert _{W^{1,\infty }} \Vert R_{v,k}\Vert _{W^{1,\infty }(L^\infty )} \Vert \Omega _k (\sigma _v \cdot )\Vert _{L^1} \\&\quad \lesssim \sigma _v^{-1} \nu _v \kappa _v^{1/q_2} \mu _v^{-3/2} \lesssim \lambda _v^{-\gamma _v}. \end{aligned}$$

$\square$

6.3.2 Advective Terms

Lemma 6.6

For the advective part of the linear error, the following bounds hold

$$\begin{aligned} \Vert R_{\textrm{lin},x,h} \Vert _{L^1 (L^1)}&\lesssim \lambda _h^{-\gamma _h} + \lambda _v^{-\gamma _v}, \\ \Vert R_{\textrm{lin},x,v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \lambda _h^{-\gamma _h} + \lambda _v^{-\gamma _v}. \end{aligned}$$

Proof

Lemmas 3.4, 5.2, 5.5, 5.7 and 5.8 yield

$$\begin{aligned}&\Vert R_{\textrm{lin},x,h} \Vert _{L^1 (L^1)} \\&\quad =\bigg \Vert {\mathcal {R}}_h \bigg [ \nabla _h \cdot \bigg ( {\overline{u}} \otimes \big ( u_{p,h} + u_{c,h} + u_{t,h} \big ) + \overline{u \otimes \big ( u_{p,v} + u_{c,v} + u_{t,v} \big ) } \\&\qquad + \big ( u_{p,h} + u_{c,h} + u_{t,h} \big ) \otimes {\overline{u}} + \overline{\big ( u_{p,v} + u_{c,v} + u_{t,v} \big ) \otimes u}\bigg ) \bigg ] \bigg \Vert _{L^1 (L^1)} \\&\quad \lesssim \bigg \Vert {\overline{u}} \otimes \big ( u_{p,h} + u_{c,h} + u_{t,h} \big ) + \overline{u \otimes \big ( u_{p,v} + u_{c,v} + u_{t,v} \big ) } \\&\qquad + \big ( u_{p,h} + u_{c,h} + u_{t,h} \big ) \otimes {\overline{u}} + \overline{\big ( u_{p,v} + u_{c,v} + u_{t,v} \big ) \otimes u} \bigg \Vert _{L^1 (L^2)} \\&\quad \lesssim \Vert u\Vert _{L^\infty (L^\infty )} \Big ( \Vert u_{p,h} \Vert _{L^1(L^2)} + \Vert u_{c,h} \Vert _{L^1(L^2)} + \Vert u_{t,h} \Vert _{L^1(L^2)} \Big ) \\&\qquad + \Vert u\Vert _{L^\infty (L^\infty )} \Big ( \Vert u_{p,v} \Vert _{L^1(L^2)} + \Vert u_{c,v} \Vert _{L^1(L^2)} + \Vert u_{t,v} \Vert _{L^1(L^2)} \Big ) \\&\quad \lesssim \lambda _h^{-\gamma _h} + \lambda _v^{-\gamma _v}. \end{aligned}$$

For the vertical advective terms, we have according to Lemmas 3.8, 5.2, 5.5, 5.7, 5.8 and A.3

$\square$

7 The Viscous Primitive Equations

In this section, we consider the viscous primitive equations (1.24)–(1.26). We begin by stating the viscous primitive-Reynolds system

$$\begin{aligned} \partial _t u - \nu _h^* \Delta _h u - \nu _v^* \partial _{zz} u + u \cdot \nabla _h u + w \partial _z u + \nabla _h p&= \nabla _h \cdot R_h + \partial _z R_v, \end{aligned}$$

(7.1)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(7.2)

$$\begin{aligned} \nabla _h \cdot u + \partial _z w&=0 . \end{aligned}$$

(7.3)

We prove the following version of Proposition 2.4. Theorem 1.13 can be proved in exactly the same fashion as Theorem 1.4.

Proposition 7.1

Suppose $(u,w,p,R_h,R_v)$ is a smooth solution of the viscous primitive-Reynolds system (7.1)–(7.3), which is well-prepared with associated time interval I and parameter $\tau >0$. Moreover, consider parameters $1\le q_1,q_2,q_3\le \infty$ and $0<s_1,s_3$ which satisfy the following constraints¹⁴

$$\begin{aligned} q_2> 2, \quad \frac{2}{q_1}> s_1 + 1, \quad \frac{2}{q_3}> s_3 + \frac{2}{q_2}, \quad s_3> \frac{1}{2\left( 1-\frac{1}{q_2}\right) } \left( \frac{1}{q_3} - \frac{1}{q_2}\right) . \end{aligned}$$

(7.4)

Finally, let $\delta ,\epsilon >0$ be arbitrary. Then there exists another smooth solution $(u+{\overline{u}}_p+{\widetilde{u}}_p,w+w_p,p+P,R_{h,1},R_{v,1})$ of the viscous primitive-Reynolds system (7.1)–(7.3) which is well-prepared with respect to the same time interval I and parameter $\tau /2$, and has the following properties:

• 1. $({\overline{u}}_p,{\widetilde{u}}_p,w_p)(x,t)=(0,0,0)$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

• 2. The perturbation and Reynolds stress tensors satisfy the following estimates

  $$\begin{aligned} \Vert R_{h,1} \Vert _{L^1 (L^1)}&\le \epsilon , \end{aligned}$$

  (7.5)
  $$\begin{aligned} \Vert R_{v,1} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \epsilon , \end{aligned}$$

  (7.6)
  $$\begin{aligned} \Vert {\overline{u}}_p \Vert _{L^1 (W^{1,1})}&\le \epsilon , \end{aligned}$$

  (7.7)
  $$\begin{aligned} \Vert {\overline{u}}_p \Vert _{L^{q_1} (H^{s_1})}&\le \epsilon , \end{aligned}$$

  (7.8)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^1 (W^{1,1})}&\le \epsilon , \end{aligned}$$

  (7.9)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_2-} (L^2)}&\le \epsilon , \end{aligned}$$

  (7.10)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_3-} (H^{s_3})}&\le \epsilon , \end{aligned}$$

  (7.11)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_2'-} (L^2)}&\le \epsilon , \end{aligned}$$

  (7.12)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_3'-} (H^{-s_3})}&\le \epsilon . \end{aligned}$$

  (7.13)

• 3. Moreover, we have the following bounds

  $$\begin{aligned} \Vert {\overline{u}}_p \Vert _{L^{2} (L^2)}&\lesssim \Vert R_h \Vert _{L^1 (L^1)}^{1/2}, \end{aligned}$$

  (7.14)
  $$\begin{aligned} \Vert w_p {\widetilde{u}}_p + w {\widetilde{u}}_p + w_p u \Vert _{L^1 (B^{-1}_{1,\infty })}&\lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

  (7.15)

In order to prove Proposition 7.1, we need the following version of Lemma 3.1.

Lemma 7.2

Let $1 \le q_1, q_2, q_3 \le \infty$ and $0<s_1, s_3$ satisfy the conditions (7.4). Then we can choose $a_i,b_i,c_i>0$ for $i=h,v$ in (3.1) with the property that there exist $\gamma _h, \gamma _v > 0$ such that

$$\begin{aligned} \kappa _h^{-1/2} \sigma _h \mu _h^{1/2}&\le \lambda _h^{-\gamma _h}, \end{aligned}$$

(7.16)

$$\begin{aligned} \kappa _v^{1/q_2 - 1} \sigma _v \mu _v^{1/2}&\le \lambda _v^{-\gamma _v}, \end{aligned}$$

(7.17)

in addition to (3.3)–(3.7) and $\mu _i,\sigma _i,\kappa _i,\nu _i\ge \lambda _i^{\gamma _i}$ for $i=h,v$.

Proof

Similar to the proof of Lemma 3.1, it suffices to show that there is a choice of $a_i,b_i,c_i>0$ for $i=h,v$ such that

$$\begin{aligned} -\bigg ( \frac{1}{2} - \frac{1}{q_1} \bigg ) c_h - s_1 (b_h+1)&>0, \end{aligned}$$

(7.18)

$$\begin{aligned} b_h - a_h - \frac{1}{2} c_h + 1&>0, \end{aligned}$$

(7.19)

$$\begin{aligned} -\bigg ( \frac{1}{q_2} - \frac{1}{q_3} \bigg ) c_v - s_3 (b_v+1)&=0 , \end{aligned}$$

(7.20)

$$\begin{aligned} b_v - a_v - \frac{1}{2} c_v + 1&>0, \end{aligned}$$

(7.21)

$$\begin{aligned} \frac{1}{2} c_h - b_h - \frac{1}{2}&>0 , \end{aligned}$$

(7.22)

$$\begin{aligned} -\left( \frac{1}{q_2} - 1\right) c_v - b_v - \frac{1}{2}&>0 . \end{aligned}$$

(7.23)

Let us first fix $0<a_h<1/2$, and then $b_h > 0$ such that

$$\begin{aligned} b_h \left( s_1 + 1 - \frac{2}{q_1} \right) < -s_1 + \left( 2-2a_h\right) \left( \frac{1}{q_1}- \frac{1}{2} \right) . \end{aligned}$$

(7.24)

Note that such a choice is possible since $s_1 + 1 - \frac{2}{q_1} <0$ according to (7.4). Because (7.4) implies $q_1<2$, (7.24) is equivalent to

$$\begin{aligned} \frac{s_1 (b_h + 1)}{\frac{1}{q_1} - \frac{1}{2}} < 2 b_h + 2 - 2a_h. \end{aligned}$$

(7.25)

Moreover, as $a_h<1/2$, we have

$$\begin{aligned} 2b_h + 1 < 2b_h + 2 - 2a_h. \end{aligned}$$

(7.26)

From (7.25) and (7.26), we deduce that there exists $c_h>0$ with

$$\begin{aligned} \frac{s_1 (b_h + 1)}{\frac{1}{q_1} - \frac{1}{2}}&< c_h, \\ 2b_h + 2 - 2a_h&> c_h, \\ 2b_h +1&< c_h, \end{aligned}$$

which are equivalent to (7.18), (7.19) and (7.22), respectively.

Next we choose $a_v,b_v,c_v>0$. We simply deduce from (7.4) that

$$\begin{aligned} \frac{s_3 \left( 1-\frac{1}{q_2} \right) }{\frac{1}{q_3} - \frac{1}{q_2}} > \frac{1}{2}. \end{aligned}$$

Thus, we can choose $0<b_v \ll 1$ such that

$$\begin{aligned} b_v \left( - 1 + \frac{s_3 \left( 1-\frac{1}{q_2} \right) }{\frac{1}{q_3} - \frac{1}{q_2}} \right) - \frac{1}{2} + \frac{s_3 \left( 1-\frac{1}{q_2} \right) }{\frac{1}{q_3} - \frac{1}{q_2}} > 0. \end{aligned}$$

(7.27)

Then we fix

$$\begin{aligned} c_v := \frac{s_3 \left( b_v +1\right) }{\frac{1}{q_3} - \frac{1}{q_2}}, \end{aligned}$$

which is positive as $\frac{1}{q_3} - \frac{1}{q_2}>0$ which in turn follows from (7.4). The choice of $c_v$ immediately implies (7.20), while (7.27) is equivalent to (7.23). Finally, (7.4) ensures

$$\begin{aligned} (b_v+1) \left( 1 - \frac{s_3}{2 \left( \frac{1}{q_3} - \frac{1}{q_2}\right) }\right) >0. \end{aligned}$$

This is equivalent to

$$\begin{aligned} b_v - \frac{1}{2} c_v + 1 > 0, \end{aligned}$$

which in turn allows for the choice of a small $a_v>0$ such that (7.21) holds. $\square$

Now we can prove Proposition 7.1.

Proof of Proposition 7.1

We make the same choice of perturbations ${\overline{u}}_p$, ${\widetilde{u}}_p$, $w_p$, P as we did in the inviscid case, see Sect. 4. The only errors that change compared to the inviscid case are the linear errors, which now contain the additional terms

$$\begin{aligned} {\mathcal {R}}_h (\nu _h^* \Delta _h {\overline{u}}_p), \qquad {\mathcal {R}}_v \big (\nu _h^* \Delta _h {\widetilde{u}}_p + \nu _v^* \partial _{zz} ({\overline{u}}_p + {\widetilde{u}}_p) \big ), \end{aligned}$$

(7.28)

respectively. Thus, the validity of (7.8), (7.10)–(7.15) follows immediately from Sects. 4–6.

As in the proof of (5.4), one can deduce from (7.16) that

$$\begin{aligned} \Vert u_{p,h} \Vert _{L^1(W^{1,1})}&\le \sum _{k \in \Lambda } \Vert a_k \Vert _{L^{1} (W^{1,\infty })} \Vert W_k(\sigma _h\cdot ) \Vert _{W^{1,1}} \lesssim \kappa _h^{-1/2} \sigma _h \mu _h^{1/2} \le \lambda _h^{-\gamma _h}. \end{aligned}$$

Analogously, we find

$$\begin{aligned} \Vert u_{c,h} \Vert _{L^{1} (W^{1,1})}&\le \sigma _h^{-1} \sum _{k \in \Lambda } \Vert \nabla a_k \Vert _{L^{1} (W^{1,\infty })} \Vert \Omega _k (\sigma _h \cdot ) \Vert _{W^{1,1}} \lesssim \sigma _h^{-1} \kappa _h^{-1/2} \sigma _h \mu _h^{-1/2} \lesssim \lambda _h^{-\gamma _h}. \end{aligned}$$

Similarly we obtain from (7.17)

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{1} (W^{1,1})}&\le \frac{1}{\Vert R_h \Vert _{L^1 (L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{1}} \Vert R_v \Vert _{L^\infty (W^{1,\infty })} \Vert W_k (\sigma _v \cdot ) \Vert _{W^{1,1}} \\&\lesssim \kappa _v^{1/q_2 - 1} \sigma _v \mu _v^{1/2} \le \lambda _v^{-\gamma _v} \end{aligned}$$

and

$$\begin{aligned} \Vert u_{c,v} \Vert _{L^{1} (W^{1,1})}&\le \frac{\sigma _v^{-1}}{\Vert R_h\Vert _{L^1(L^1)}} \sum _{k=1}^2 \Vert g_{v,k}^- (\nu _v \cdot ) \Vert _{L^{1}} \Vert \nabla _h R_{v,k} \Vert _{L^\infty (W^{1,\infty })} \Vert \Omega _k (\sigma _v \cdot ) \Vert _{W^{1,1}} \\&\lesssim \sigma _v^{-1} \kappa _v^{1/q_2 - 1} \sigma _v \mu _v^{-1/2} \le \lambda _v^{-\gamma _v}. \end{aligned}$$

Since $\Vert u_{t,h} \Vert _{L^1(W^{1,1})}\lesssim \lambda _h^{-\gamma _h}$ and $\Vert u_{t,v} \Vert _{L^1(W^{1,1})}\lesssim \lambda _v^{-\gamma _v}$, which follow immediately from Lemma 5.8, we deduce (7.7) and (7.9) by choosing $\lambda _h,\lambda _v$ sufficiently large (depending on $R_h$ and $R_v$, respectively).

In order to show (7.5) and (7.6), it remains to estimate the terms in (7.28). Using Lemmas 3.4, 3.8 and A.3, as well as $\Vert {\overline{u}}_p \Vert _{L^1 (W^{1,1})} \lesssim \lambda _h^{-\gamma _h}$ and $\Vert {\widetilde{u}}_p \Vert _{L^1 (W^{1,1})} \lesssim \lambda _v^{-\gamma _v}$ which we established above, we find

$$\begin{aligned}&\Vert {\mathcal {R}}_h (\nu _h^* \Delta _h {\overline{u}}_p ) \Vert _{L^1 (L^1) } = \nu _h^* \Vert \nabla _h {\overline{u}}_p + \nabla _h {\overline{u}}_p^T \Vert _{L^1 (L^1)} \lesssim \Vert {\overline{u}}_p \Vert _{L^1 (W^{1,1})} \lesssim \lambda _h^{-\gamma _h}, \\&\Vert {\mathcal {R}}_v (\nu _h^* \Delta _h {\widetilde{u}}_p + \nu _v^* \partial _{zz} ({\overline{u}}_p + {\widetilde{u}}_p) ) \Vert _{L^1 (B^{-1}_{1,\infty }) }\\&\quad \lesssim \Vert \Delta _h {\widetilde{u}}_p \Vert _{L^1 (B^{-1}_{1,\infty }) } + \Vert \partial _{z} ({\overline{u}}_p + {\widetilde{u}}_p) \Vert _{L^1 (B^{-1}_{1,\infty }) } \\&\quad \lesssim \Vert \nabla _h {\widetilde{u}}_p \Vert _{L^1 (L^1) } + \Vert {\overline{u}}_p + {\widetilde{u}}_p \Vert _{L^1 (L^1) } \\&\quad \lesssim \Vert {\overline{u}}_p \Vert _{L^1 (W^{1,1} )} + \Vert {\widetilde{u}}_p \Vert _{L^1 (W^{1,1} )} \lesssim \lambda _h^{-\gamma _h} + \lambda _v^{-\gamma _v}. \end{aligned}$$

$\square$

8 Two-Dimensional Hydrostatic Euler Equations

In this section, we will develop a convex integration scheme for the two-dimensional hydrostatic Euler equations (1.28)–(1.30), which is somewhat different in nature to the scheme in the three-dimensional case. A similar scheme is established in Sect. 9 for the (two-dimensional) Prandtl equations (1.32)–(1.34).

In two dimensions, the hydrostatic Euler–Reynolds system (2.1)–(2.3) reduces to

$$\begin{aligned} \partial _t u + u \partial _{x_1} u + w \partial _z u + \partial _{x_1} p&= \partial _{x_1} R_h + \partial _z R_v, \\ \partial _z p&= 0, \\ \partial _{x_1} u + \partial _z w&=0 . \end{aligned}$$

We observe that in contrast with the three-dimensional case, u, $R_h$ and $R_v$ are now just scalar quantities, where $R_h$ does not depend on z and $R_v$ is mean-free with respect to z. The former allows to include $R_h$ as part of the pressure. In other words, by setting

$$\begin{aligned} p' = p - R_h \end{aligned}$$

we may assume without loss of generality that $R_h = 0$ (up to a redefinition of the pressure). So all in all the two-dimensional hydrostatic Euler–Reynolds system we will work with reads

$$\begin{aligned} \partial _t u + u \partial _{x_1} u + w \partial _z u + \partial _{x_1} p&= \partial _z R_v, \end{aligned}$$

(8.1)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(8.2)

$$\begin{aligned} \partial _{x_1} u + \partial _z w&=0 , \end{aligned}$$

(8.3)

with unknowns u, w, p and $R_v$. As $R_h$ is no longer there, the only task is to minimise $R_v$. For this reason, we will only have a baroclinic perturbation ${\widetilde{u}}_p$ in Proposition 8.1.

In this section, we will prove the following version of the inductive proposition (cf. Proposition 2.4). Theorem 1.22 then follows exactly as Theorem 1.4.

Proposition 8.1

Suppose $(u,w,p,R_v)$ is a smooth solution of the two-dimensional hydrostatic Euler–Reynolds system (8.1)–(8.3), which is well-prepared with associated time interval I and parameter $\tau >0$. Moreover, consider parameters $1 \le q_2, q_3 \le \infty$ and $0<s_3$ which satisfy the constraints in (1.31). Finally, let $\delta ,\epsilon > 0$ be arbitrary. Then there exists another smooth solution $(u + {\widetilde{u}}_p, w + w_p, p+P, R_{v,1} )$ of the two-dimensional hydrostatic Euler–Reynolds system (8.1)–(8.3) which is well-prepared with respect to the same time interval I and parameter $\tau /2$, and has the following properties:

• 1. $({\widetilde{u}}_p,w_p)(x,t)=(0,0)$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

• 2. It satisfies the following estimates

  $$\begin{aligned} \Vert R_{v,1} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \epsilon , \end{aligned}$$

  (8.4)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_2-} (L^2)}&\le \epsilon , \end{aligned}$$

  (8.5)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_3-} (H^{s_3})}&\le \epsilon , \end{aligned}$$

  (8.6)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_2'-} (L^2)}&\le \epsilon , \end{aligned}$$

  (8.7)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_3'-} (H^{-s_3})}&\le \epsilon . \end{aligned}$$

  (8.8)

• 3. Moreover, we have that

  $$\begin{aligned} \Vert w_p {\widetilde{u}}_p + w {\widetilde{u}}_p + w_p u \Vert _{L^1 (B^{-1}_{1,\infty })} \lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

  (8.9)

8.1 Preliminaries

Due to the fact that there is no longer an error $R_h$ in the two-dimensional hydrostatic Euler–Reynolds system (8.1)–(8.3), there is no need for a barotropic perturbation ${\overline{u}}_p$. Thus, there is only a baroclinic perturbation ${\widetilde{u}}_p$ in Proposition 8.1. For this reason only the ‘vertical’ parameters $\mu _v,\sigma _v,\kappa _v,\nu _v$ are used in Sects. 8 and 9. Regarding the two-dimensional hydrostatic Euler equations (1.28)–(1.30), we need the following version of Lemma 3.1.

Lemma 8.2

Let $1 \le q_2, q_3 \le \infty$ and $0<s_3$ satisfy the constraints (1.31). Then we can choose $a_v,b_v,c_v>0$ in (3.1) with the property that there exists $\gamma _v>0$ such that

$$\begin{aligned} \kappa _v^{2/q_2 - 1} \sigma _v \mu _v&\le \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.10)

$$\begin{aligned} \sigma _v^{-1} \nu _v \kappa _v^{1/q_2} \mu _v^{-3/2}&\le \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.11)

in addition to (3.5), (3.7) and $\mu _v,\sigma _v,\kappa _v,\nu _v\ge \lambda _v^{\gamma _v}$.

Proof

Similar to the proof of Lemma 3.1, it suffices to show that there is a choice of $a_v,b_v,c_v>0$ such that

$$\begin{aligned} -\bigg ( \frac{1}{q_2} - \frac{1}{q_3} \bigg ) c_v - s_3 (b_v+1)&=0 , \end{aligned}$$

(8.12)

$$\begin{aligned} b_v - a_v - \frac{1}{q_2} c_v + \frac{3}{2}&>0, \end{aligned}$$

(8.13)

$$\begin{aligned} -\left( \frac{2}{q_2} - 1\right) c_v - b_v - 1&>0 . \end{aligned}$$

(8.14)

We know from (1.31) that

$$\begin{aligned} \frac{3}{2} - \frac{s_3}{q_2\left( \frac{1}{q_3} - \frac{1}{q_2}\right) } > 0. \end{aligned}$$

Hence, we can choose $0<b_v \ll 1$ such that

$$\begin{aligned} b_v \left( 1 -\frac{s_3}{q_2\left( \frac{1}{q_3} - \frac{1}{q_2}\right) }\right) + \frac{3}{2} - \frac{s_3}{q_2\left( \frac{1}{q_3} - \frac{1}{q_2}\right) } > 0. \end{aligned}$$

By setting

$$\begin{aligned} c_v := \frac{s_3 \left( b_v +1\right) }{\frac{1}{q_3} - \frac{1}{q_2}}, \end{aligned}$$

the latter implies that

$$\begin{aligned} b_v - \frac{1}{q_2} c_v + \frac{3}{2} >0, \end{aligned}$$

which in turn allows for the choice of a small $a_v>0$ such that (8.13) holds. The definition of $c_v$ immediately implies (8.12). Finally, (1.31) guarantees that

$$\begin{aligned} \left( 1-\frac{2}{q_2}\right) \frac{s_3}{\left( \frac{1}{q_3} - \frac{1}{q_2}\right) } - 1 > 0; \end{aligned}$$

hence,

$$\begin{aligned} (b_v+1) \left( \left( 1-\frac{2}{q_2}\right) \frac{s_3}{\left( \frac{1}{q_3} - \frac{1}{q_2}\right) } - 1 \right) > 0, \end{aligned}$$

which is equivalent to (8.14). $\square$

Remark 8.3

Notice that compared to Lemma 3.1 we have replaced (3.6) by (8.11), where the latter is a weaker restriction than the former. Indeed, it is simple to see that (3.6) implies (8.11) provided $q_2>2$. Note furthermore that (8.11) suffices to prove Lemma 6.5. Moreover, the additional inequality (8.10) is needed to deal with an additional spatial corrector for the vertical velocity.

Moreover, we need a two-dimensional version of the vertical inverse divergence operator ${\mathcal {R}}_v$, cf. Definition 3.7.

Definition 8.4

We define the map¹⁵${\mathcal {R}}_v: C^\infty _{0,z}({\mathbb {T}}^2;{\mathbb {R}}) \rightarrow C^\infty ({\mathbb {T}}^2;{\mathbb {R}})$ by

$$\begin{aligned} ({\mathcal {R}}_v v) (x_1,z) := \int _0^z v (x_1,z') \,\textrm{d} z' - \int _0^1 \int _0^{z'} v (x_1,z'') \,\textrm{d} z'' \,\textrm{d} z'. \end{aligned}$$

(8.15)

Note that the vertical inverse divergence operator defined in Definition 8.4 has the same properties as stated in Lemma 3.8.

Regarding the building blocks, we will use the following version of Proposition 3.9.

Proposition 8.5

There exists a function $\phi \in C^\infty ({\mathbb {T}};{\mathbb {R}})$ (referred to as the Mikado density) depending on a parameter $\mu _v$, with the following properties.

• 1. The function $\phi$ has zero mean. Moreover, $\int _{{\mathbb {T}}} \phi ^2 \,\textrm{d} x= 1$.

• 2. There exists $\Omega \in C^\infty ({\mathbb {T}};{\mathbb {R}})$ with zero mean such that $\phi =\partial _{x_1} \Omega$.

• 3. For all $s\ge 0$ and $1\le p\le \infty$, the following estimates hold:

  $$\begin{aligned} \Vert \phi \Vert _{W^{s,p}({\mathbb {T}})}&\lesssim \mu _v^{\frac{1}{2}-\frac{1}{p} + s} , \\ \Vert \Omega \Vert _{W^{s,p}({\mathbb {T}})}&\lesssim \mu _v^{-\frac{1}{2}-\frac{1}{p}+s}. \end{aligned}$$

  Here the implicit constant may depend on s, p, but it does not depend on $\mu _v$.

Similar to Proposition 3.9, Proposition 8.5 can be proved as in Cheskidov and Luo (2022, Section 4.1). In fact, the function $\phi$ in Proposition 8.5 coincides with the function $\phi _2$ from Proposition 3.9.

Analogously to Lemma 3.11, the estimates

$$\begin{aligned} \Vert \phi (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _v)^s \mu _v^{\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(8.16)

$$\begin{aligned} \Vert \Omega (\sigma \cdot ) \Vert _{W^{s,p}}&\lesssim (\sigma \mu _v)^s \mu _v^{-\frac{1}{2} - \frac{1}{p}}, \end{aligned}$$

(8.17)

hold for all $\sigma \in {\mathbb {N}}$, $s\ge 0$ and $1\le p\le \infty$.

In what follows, we will always write $\phi (x)$, but let us clarify that actually $\phi$ only depends on $x_1$.

Finally, we remark that for the two-dimensional convex integration scheme developed in this section, we will work with the same temporal intermittency functions as defined in Sect. 3.5.2.

8.2 Definition of the Perturbation

The velocity perturbation will be written as

$$\begin{aligned} {\widetilde{u}}_p&= u_{p,v} + u_{t,v}, \\ w_p&= w_{p,v} + w_{c,v} + w_{t,v}, \end{aligned}$$

so in contrast with the three-dimensional case, a spatial corrector $u_{c,v}$ is not needed; however, we will have a spatial corrector $w_{c,v}$ for the vertical velocity.

We make the following choice for the principal part of the perturbation

$$\begin{aligned} u_{p,v}(x,t)&\,:= - g_{v,2}^- (\nu _v t) \theta (t) R_{v}(x,t) \phi (\sigma _v x), \end{aligned}$$

(8.18)

$$\begin{aligned} w_{p,v}(x,t)&\,:= g_{v,2}^+ (\nu _v t) \theta (t) \phi (\sigma _v x) . \end{aligned}$$

(8.19)

Remark 8.6

In order to achieve endpoint time integrability for w (cf. Remark 1.24), one has to multiply the right-hand side of (8.19) by $\Vert R_v \Vert _{L^1(B^{-1}_{1,\infty })}$ and divide the right-hand side of (8.18) by the same factor. Further modifications are straightforward. In order to get endpoint time integrability for u, one proceeds as described in Remarks 1.5, 2.6 and 5.6.

Then we introduce a corrector for the vertical velocity in order to get a divergence-free perturbation:

$$\begin{aligned} w_{c,v}\, {:=}\, {\mathcal {R}}_v \Big ( g_{v,2}^- (\nu _v t) \theta \partial _{x_1} (R_{v} \phi (\sigma _v x)) \Big ), \end{aligned}$$

(8.20)

where ${\mathcal {R}}_v$ is now given by Definition 8.4. Note that $R_v$ is mean-free with respect to z, and hence, the operator ${\mathcal {R}}_v$ can be applied to the expression in (8.20). With Lemma 3.8, it is simple to see that $\partial _{x_1} u_{p,v} + \partial _z w_{p,v} = 0$.

Finally, we introduce a temporal corrector of the form

$$\begin{aligned} u_{t,v} := \nu _v^{-1} h_{v,2} (\nu _v t) \partial _z R_v. \end{aligned}$$

(8.21)

To keep the whole velocity field divergence-free, we now must introduce a temporal corrector for the vertical velocity

$$\begin{aligned} w_{t,v} := -\nu _v^{-1} h_{v,2} (\nu _v t) \partial _{x_1} R_v. \end{aligned}$$

(8.22)

We observe that $\partial _{x_1} {\widetilde{u}}_p + \partial _z w_p = 0$, item 1 of Proposition 8.1 holds, and $u_{p,v}$ and $u_{t,v}$ are indeed mean-free with respect to z.

8.3 The New Reynolds Stress Tensor

As in the three-dimensional scheme, the new Reynolds stress tensor $R_{v,1}$ will be written as

$$\begin{aligned} R_{v,1} = R_{\textrm{osc},v} + R_{\textrm{lin},v} + R_{\textrm{cor},v}. \end{aligned}$$

First, we set $R_{\textrm{osc},v}=R_{\textrm{osc},x,v}+R_{\textrm{osc},t,v}$, where

$$\begin{aligned} \begin{aligned} R_{\text {osc},x,v}&\,{:=} - g_{v,2}^- (\nu _v t) g_{v,2}^+ (\nu _v t) \theta ^2 R_v \bigg ( \phi ^2 (\sigma _v x) - \int _{{\mathbb {T}}} \phi ^2 (x) \,\text {d} x\bigg ), \\ R_{\text {osc},t,v}&\,{:=}\, \nu _v^{-1} h_{v,2} (\nu _v t) \partial _t R_v. \end{aligned} \end{aligned}$$

Exactly as in Lemma 4.3 one can show that

$$\begin{aligned} \partial _t u_{t,v} + \partial _z ( w_{p,v} u_{p,v} + R_v) = \partial _z R_{\textrm{osc},v}. \end{aligned}$$

(8.23)

Next, we define the linear and corrector errors by

and

Note that the arguments of the operator ${\mathcal {R}}_v$ are indeed mean-free with respect to z.

In the three-dimensional convex integration scheme, we introduced a pressure perturbation, see Sect. 4.2. An analogue of this perturbation is not needed in two dimensions as there is no barotropic perturbation. However, the pressure has to absorb some terms that were covered by the horizontal Reynolds stress tensor in the three-dimensional scheme. To this end, we define

$$\begin{aligned} P \,{:=}\, - 2 \overline{u \big (u_{p,v}+u_{t,v}\big )} - \overline{\big (u_{p,v}+u_{t,v}\big )^2}. \end{aligned}$$

(8.24)

We observe that $\partial _z P = 0$.

Let us finally remark that it is straightforward to see that $R_{v,1}$ is mean-free with respect to z, that $R_{v,1}(x,t)=0$ whenever ${{\,\textrm{dist}\,}}(t,I^c)\le \frac{\tau }{2}$, and that $(u + {\widetilde{u}}_p, w + w_p, p+P, R_{v,1} )$ solves (8.1).

8.4 Estimates on the Perturbation

Now we claim the following estimates on the perturbation.

Lemma 8.7

If $\lambda _v$ is chosen sufficiently large (depending on $R_v$), then we have that

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_2-} (L^2)}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.25)

$$\begin{aligned} \Vert u_{p,v} \Vert _{L^{q_3-} (H^{s_3})}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.26)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_2'-} (L^2)}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.27)

$$\begin{aligned} \Vert w_{p,v} \Vert _{L^{q_3'-} (H^{-s_3})}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.28)

$$\begin{aligned} \Vert w_{p,v} u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty } )}&\lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

(8.29)

Proof

In fact, the proof of (8.25)–(8.28) can be taken verbatim from Lemma 5.5. Estimate (8.29) can also be proved in a similar way as in Lemma 5.5. To this end, we need a version of Lemma 5.3, namely the estimate

$$\begin{aligned} \bigg \Vert \bigg ( \phi ^2 (\sigma _v \cdot ) - \int _{{\mathbb {T}}} \phi ^2 \,\textrm{d} x\bigg ) R_v \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \lesssim \sigma _v^{-1}, \end{aligned}$$

(8.30)

which holds for any $1\le p\le \infty$. To show (8.30), we mimic the proof of Lemma 5.3. This requires to introduce a one-dimensional horizontal inverse divergence operator ${\mathcal {R}}_h$, which is defined analogously to the vertical inverse divergence, cf. Definition 8.4, specifically

$$\begin{aligned} {\mathcal {R}}_h v (x) \,{:=}\, \int _0^x v (x') \,\textrm{d} x' - \int _0^1 \int _0^{x'} v (x'') \,\textrm{d} x'' \,\textrm{d} x'. \end{aligned}$$

Consequently, ${\mathcal {R}}_h$ has the properties stated in Lemma 3.8. Using property (3.30), we are able to prove (8.30), and thus, (8.29) follows. $\square$

The spatial and temporal correctors can be estimated as follows.

Lemma 8.8

The spatial and temporal correctors satisfy the following estimates

$$\begin{aligned} \Vert w_{c,v} \Vert _{L^{q_2'} (L^2)} + \Vert w_{c,v} \Vert _{L^{q_3'} (H^{-s_3} )}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.31)

$$\begin{aligned} \Vert u_{t,v} \Vert _{L^\infty (W^{n,\infty } )}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.32)

$$\begin{aligned} \Vert w_{t,v} \Vert _{L^\infty (W^{n,\infty } )}&\lesssim \lambda _v^{-\gamma _v}, \end{aligned}$$

(8.33)

where $n\in {\mathbb {N}}$ is arbitrary, and the implicit constant may depend on n.

Proof

We only prove (8.31), as the proof of the estimates for the temporal correctors (8.32), (8.33) is similar to the proof of Lemma 5.8. We obtain using Lemmas 3.8 and 8.2, and equations (3.51), (8.16)

$$\begin{aligned} \Vert w_{c,v} \Vert _{L^{q_2'} (L^2)}&\lesssim \Vert g_{v,2}^- (\nu _v \cdot ) \Vert _{L^{q_2'}} \Vert \theta \Vert _{L^\infty } \Vert R_v \Vert _{L^\infty (W^{1,\infty })} \Vert \phi (\sigma _v \cdot ) \Vert _{H^{1}} \\&\lesssim \kappa _v^{1/q_2 - 1/q_2'} \sigma _v \mu _v = \kappa _v^{2/q_2 - 1} \sigma _v \mu _v \le \lambda _v^{-\gamma _v}. \end{aligned}$$

Analogously

$$\begin{aligned} \Vert w_{c,v} \Vert _{L^{q_3'} (H^{-s_3})}&\lesssim \Vert g_{v,2}^- (\nu _v \cdot ) \Vert _{L^{q_3'}} \Vert \theta \Vert _{L^\infty } \Vert R_v \Vert _{L^\infty (W^{1,\infty })} \Vert \phi (\sigma _v \cdot ) \Vert _{H^{1-s_3}} \\&\lesssim \kappa _v^{1/q_2 - 1/q_3'} (\sigma _v \mu _v)^{1-s_3} \\&= \big (\kappa _v^{2/q_2 - 1} \sigma _v \mu _v \big )\big (\kappa _v^{1/q_3 - 1/q_2} (\sigma _v\mu _v)^{-s_3} \big )\le \lambda _v^{-\gamma _v}. \end{aligned}$$

Here we have used that ${\mathcal {R}}_v$ is bounded in $H^{-s_3}$, see (3.29) and keeping in mind that $H^{-s_3}=B^{-s_3}_{2,2}$, and $s_3\le 1$. $\square$

Lemmas 8.7 and 8.8 show that estimates (8.5)–(8.9) hold.

8.5 Estimates on the Reynolds Stress Tensor

In order to finish the proof of Proposition 8.1, it remains to show (8.4).

Lemma 8.9

If $\lambda _v$ is chosen sufficiently large (depending on $R_v$), then the errors satisfy the following estimates

$$\begin{aligned} \Vert R_{\textrm{osc},v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \frac{\epsilon }{3}, \\ \Vert R_{\textrm{cor},v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \frac{\epsilon }{3}, \\ \Vert R_{\textrm{lin},v} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \frac{\epsilon }{3}. \end{aligned}$$

Proof

Lemma 8.9 can be proved similarly to Lemmas 6.2–6.6, with only small modifications: To obtain the required estimate for $R_{\textrm{osc},x,v}$, one has to use (8.30) rather than Lemma 5.3. Moreover, since in two dimensions there is no spatial corrector $u_{c,v}$, the time derivative part of the linear error $R_{\textrm{lin},t,v}$ must be estimated slightly differently compared to Lemma 6.5. To this end, we write

$$\begin{aligned} u_{p,v}&= - g_{v,2}^- (\nu _v t) \theta R_{v} \phi (\sigma _v x) \\&= - \sigma _v^{-1} g_{v,2}^- (\nu _v t) \theta R_{v} \partial _{x_1} \big [\Omega (\sigma _v x)\big ] \\&= \partial _{x_1} \big [ - \sigma _v^{-1} g_{v,2}^- (\nu _v t) \theta R_{v} \Omega (\sigma _v x) \big ] + \sigma _v^{-1} g_{v,2}^- (\nu _v t) \theta (\partial _{x_1} R_{v}) \Omega (\sigma _v x) . \end{aligned}$$

Hence, we find (by using inequality (8.11))

$$\begin{aligned}&\Big \Vert {\mathcal {R}}_v \big [ \partial _t u_{p,v} \big ] \Big \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \Vert \partial _t u_{p,v} \Vert _{L^1 (B^{-1}_{1,\infty })} \\&\quad \lesssim \Big \Vert \partial _t \big [ \sigma _v^{-1} g_{v,2}^- (\nu _v \cdot ) \theta R_{v} \Omega (\sigma _v \cdot ) \big ] \Big \Vert _{L^1(L^1)}\\&\qquad + \Big \Vert \partial _t \big [ \sigma _v^{-1} g_{v,2}^- (\nu _v \cdot ) \theta (\partial _{x_1} R_{v}) \Omega (\sigma _v \cdot ) \big ] \Big \Vert _{L^1(L^1)} \\&\quad \lesssim \sigma _v^{-1} \Vert g_{v,2}^- (\nu _v \cdot ) \Vert _{W^{1,1}} \Vert \theta \Vert _{W^{1,\infty }} \Vert R_v\Vert _{W^{1,\infty }(W^{1,\infty })} \Vert \Omega (\sigma _v \cdot ) \Vert _{L^1} \\&\quad \lesssim \sigma _v^{-1} \nu _v \kappa _v^{1/q_2} \mu _v^{-3/2} \le \lambda _v^{\gamma _v}. \end{aligned}$$

$\square$

This finishes the proof of Proposition 8.1.

9 The Two-Dimensional Prandtl Equations

In this section, we will study the following Prandtl–Reynolds system

$$\begin{aligned} \partial _t u - \nu _v^* \partial _{zz} u + u \partial _{x_1} u + w \partial _z u + \partial _{x_1} p&= \partial _z R_v, \end{aligned}$$

(9.1)

$$\begin{aligned} \partial _z p&= 0, \end{aligned}$$

(9.2)

$$\begin{aligned} \partial _{x_1} u + \partial _z w&=0 , \end{aligned}$$

(9.3)

with unknowns $(u,w,p,R_v)$. As in Sect. 8, there is no horizontal Reynolds stress tensor $R_h$. In this setting, we have the following version of the inductive proposition (cf. Proposition 2.4). As before, the proof of Theorem 1.29 then works in the same way as the proof of Theorem 1.4.

Proposition 9.1

Suppose $(u,w,p,R_v)$ is a smooth solution of the Prandtl-Reynolds system (9.1)–(9.3), which is well-prepared with associated time interval I and parameter $\tau >0$. Moreover, consider parameters $1 \le q_2, q_3 \le \infty$ and $0<s_3$ which satisfy the constraints in (1.35). Finally, let $\delta ,\epsilon > 0$ be arbitrary. Then there exists another smooth solution $(u + {\widetilde{u}}_p, w + w_p, p+P, R_{v,1} )$ of the Prandtl–Reynolds system (9.1)–(9.3), which is well-prepared with respect to the same time interval I and parameter $\tau /2$ and has the following properties:

• 1. $({\widetilde{u}}_p,w_p)(x,t)=(0,0)$ whenever ${{\,\textrm{dist}\,}}(t, I^c) \le \tau /2$.

• 2. The perturbation and the new Reynolds stress tensor satisfy the following estimates

  $$\begin{aligned} \Vert R_{v,1} \Vert _{L^1 (B^{-1}_{1,\infty })}&\le \epsilon , \end{aligned}$$

  (9.4)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{1} (W^{1,1})}&\le \epsilon , \end{aligned}$$

  (9.5)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_2-} (L^2)}&\le \epsilon , \end{aligned}$$

  (9.6)
  $$\begin{aligned} \Vert {\widetilde{u}}_p \Vert _{L^{q_3-} (H^{s_3})}&\le \epsilon , \end{aligned}$$

  (9.7)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_2'-} (L^2)}&\le \epsilon , \end{aligned}$$

  (9.8)
  $$\begin{aligned} \Vert w_p \Vert _{L^{q_3'-} (H^{-s_3})}&\le \epsilon , \end{aligned}$$

  (9.9)

• 3. Finally, the products of the vertical and horizontal perturbations satisfy that

  $$\begin{aligned} \Vert w_p {\widetilde{u}}_p + w {\widetilde{u}}_p + w_p u \Vert _{L^1 (B^{-1}_{1,\infty })} \lesssim \Vert R_v \Vert _{L^1 (B^{-1}_{1,\infty })}. \end{aligned}$$

  (9.10)

Proof

In order to prove Proposition 9.1, we modify the proof of Proposition 8.1 in the same way as we did in the three-dimensional case, cf. proof of Proposition 7.1. Specifically we choose ${\widetilde{u}}_p$, $w_p$ and P as in the proof of Proposition 8.1, while the linear error now contains the additional term

$$\begin{aligned} {\mathcal {R}}_v (\nu _v^* \partial _{zz} {\widetilde{u}}_p ). \end{aligned}$$

(9.11)

Then it follows from Sect. 8 that estimates (9.6)–(9.10) hold. In order to show (9.5), we proceed as in the proof of Proposition 7.1. To this end, we need the additional parameter estimate

$$\begin{aligned} \kappa _v^{1/q_2 - 1} \sigma _v \mu _v^{1/2} \le \lambda _v^{-\gamma _v}. \end{aligned}$$

We can achieve this as soon as

$$\begin{aligned} \frac{s_3\left( 1 - \frac{1}{q_2}\right) }{\frac{1}{q_3} - \frac{1}{q_2}} > \frac{1}{2}, \end{aligned}$$

(9.12)

see the proof of Lemma 7.2. Using

$$\begin{aligned} \frac{1}{1 - \frac{2}{q_2}} > \frac{1}{2\left( 1 - \frac{1}{q_2}\right) } \end{aligned}$$

we see that (9.12) holds according to (1.35).

It remains to estimate the additional term in (9.11). This works exactly as in the proof of Proposition 7.1. $\square$

Appendices

Littlewood–Paley Theory, Besov Spaces and Paradifferential Calculus

In this appendix, we state some basic definitions from Littlewood–Paley theory and paradifferential calculus, which will be used throughout the paper. More details can be found in Bahouri et al. (2011), Mourrat and Weber (2017), Leoni (2017) and Sawano (2018).

1.1 Littlewood–Paley Theory and Besov Spaces

We first introduce a dyadic partition of unity $\{ \rho _j \}_{j=-1}^\infty$ as follows

$$\begin{aligned} \rho _0 (\xi ) = \rho (\xi ), \quad \rho _j (\xi ) = \rho (2^{-j} \xi ) \text { for }j=1,2,\ldots , \quad \rho _{-1} (\xi ) = 1 - \sum _{j=0}^\infty \rho _j (\xi ). \end{aligned}$$

Then for $f \in {\mathcal {S}}' ({\mathbb {T}}^3)$ the Littlewood–Paley blocks are given by

$$\begin{aligned} \widehat{\Delta _j f} (\xi ) = \rho _j (\xi ) {\widehat{f}} (\xi ), \quad j=-1,0, \ldots . \end{aligned}$$

The Besov space $B^{s}_{p,q} ({\mathbb {T}}^3)$ is then defined in terms of the following norm (based on the Littlewood-Paley decomposition), which reads for $q < \infty$

$$\begin{aligned} \Vert f \Vert _{B^{s}_{p,q}} \,{:=}\, \Vert \Delta _{-1} f \Vert _{L^p} + \bigg ( \sum _{j=0}^\infty 2^{s j q} \Vert \Delta _j f \Vert _{L^p}^q \bigg )^{1/q}. \end{aligned}$$

If $q = \infty$, the norm is defined as follows

$$\begin{aligned} \Vert f \Vert _{B^s_{p,\infty }} \,{:=}\, \Vert \Delta _{-1} f \Vert _{L^p} + \sup _{j \ge 0} \big ( 2^{s j } \Vert \Delta _j f \Vert _{L^p} \big ). \end{aligned}$$

It is also possible to define Besov spaces using difference quotients. We first define the forward finite difference operator

$$\begin{aligned} \Delta ^1_h f (x) = f (x+h) - f(x). \end{aligned}$$

The higher-order finite differences are defined inductively. For $m \ge 2$, we define that

$$\begin{aligned} \Delta ^m_h f(x) = \Delta ^1_h ( \Delta _h^{m-1} f (x)). \end{aligned}$$

Let $s > 0$ and $1 \le p, q \le \infty$ and let $\lfloor s \rfloor$ denote the integer part of s, then the Besov norm may be defined as follows (if $q < \infty$)

$$\begin{aligned} \Vert f \Vert _{B^s_{p,q}} = \Vert f \Vert _{L^p} + \bigg ( \int _{{\mathbb {T}}^3} \Vert \Delta _h^{\lfloor s \rfloor + 1} f \Vert _{L^p}^q \frac{\textrm{d}h}{|h |^{3 + s q}} \bigg )^{1/q}. \end{aligned}$$

If $q = \infty$, the Besov norm is given by

$$\begin{aligned} \Vert f \Vert _{B^s_{p,\infty }} \,{:=}\, \Vert f \Vert _{L^p} + \sup _{h \in {\mathbb {R}}^3 \backslash \{ 0 \}} \frac{\Vert \Delta _h^{\lfloor s \rfloor + 1} f \Vert _{L^p}}{|h |^s}. \end{aligned}$$

These two different definitions of the Besov norm are equivalent.

Remark A.1

Note that the index h in the notation for the finite differences $\Delta _h^m$ represents the shift. This is in contrast with the main body of this paper where the index h always means “horizontal”.

Remark A.2

The reader should note that $B^s_{p,p} ({\mathbb {T}}^3) = W^{s,p} ({\mathbb {T}}^3)$ when $s \in {\mathbb {R}} \backslash {\mathbb {Z}}$ and $1 \le p \le \infty$. This is stated in Amann (2000, Equation 3.5), for example. The equivalence of function spaces in the case $0< s < 1$ can also be found in Tartar (2007, Definition 32.2) (where the interpolation space-based definition of Besov spaces is used) and in Simon (1990, Proposition 2) and Brasseur (2018, Page 1686) (where the Sobolev spaces are defined using the Sobolev–Slobodeckij seminorm). We note that the case $0< s < 1$ can easily be extended to all $s > 0$ with $s \notin {\mathbb {N}}$ by using Theorem 2.3 in Sawano (2018). Finally, we recall that $B^s_{2,2} ({\mathbb {T}}^3) = H^s ({\mathbb {T}}^3)$ for all $s \in {\mathbb {R}}$ (even when s is an integer), see (Bahouri et al. 2011, Page 99). The latter is used several times in this paper.

Next we recall some essential estimates regarding the Besov norm.

Lemma A.3

For any $1\le p,q,q_1,q_2\le \infty$, $\alpha \in {\mathbb {R}}$, $\delta >0$ the following estimates hold

$$\begin{aligned} \Vert f \Vert _{B^0_{p,\infty }}&\lesssim \Vert f \Vert _{L^p} \lesssim \Vert f \Vert _{B^0_{p,1}} , \end{aligned}$$

(A.1)

$$\begin{aligned} \Vert f \Vert _{B^\alpha _{p,q_1}}&\lesssim \Vert f \Vert _{B^{\alpha +\delta }_{p,q_2}}, \end{aligned}$$

(A.2)

$$\begin{aligned} \Vert f \Vert _{B^\alpha _{p,q_1}}&\lesssim \Vert f \Vert _{B^{\alpha }_{p,q_2}}, \quad \text { if }q_1\ge q_2 \end{aligned}$$

(A.3)

$$\begin{aligned} \Vert \partial _i f \Vert _{B^{\alpha -1}_{p,q}}&\lesssim \Vert f \Vert _{B^\alpha _{p,q}}. \end{aligned}$$

(A.4)

Proof

Estimates (A.1), (A.2), (A.3) and (A.4) can be found in Sawano (2018) Propositions 2.1, 2.2, 2.3 and Theorem 2.2, respectively. $\square$

Note that (A.2) implies that

$$\begin{aligned} \Vert f \Vert _{B^\alpha _{p,q}} \lesssim \Vert f \Vert _{B^{\beta }_{p,q}} \end{aligned}$$

for any $\alpha \le \beta$.

1.2 Paradifferential Calculus

We recall Bony’s product decomposition

$$\begin{aligned} f g = T_f g + T_g f + R(f,g). \end{aligned}$$

(A.5)

The terms $T_f g$ and $T_g f$ are called paraproducts and are given by

$$\begin{aligned} T_f g = \sum _{j=-1}^\infty \sum _{i=-1}^{j-2} \Delta _i f \Delta _j g, \quad T_g f = \sum _{j=-1}^\infty \sum _{i=-1}^{j-2} \Delta _i g \Delta _j f. \end{aligned}$$

(A.6)

The term R(f, g) is referred to as the resonance term and is defined as

$$\begin{aligned} R(f,g) = \sum _{|k - j |\le 1} \Delta _k f \Delta _j g. \end{aligned}$$

(A.7)

We will also use the notation $T(f,g)\,{:=}\, T_f (g)$ and $T(g,f)\,{:=}\, T_g(f)$.

One can estimate the three terms of the product decomposition separately. For the paraproducts, we have the following estimates.

Lemma A.4

(Lemma 2.1 in Prömel and Trabs 2016) Let $\alpha ,\beta \in {\mathbb {R}}$ and $1\le p,p_1,p_2,q,q_1,q_2\le \infty$ with

$$\begin{aligned} \frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}, \qquad \frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}. \end{aligned}$$

• For any $f \in L^{p_1} ({\mathbb {T}}^3)$ and $g \in B^{\beta }_{p_2,q} ({\mathbb {T}}^3)$, we have that

  $$\begin{aligned} \Vert T_f (g) \Vert _{B^\beta _{p,q}} \lesssim \Vert f \Vert _{L^{p_1}} \Vert g \Vert _{B^\beta _{p_2,q}}. \end{aligned}$$

• If $\alpha < 0$ then for any $f \in B^{\alpha }_{p_1, q_1} ({\mathbb {T}}^3)$ and $g \in B^\beta _{p_2,q_2} ({\mathbb {T}}^3)$, we have

  $$\begin{aligned} \Vert T_f (g) \Vert _{B^{\alpha + \beta }_{p,q}} \lesssim \Vert f \Vert _{B^\alpha _{p_1,q_1}} \Vert g \Vert _{B^\beta _{p_2,q_2}}. \end{aligned}$$

We recall the following estimate on the resonance term.

Lemma A.5

(Theorem 2.85 in Bahouri et al. 2011) Let $\alpha ,\beta \in {\mathbb {R}}$ and $1\le p,p_1,p_2,q,q_1,q_2\le \infty$ with

$$\begin{aligned} \frac{1}{p} = \frac{1}{p_1} + \frac{1}{p_2}, \qquad \frac{1}{q} = \frac{1}{q_1} + \frac{1}{q_2}. \end{aligned}$$

• If $\alpha +\beta >0$, then we have for any $f \in B^{\alpha }_{p_1,q_1} ({\mathbb {T}}^3)$ and $g \in B^{\beta }_{p_2,q_2} ({\mathbb {T}}^3)$

  $$\begin{aligned} \Vert R(f,g) \Vert _{B^{\alpha +\beta }_{p,q}} \lesssim \Vert f \Vert _{B^{\alpha }_{p_1,q_1}} \Vert g \Vert _{B^{\beta }_{p_2,q_2}}. \end{aligned}$$

• If $\alpha +\beta =0$ and $q=1$, then we have for any $f \in B^{\alpha }_{p_1,q_1} ({\mathbb {T}}^3)$ and $g \in B^{\beta }_{p_2,q_2} ({\mathbb {T}}^3)$

  $$\begin{aligned} \Vert R(f,g) \Vert _{B^{0}_{p,\infty }} \lesssim \Vert f \Vert _{B^{\alpha }_{p_1,q_1}} \Vert g \Vert _{B^{\beta }_{p_2,q_2}}. \end{aligned}$$

Combining these estimates leads to the following result.

Lemma A.6

Let $\alpha< 0 < \beta$, $\beta + \alpha > 0$, and $1 \le p_1, p_2, p, q_1,q_2 \le \infty$ with

$$\begin{aligned} \frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{p}. \end{aligned}$$

• We have for any $f \in B^\alpha _{p_1,q_1} ({\mathbb {T}}^3)$ and $g \in B^\beta _{p_2,q_2} ({\mathbb {T}}^3)$

  $$\begin{aligned} \Vert fg \Vert _{B^\alpha _{p,q_1}} \lesssim \Vert f \Vert _{B^\alpha _{p_1,q_1}} \Vert g \Vert _{B^\beta _{p_2,q_2}}. \end{aligned}$$

  (A.8)

• Let $\{f_n\},\{g_n\}$ be sequences such that $f_n \rightarrow f$ in $B^\alpha _{p_1,q_1} ({\mathbb {T}}^3)$ and $g_n \rightarrow g$ in $B^\beta _{p_2,q_2} ({\mathbb {T}}^3)$. Then $f_n g_n\rightarrow fg$ in $B^\alpha _{p,q_1}$.

Proof

Estimate (A.8) is a simple consequence of Lemmas A.4 and A.5, see also (Mourrat and Weber 2017, Prop. A.7) for a similar proof.

The convergence claimed in the second bullet point can be easily deduced from (A.8). Indeed, we have

$$\begin{aligned} \Vert f_n g_n - fg \Vert _{B^\alpha _{p,q_1}}&\lesssim \Vert f_n (g_n - g) \Vert _{B^\alpha _{p,q_1}} + \Vert (f_n - f) g \Vert _{B^\alpha _{p,q_1}} \\&\lesssim \Vert f_n \Vert _{B^\alpha _{p_1,q_1}} \Vert g_n-g \Vert _{B^\beta _{p_2,q_2}} + \Vert f_n-f \Vert _{B^\alpha _{p_1,q_1}} \Vert g \Vert _{B^\beta _{p_2,q_2}} \rightarrow 0. \end{aligned}$$

$\square$

1.3 Alternative Proof of Lemma 5.3

In this section, we present an alternative proof of Lemma 5.3 using a paraproduct decomposition and Lemmas A.4 and A.5.

Alternative Proof of Lemma 5.3

We start by decomposing the term under consideration in terms of Bony’s decomposition, i.e.

$$\begin{aligned}&\bigg \Vert \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\quad \le \bigg \Vert T \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\qquad + \bigg \Vert T \bigg ( R_{v,k}, \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\qquad + \bigg \Vert R \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })}. \end{aligned}$$

Using Lemma A.4, we find

$$\begin{aligned}&\bigg \Vert T \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \nonumber \\&\lesssim \bigg \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{B^{-1}_{1,\infty }} \Vert R_{v,k} \Vert _{L^p(B^0_{\infty ,\infty })} . \end{aligned}$$

(A.9)

Another application of Lemma A.4 yields

$$\begin{aligned}&\bigg \Vert T \bigg ( R_{v,k}, \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\lesssim \bigg \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{B^{-1}_{1,\infty }} \Vert R_{v,k} \Vert _{L^p (L^\infty )} . \end{aligned}$$

Finally, Lemmas A.3 and A.5 lead to

$$\begin{aligned}&\bigg \Vert R \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{-1}_{1,\infty })} \\&\quad \lesssim \bigg \Vert R \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{s}_{1,\infty })} \\&\quad \lesssim \bigg \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{B^{-1}_{1,\infty }} \Vert R_{v,k} \Vert _{L^p (B^{1+s}_{\infty ,\infty })} \end{aligned}$$

where $0<s\ll 1$ can be chosen arbitrary.

To conclude, we proceed similarly to the original proof. Due to Lemmas 3.4 and A.3, and estimate (5.6)

$$\begin{aligned}&\bigg \Vert \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{B^{-1}_{1,\infty }} \\&\quad = \bigg \Vert \nabla _h \cdot \bigg [ {\mathcal {R}}_h\bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg )\bigg ]\bigg \Vert _{B^{-1}_{1,\infty }} \\&\quad \lesssim \bigg \Vert {\mathcal {R}}_h\bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg )\bigg \Vert _{B^{0}_{1,\infty }} \\&\quad \lesssim \bigg \Vert {\mathcal {R}}_h\bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg )\bigg \Vert _{L^1} \\&\quad \lesssim \sigma _v^{-1} \bigg \Vert \phi _k W_k - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{L^1} \lesssim \sigma _v^{-1} . \end{aligned}$$

This finishes the proof of Lemma 5.3. $\square$

Other Estimates

1.1 Improved Hölder Inequality

Let us recall the following estimate from Modena and Székelyhidi (2018).

Lemma B.1

(Improved Hölder inequality) For any $\sigma \in {\mathbb {N}}$, $1 \le p \le \infty$ and all functions $f\in C^1({\mathbb {T}}^d)$, $g\in L^p({\mathbb {T}}^d)$ it holds that

$$\begin{aligned} \Big |\Vert f (\cdot ) g (\sigma \cdot ) \Vert _{L^p} - \Vert f \Vert _{L^p} \Vert g \Vert _{L^p} \Big |\lesssim \sigma ^{-1/p} \Vert f \Vert _{C^1} \Vert g \Vert _{L^p}. \end{aligned}$$

(B.1)

The proof can be found in Modena and Székelyhidi (Modena and Székelyhidi 2018, Lemma 2.1).

1.2 Oscillatory Paraproduct Estimate

We are going to prove a version of Lemma B.1 in the case of Besov spaces.

Lemma B.2

For any $\sigma \in {\mathbb {N}}$, $1\le p,q\le \infty$, $s\in {\mathbb {R}}$, $0<\epsilon \le 1$ and all functions $f \in L^p ({\mathbb {T}}^3)$, $g \in B^{s+\epsilon }_{p,q} ({\mathbb {T}}^3) \cap B^{s+1+\epsilon }_{\infty ,q} ({\mathbb {T}}^3)$ it holds that

$$\begin{aligned} \Vert T (f_\sigma , g) \Vert _{B^{s}_{p,q}} \lesssim \Vert f \Vert _{L^p} \Vert g \Vert _{B^{s+\epsilon }_{p,q}} + \sigma ^{-1/p} \Vert f \Vert _{L^p} \Vert g \Vert _{B^{s+1+\epsilon }_{\infty ,q}}, \end{aligned}$$

(B.2)

where we define $f_\sigma (x) {:=} f(\sigma x)$.

Proof

We first recall the low-frequency cut-off operator $S_j$ from Bahouri et al. (2011):

$$\begin{aligned} S_j f\, {:=}\,\sum _{i=-1}^{j-1} \Delta _i f. \end{aligned}$$

Hence, we may write $T (f_\sigma , g) = \sum _{j=-1}^\infty S_{j-1} f_\sigma \Delta _j g = \sum _{j=1}^\infty S_{j-1} f_\sigma \Delta _j g$, where the latter equation follows from the fact that $S_{-2}f_\sigma =S_{-1}f_\sigma =0$. In order to estimate the Besov norm of $T (f_\sigma , g)$, we will use (Bahouri et al. 2011, Lemma 2.69). To be able to use this lemma, we need to show that $\textrm{supp}\,\big ({\mathcal {F}} (S_{j-1} f_\sigma \Delta _j g) \big )$ lies in $2^j {\mathcal {C}}$ for any $j\in {\mathbb {N}}_0$, where ${\mathcal {C}}$ is a fixed annulus.

In order to show this, let $j\in {\mathbb {N}}$ and $i\in \{-1,\ldots ,j-2\}$. By construction of the dyadic partition of unity, there exist radii $0<r<r_0<R$ such that the support of $\rho _{-1}$ is contained in the ball with radius $r_0$, the support of $\rho _0$ is contained in the annulus with inner radius r and outer radius R, and $r_0<2r< R< 4r$. Next we observe

$$\begin{aligned} \textrm{supp}\,\big ( {\mathcal {F}} (\Delta _i f_\sigma \Delta _j g) \big ) = \textrm{supp}\,\big ( (\rho _i {\widehat{f}}_\sigma ) * (\rho _j {\widehat{g}}) \big ). \end{aligned}$$

For $i=-1$, this yields that $\textrm{supp}\,\big ( {\mathcal {F}} (\Delta _i f_\sigma \Delta _j g) \big )$ is contained in the annulus with inner radius $2^j(r-2^{-j}r_0)$ and outer radius $2^j(R+2^{-j}r_0)$, which is in turn a subset of the annulus with inner and outer radii $2^j(r-\frac{1}{2}r_0)$ and $2^j(R+\frac{1}{2}r_0)$. Note that the inner radius is positive due to $2r>r_0$. Similarly for $i\ge 0$, we obtain that $\textrm{supp}\,\big ( {\mathcal {F}} (\Delta _i f_\sigma \Delta _j g) \big )$ is contained in the annulus with inner radius $2^j(r-2^{i-j}R)$ and outer radius $2^j(R+2^{i-j}R)$, which is in turn a subset of the annulus with inner and outer radii $2^j(r-\frac{1}{4}R)$ and $2^j(R+\frac{1}{4}R)$. Again note that $4r>R$ implies that the inner radius is positive. Hence, there exists an annulus ${\mathcal {C}}$ such that

$$\begin{aligned} \textrm{supp}\,\big ({\mathcal {F}} (S_{j-1} f_\sigma \Delta _j g) \big ) \subset 2^j {\mathcal {C}}. \end{aligned}$$

Hence, we may apply Lemma 2.69 from Bahouri et al. (2011) to conclude that

$$\begin{aligned} \Vert T (f_\sigma , g) \Vert _{B^{s}_{p,q}} \lesssim \Big \Vert \big (2^{js} \Vert S_{j-1} f_\sigma \Delta _j g \Vert _{L^p}\big )_{j\in {\mathbb {N}}} \Big \Vert _{l^q({\mathbb {N}})}. \end{aligned}$$

(B.3)

In order to estimate $\Vert S_{j-1} f_\sigma \Delta _j g \Vert _{L^p}$ we define $\phi _i\,{:=}\, {\mathcal {F}}^{-1} \rho _i$. Hence we have $\Delta _i f = \phi _i * f$. A direct computation shows $\Vert \phi _i\Vert _{L^1} = \Vert \phi _0 \Vert _{L^1}$ for any $i\in {\mathbb {N}}$. Hence, we obtain for $1\le p <\infty$ and for any i, j by Minkowski’s integral inequality and Lemma B.1

$$\begin{aligned} \Vert \Delta _{i} f_\sigma \Delta _j g \Vert _{L^p}&= \bigg ( \int _{{\mathbb {R}}^n} \bigg |\Delta _j g (x) \bigg ( \int _{{\mathbb {R}}^n } \phi _i (y) f_\sigma (x - y) \,\textrm{d} y\bigg ) \bigg |^p \,\textrm{d} x\bigg )^{1/p} \\&\le \int _{{\mathbb {R}}^n} \bigg ( \int _{{\mathbb {R}}^n} |f_\sigma (x-y) \phi _i (y) \Delta _j g (x) |^p \,\textrm{d} x\bigg )^{1/p} \,\textrm{d} y\\&= \int _{{\mathbb {R}}^n} |\phi _i (y) |\Vert f(\cdot \sigma - \sigma y) \Delta _j g ( \cdot ) \Vert _{L^p} \,\textrm{d} y\\&\lesssim \int _{{\mathbb {R}}^n} |\phi _i (y) |\bigg ( \Vert f (\cdot - \sigma y) \Vert _{L^p} \Vert \Delta _j g \Vert _{L^p} + \sigma ^{-1/p} \Vert \Delta _j g \Vert _{C^1} \Vert f (\cdot - \sigma y ) \Vert _{L^p} \bigg ) \,\textrm{d} y\\&\le \Vert \phi _0 \Vert _{L^1} \Big (\Vert f \Vert _{L^p} \Vert \Delta _j g \Vert _{L^p} + \sigma ^{-1/p} \Vert \Delta _j g \Vert _{C^1} \Vert f \Vert _{L^p}\Big ) \\&\lesssim \Vert f \Vert _{L^p} \Vert \Delta _j g \Vert _{L^p} + \sigma ^{-1/p} \Vert \Delta _j g \Vert _{C^1} \Vert f \Vert _{L^p}. \end{aligned}$$

For the case $p=\infty$, one obtains the same result; the details are left to the reader. Hence,

$$\begin{aligned} \Vert S_{j-1} f_\sigma \Delta _j g \Vert _{L^p}&\le \sum _{i=-1}^{j-2} \Vert \Delta _{i} f_\sigma \Delta _j g \Vert _{L^p} \nonumber \\&\lesssim j \Big (\Vert f \Vert _{L^p} \Vert \Delta _j g \Vert _{L^p} + \sigma ^{-1/p} \Vert \Delta _j g \Vert _{C^1} \Vert f \Vert _{L^p} \Big ) \end{aligned}$$

(B.4)

for any $j\in {\mathbb {N}}$. Combining (B.3) and (B.4), we get for any $1\le q<\infty$

$$\begin{aligned}&\Vert T (f_\sigma , g) \Vert _{B^{s}_{p,q}} \\&\quad \lesssim \bigg (\sum _{j=1}^\infty 2^{jsq} \Vert S_{j-1}f_\sigma \Delta _j g\Vert _{L^p}^q \bigg )^{1/q} \\&\quad \lesssim \bigg (\sum _{j=1}^\infty 2^{jsq} j^q \Big (\Vert f \Vert _{L^p} \Vert \Delta _j g \Vert _{L^p} + \sigma ^{-1/p} \Vert \Delta _j g \Vert _{C^1} \Vert f \Vert _{L^p} \Big )^q \bigg )^{1/q} \\&\quad \lesssim \Vert f \Vert _{L^p} \bigg (\sum _{j=1}^\infty 2^{j(s+\epsilon )q} j^q 2^{-j\epsilon q} \Vert \Delta _j g \Vert _{L^p}^q \bigg )^{1/q} \\&\qquad + \sigma ^{-1/p} \Vert f \Vert _{L^p} \bigg (\sum _{j=1}^\infty 2^{j(s+\epsilon )q} j^q 2^{-j\epsilon q} \Vert \Delta _j g \Vert _{C^1}^q \bigg )^{1/q} \\&\quad \lesssim \Vert f \Vert _{L^p} \Vert g \Vert _{B_{p,q}^{s+\epsilon }} + \sigma ^{-1/p} \Vert f \Vert _{L^p} \Vert g \Vert _{B_{\infty ,q}^{s+1+\epsilon }}, \end{aligned}$$

where we have used that $\Vert \Delta _j g \Vert _{C^1} \le \Vert \nabla \Delta _j g \Vert _{L^\infty } + \Vert \Delta _j g \Vert _{L^\infty }$, $\nabla \Delta _j g= \Delta _j \nabla g$ and Lemma A.3. For the case $q=\infty$, we proceed analogously. $\square$

Lemma B.2 can be used to prove the following slightly weaker version of Lemma 5.3.

Lemma B.3

For all $1\le p\le \infty$, $\delta >0$ and $k\in \{1,2\}$, we have

$$\begin{aligned} \bigg \Vert \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ) R_{v,k} \bigg \Vert _{L^p (B^{-1-\delta }_{1,\infty })} \lesssim \sigma _v^{-1} + \Vert R_{v} \Vert _{L^p (B^{-1}_{1,\infty })}.\nonumber \\ \end{aligned}$$

(B.5)

Proof

Compared to the proof presented in Sect. A.3, the only difference is how we handle the paraproduct in (A.9). We use Lemma B.2 and estimate (5.6) to find

$$\begin{aligned}&\bigg \Vert T \bigg ( \bigg ( \phi _k (\sigma _v \cdot ) W_k (\sigma _v \cdot ) - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg ), R_{v,k} \bigg ) \bigg \Vert _{L^p (B^{-1-\delta }_{1,\infty })} \\&\quad \lesssim \bigg \Vert \phi _k W_k - \int _{{\mathbb {T}}^2} \phi _k ( x) W_k (x) \,\textrm{d} x\bigg \Vert _{L^1} \Vert R_{v} \Vert _{L^p (B^{-1}_{1,\infty })} \\&\qquad + \sigma _v^{-1} \bigg \Vert \phi _k W_k - \int _{{\mathbb {T}}^2} \phi _k (x) W_k (x) \,\textrm{d} x\bigg \Vert _{L^1} \Vert R_{v} \Vert _{L^p (B^{0}_{\infty ,\infty })}\\&\quad \lesssim \Vert R_{v} \Vert _{L^p (B^{-1}_{1,\infty })} + C_{R_v} \sigma _v^{-1} . \end{aligned}$$

$\square$

With Lemma B.3, one can show

$$\begin{aligned} \Vert w_{p,v} u_{p,v} \Vert _{L^1(B^{-1-\delta }_{1,\infty })} \lesssim \Vert R_v \Vert _{L^1(B^{-1}_{1,\infty })} \end{aligned}$$

which is slightly weaker than (5.13). This finally allows to prove a version of Theorem 1.4 where the regularity parameter is $s=1+$.