Abstract
This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order \( \mathcal {O}(\varepsilon ) \), as \( \varepsilon \rightarrow 0^+ \), where \( \varepsilon \) represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.
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1 Introduction
1.1 The Compressible Primitive Equations
The compressible primitive equations [see (CPE)] are used by meteorologists to perform theoretical investigations and practical weather predictions (see, e.g., [45]). In comparison with the general hydrodynamic and thermodynamic equations, the vertical component of the momentum equations is missing in the compressible primitive equations. Instead, it is replaced by the hydrostatic balance equation [see (CPE)\(_{3}\)], which is also known as the quasi-static equilibrium equation. From the meteorologists’ point of view, such an approximation is reliable and useful for two reasons: the balance of gravity and pressure dominates the dynamics in the vertical direction, and the vertical velocity is usually hard to observe in reality (see, e.g., [59, Chapter 4]). On the other hand, by formally taking the zero limit of the aspect ratio between the vertical scale and the planetary horizontal scale, the authors in [22] derive the compressible primitive equations from the compressible hydrodynamic equations. Such a deviation is very common in planetary scale geophysical models, which represents the fact that the vertical scale of the atmosphere (or ocean) is significantly smaller than the planetary horizontal scale. We refer, for more comprehensive meteorological studies, to [59, 66].
As far as we know, there are very few mathematical studies concerning the compressible primitive equations (referred to as CPE hereafter). Lions, Temam, and Wang first introduced CPE into the mathematical community in [45]. They formulated the commonly known primitive equations (referred to as PE hereafter) with the incompressibility condition as the representation of the compressible primitive equations in the pressure coordinates (p-coordinates) instead of the physical ones with the vertical spatial coordinate. On the other hand, as mentioned before, the authors in [22] introduce these equations with a formal deviation, and a rigorous justification is still an open question for now. In [22], the stability of weak solutions is also investigated (see also [63]). The stability is meant in the sense that a sequence of weak solutions, satisfying some entropy conditions, contains a subsequence converging to another weak solution, i.e., a very weak sense of stability. The existence of such weak solutions is recently constructed in [49, 65] (see also [21, 30] for the existence of global weak solutions to some variant of compressible primitive equations in two spatial dimension). In [50], we also construct local strong solutions to CPE in two cases: with gravity but no vacuum; with vacuum but no gravity.
In analogy to the low Mach number limit in the study of compressible hydrodynamic equations, this and our subsequent works are aiming to study the low Mach number limit of the compressible primitive equations without gravity and Coriolis force. It is worth mentioning that while taking into account the Coriolis force would not change much our proof, considering gravity in our system causes challenging difficulties. Let \( \varepsilon \) denote the Mach number, and let \( \rho ^\varepsilon \in {\mathbb {R}}, v^\varepsilon \in {\mathbb {R}}^2, w^\varepsilon \in {\mathbb {R}} \) be the density, the horizontal and the vertical velocities, respectively. System (CPE), is obtained by rescaling the original CPE, which is similar to the rescaling of the compressible Navier–Stokes equations (see, e.g., [27]): in \( \Omega _h \times 2 {\mathbb {T}} \):
where \( P(\rho ^\varepsilon ) = (\rho ^\varepsilon )^\gamma \) and \( {\mathbb {S}} (v^\varepsilon ) = \mu (\nabla _hv^\varepsilon + \nabla _h^\top v^\varepsilon ) + (\lambda - \mu ) \mathrm {div}_h\,v^\varepsilon {\mathbb {I}}_2 \) represent the pressure potential and the viscous stress tensor, respectively, with the shear and bulk viscosity coefficients \( \mu \) and \( \lambda -\mu + \frac{2}{3}\mu = \lambda - \frac{1}{3} \mu \). The physical requirements of \( \mu , \lambda , \gamma \) are \( \lambda - \frac{1}{3} \mu > 0\), \( \mu > 0 \) and \( \gamma > 1 \). Moreover, we focus our study on the case when \( \Omega _h := {\mathbb {T}}^2 \subset {\mathbb {R}}^2 \), and we study (CPE) subject to the following symmetry:
Consequently, solutions to (CPE) satisfy the following physical stress-free and impermeability physical boundary conditions:
Hereafter, we have and will use \( \nabla _h, \mathrm {div}_h\,\) and \( \Delta _h \) to represent the horizontal gradient, the horizontal divergence, and the horizontal Laplace operator, respectively; that is,
We recall the incompressible primitive equations: in \(\Omega _h \times 2{\mathbb {T}} \),
subject to the following symmetry:
We will show that the asymptotic system of (CPE), as \( \varepsilon \rightarrow 0^+ \), is the incompressible primitive equations (PE), with \( c^2_s = \gamma \rho _0^{\gamma -1} \) and \( \rho _0 = \text {constant} \). Here \( \rho _1 \) is the Lagrangian multiplier for the constraint (PE)\(_{1}\) satisfying
In addition, due to the conservation of linear momentum of (PE), we can impose the following condition for \( v_p \):
for any time \( t \geqq 0 \) as long as the solution exists.
Historically, the limit system (PE), besides acting as the representation of the CPE in the p-coordinates, is introduced as the limit system of Boussinesq equations (referred to as BE hereafter) when the aspect ratio between the vertical scale and the horizontal scale is very small, while the Boussinesq equations are the limit equations of the full compressible hydrodynamic equations with small Mach number and low stratification (see, e.g., [44]). That is to say, starting from the compressible hydrodynamic equations, by taking the low Mach number limit and then the small aspect ratio limit (referred to as LMSAR), one will arrive, formally, at the BE and then at the PE. On the other hand, by taking the small aspect ratio limit and then the low Mach number limit (referred to as SARLM), at least formally, the limit system of the compressible hydrodynamic equations is also the PE with the CPE as a middle state. Depending on the order of asymptotic limits, this gives us two directions from the hydrodynamic equations to the PE, which we will refer to as the PE diagram (see Fig. 1). The LMSAR part of the PE diagram has been shown to hold on solid ground in various settings (see, e.g., [3, 28, 41, 55, 57]). However, the validity of the SARLM part is relatively open. In order to fully justify the PE diagram, we investigate the low Mach number limit of the CPE in this work, which, as mentioned above, leads to the PE as the limit system. We remark that, the stratification effect of the gravity has been neglected in this work.
The PE diagram
Each of the equations in the PE diagram has its own significance and has been studied separately in a large amount of the literature. It would certainly be too ambitious to review all of those works. We refer readers to the study of compressible hydrodynamic equations in, e.g., the books [23, 47, 48, 54]. As the limit system of the PE diagram, the primitive equations (PE) have been investigated intensively since they were introduced in [43,44,45]. For instance, the global weak solutions are established in [44]. Local well-posedness with general data and global well-posedness with small data of strong solutions to the PE in three spatial dimensions were studied in [33] by Guillén-González, Masmoudi and Rodríguez-Bellido. Petcu and Wirosoetisno in [56] investigated the Sobolev and Gevrey regularity of the solutions to the PE. In [36], in a domain with small depth, the authors address the global existence of strong solutions to PE. The well-posedness of unique global strong solutions was obtained by Cao and Titi in [12] (see, also, [7,8,9,10,11, 13, 15, 31, 34] and the references therein for related studies). Considering the inviscid primitive equations, or hydrostatic incompressible Euler equations, the existence of solutions in the analytic function space and in the \(H^s \) space are established in [5, 40, 53]. Renardy in [58] showed that the linearization of the equations at certain shear flows is ill-posed in the sense of Hadamard. Recently, the authors in [6, 67] constructed a finite-time blowup for the inviscid PE in the absence of rotation (i.e., without the Coriolis force).
In this work, we show that the PE can be viewed as the limit system of the CPE with the zero Mach number limit. The zero Mach number limit of the compressible hydrodynamic equations is a vast subject which has been studied for decades. Fruitful results have been obtained since the early works of Klainerman and Majda in [38, 39], where the authors investigate the vanishing Mach number limit of compressible Euler equations with well-prepared initial data (see also [61, 62]). Later Ukai [64], the theory of low Mach number limit of compressible Euler equations extended to ill-prepared initial data (or called general data in some literatures). We remark here that the difference between the well-prepared and ill-prepared initial data is that the well-prepared initial data have excluded the acoustic waves, while the ill-prepared initial data allow the interaction of the solutions with the high-frequency acoustic waves. In \( {\mathbb {R}}^n \), \( n = 2,3 \), such high-frequency acoustic waves disperse as shown in [64], which implies strong convergences as the Mach number goes to zero. This can be also proved by applying Strichartz’s estimate (see, e.g., [4, 14, 32, 37, 42]) for linear wave equations to the acoustic equations (see, e.g., [20]). In \( {\mathbb {T}}^n \), \( n = 2, 3 \), the high-frequency acoustic waves interact with each other and lead to fast oscillations and weak convergences when taking the low Mach number limit. Such a fast oscillation phenomenon was first systematically studied in [29, 60] for hyperbolic and parabolic systems, and by Lions and Masmoudi for compressible Navier–Stokes equations in [46]. We refer, for the comparison of the whole space case, i.e., in \( {\mathbb {R}}^n \) and the periodic domain case, i.e., in \( {\mathbb {T}}^n \), to [52]. See also [16,17,18] for the studies in the Besov spaces. We acknowledge that the discussion here barely unveils the theory of the low Mach number limit, and we refer the reader to [1, 2, 19, 24,25,26,27,28, 55] and the references therein for more comprehensive studies and recent progress.
In this work, we will focus on investigating the low Mach number limit of (CPE) with well-prepared initial data. The convergence of the solutions of the CPE to the solution of the PE is in the strong sense. Furthermore, we are also able to obtain explicit convergence rate (see Theorem 1.2). In particular, we obtain a class of global large solutions to (CPE) with \( \varepsilon \) small enough.
We remark that in an upcoming paper [51], we will consider the low Mach number limit of (CPE) with ill-prepared initial data, i.e., initial data with large, high-frequency acoustic waves.
1.2 The Low Mach Number Limit Problem and Main Theorem
In order to describe the aforementioned asymptotic limit, we study (CPE) with \( (\rho ^\varepsilon , v^\varepsilon , w^\varepsilon ) \) close to an asymptotic state \( (\rho _0, v_p, w_p) \), where \( (\rho _0, v_p, w_p) \) satisfies (PE). For any \( \varepsilon > 0 \), the following ansatz is imposed:
Recall that \( \rho _1 \) has zero average in the domain [see (1.1)]. This is motivated by [35]. The term \( \varepsilon ^2 \rho _1 \) is to capture the cancelation in system (1.4). In addition, we shall employ the notation
For the sake of convenience, from time to time hereafter, we may drop the superscript \( \varepsilon \) from the functions. Consequently, from (CPE) and (PE), the new unknown \((\xi , \psi ^h, \psi ^z) \) is governed by the following system:
Observe that, owing to the symmetry in (SYM-CPE) and (SYM-PE), the following conditions hold automatically:
for smooth enough functions. Recalling that \( c_s^2 = \gamma \rho _0^{\gamma - 1} \), we note that
where
with \( C = C(\bigl \Vert \rho ^{\gamma -2} \bigr \Vert _{L^{\infty }},\bigl \Vert \rho _0^{\gamma -2} \bigr \Vert _{L^{\infty }}) \). Therefore, by denoting
we can write system (1.4) as
In order to recover the vertical velocity perturbation \( \psi ^z \), we introduce the following notations, for any function f in \( \Omega _h \times 2{\mathbb {T}} \), which is also even in the z-variable:
The periodicity and symmetry of f imply that \( \overline{f}(x,y,t) = \int _{k}^{k+1} f(x,y,z',t)\,\mathrm{d}z' \) for any \( k \in {\mathbb {Z}} \). Notice that from (CPE)\(_{3}\), \( \rho \) is independent of the vertical variable z. Then by averaging (CPE)\(_{1}\) over the vertical direction, thanks to (1.5), one will get
In particular, from the above, the vertical velocity w is determined through \( \rho , v \) by the formula, thanks to (1.5):
where we have substituted the following identity thanks to (PE)\(_{1}\) and (1.5),
Such facts imply that in (CPE), (PE) and (1.8), the vertical velocities and the vertical perturbation, i.e., \( w^\varepsilon , w_p, \psi ^{\varepsilon ,z} \), are fully determined by \( v^\varepsilon , v_p, \rho ^\varepsilon \). Therefore, there is no need to impose initial data for \( w^\varepsilon , w_p, \psi ^{\varepsilon ,z} \).
System (1.8) (or equivalently (1.4)) is complemented with initial data,
with the compatibility conditions
where \( \rho _{in}^\varepsilon = \rho _0 + \varepsilon ^2 \rho _{1,in} + \xi _{in}, ~ v_{in}^{\varepsilon } = v_{p,in} + \psi ^h_{in}, ~ w^\varepsilon _{in} = w_{p,in} + \psi ^z_{in} \), and \( \psi ^z_{in} \) is given by
Here \( v_{p,in}, \rho _{1,in}, \rho _{1,in,1}, w_{p,in}, \mathcal G_{i,in}, {\mathcal {F}}_{i,in} \) are initial values of \( v_{p}, \rho _1, \partial _t\rho _{1}, w_{p},{\mathcal {G}}_{i}, {\mathcal {F}}_{i} \), \( i \in \lbrace 1,2 \rbrace \), respectively, while \( w_{p,in} \) is given by \( w_{p,in} = - \int _0^z \mathrm {div}_h\,v_{p,in} \,\mathrm{d}z' \).
It is worth stressing that we will choose the initial time derivatives of the perturbations, i.e., \( (\varepsilon ^{-1}\xi _{in,1}, \psi ^h_{in,1}) \) in (1.13), to be bounded, uniformly in \( \varepsilon \) [see (1.14) and Theorem 1.2]. The reason for such choices of initial data is to exclude the high-frequency acoustic waves which corresponds to the fact that our initial data are well-prepared.
We denote the initial energy functional by
Remark 1
\( (\rho _{in}^\varepsilon , v_{in}^\varepsilon ) \) is the corresponding initial datum of \( (\rho ^\varepsilon , v^\varepsilon ) \) for system (CPE). \( v_{p,in} \) is the initial datum of \( v_p \) for system (PE). Accordingly, \( \rho _{1,in} = \rho _{1,in}(x,y), \rho _{1,in,1}= \rho _{1,in,1}(x,y) \) are determined by the following elliptic problems:
where \( v_{p,in,1} \) is the initial value of \( \partial _tv_{p} \) determined by
As we stated before, we focus in this work on the asymptotic limit as \( \varepsilon \rightarrow 0^+ \). We have the following global regularity of the limit system (PE):
Theorem 1.1
(Global regularity of the PE) For \( \lambda< 4\mu < 12 \lambda \), suppose that (PE) is complemented with initial data \( v_{p,in} \in H^1(\Omega _h \times 2{\mathbb {T}}) \), which is even in the z-variable, and satisfies the compatibility conditions
Then there exists a solution \( (v_p, \rho _1) \), with \( \int _0^1 \mathrm {div}_h\,v_{p} \,\mathrm{d}z = 0 \) and \( w_p \) given by (1.11), to the primitive equations (PE). Moreover, there is a constant \( C_{p,in} \) depending only \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{1}} \) such that
Moreover,
for some positive constants c, C. In addition, if \( v_{p,in} \in H^s(\Omega _h \times 2{\mathbb {T}}) \) for any integer \( s \geqq 2 \), there is a positive constant \( C_{p,in,s} \), depending only on \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{s}} \) such that
Proof
The local well-posedness of solutions to (PE) in the function space \( H^s \) has been established in [56]. What is left is the global regularity estimate, which is a direct consequence of Proposition 3. \(\quad \square \)
Remark 2
We only have to be careful about the different estimates caused by the viscosity tensor. In particular, it is the \( L^q \) estimate of \( v_p \), below in Section 5, that requires the constraints on the viscosity coefficients. For solutions with \( H^2 \) initial data, the result can be found in [41]. The new thing we treat here is the case when \( v_{p,in} \in H^s \), for \( s > 2 \).
Remark 3
The regularity of \( \rho _1 \) can be obtained by solving elliptic problem (4.1).
Now we can state our main theorem in this work.
Theorem 1.2
(Low Mach number limit of the CPE) For \( \lambda< 4 \mu < 12 \lambda \), suppose \( v_{p,in} \in H^s(\Omega _h \times 2{\mathbb {T}}) \), with integer \( s \geqq 3 \), and it satisfies the compatibility conditions (1.15). Also, we complement (1.8) with initial data \( (\xi ^{\varepsilon }, \psi ^{\varepsilon ,h})\bigr |_{t=0} = (\xi _{in}, \psi ^h_{in}) \in H^2(\Omega _h\times 2 {\mathbb {T}}) \times H^2(\Omega _h \times 2\mathbb T) \) as in (1.12), which satisfies the compatibility conditions (1.13). Recall that we also require \( v_{p,in}, \psi ^h_{in} \) to be even in the z-variable. Then there exists a positive constant \( \varepsilon _0 \in (0,1) \) small enough, such that if \( \varepsilon \in (0,\varepsilon _0) \) and \( {\mathcal {E}}_{in} \leqq \varepsilon ^2 \), there exists a global unique strong solution \( (\xi ^\varepsilon , \psi ^{\varepsilon ,h}) \) to system (1.8). In particular, the following regularity is satisfied:
In addition, we have the following estimate:
for some positive constant C depending only on \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{3}} \), which is independent of \( \varepsilon \). In particular, \( (\rho ^\varepsilon , v^\varepsilon , w^\varepsilon ) \) as in (1.3) is a globally defined strong solution to (CPE) and the following asymptotic estimate holds:
for some positive constant C depending only on \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{3}} \), which is independent of \( \varepsilon \), where \( w^\varepsilon , w_p \) are given as in (1.9), (1.11), respectively.
Remark 4
According to (1.18), the time derivatives, in comparison to the spatial derivatives, have larger perturbations. However, thanks to the well-prepared data setting, they are bounded.
This work will be organized as follows: in Section 2, we summarize the notations which will be commonly used in later paragraphs. Section 3 focuses on the \( \varepsilon \)-independent a priori estimates, which are the foundation of the low Mach number limit. In Section 4, we focus on the proof of Theorem 1.2. This will be shown through a continuity argument. Finally in Section 5, we summarize the proof of Theorem 1.1.
2 Preliminaries
We use the notations
to represent the integrals in \( \Omega \) and \( \Omega _h \) respectively. Hereafter, \( \partial _h \in \lbrace \partial _x, \partial _y \rbrace \) represents the horizontal derivatives, and \( \partial _z\) represents the vertical derivative.
We will use \( \bigl | \cdot \bigr |_{}, \bigl \Vert \cdot \bigr \Vert _{} \) to denote norms in \( \Omega _h \subset {\mathbb {R}}^2 \) and \( \Omega _h \times 2{\mathbb {T}} \subset {\mathbb {R}}^3 \), respectively. After applying Ladyzhenskaya’s and Agmon’s inequalities in \( \Omega _h \) and \( \Omega \), directly we have
for the function f with bounded right-hand sides. Also, applying Minkowski’s and Hölder’s inequalities yields
We use \( \delta > 0 \) to denote a arbitrary constant which will be chosen later adaptively small. Correspondingly, \( C_\delta \) is some positive constant depending on \( \delta \). In addition, for any quantities A and B, \( A \lesssim B \) is used to denote that there exists a positive constant independent of the solutions such that \( A \leqq C B \).
The following energy and dissipation functionals will be employed:
Then \( {\mathcal {E}}(0) = {\mathcal {E}}_{in} \), where \( {\mathcal {E}}_{in} \) is as in (1.14). In this work, we shall use \( Q({\mathcal {E}} ) \) to denote a polynomial quantity, with positive coefficients, of \( \sqrt{{\mathcal {E}}} \) and \( Q(0) = 0 \). In general, \( Q(\cdot ) \) is a generic polynomial quantity, with positive coefficients, of the arguments and \( Q(0) = 0 \).
3 \( \varepsilon \)-Independent A Priori Estimate
This section is devoted to show the following:
Proposition 1
For any \( T > 0 \), \( t \in [0,T] \), suppose that the solution \( (v_p, \rho _1) \) with \( w_p \) given by (1.11) to (PE) satisfies
for some positive constant C, and
Then any solution \( (\psi ^h, \xi ) \) to (1.8), with initial data as in (1.12), provided that it exists in the time interval \( [0,T]\), with \( \psi ^z \) given by (1.10), satisfies
for some positive constant \( C' \) depending only on the bounds in (3.1). In particular, \( C'\) is independent of \( \varepsilon \) and T.
Remark 5
We remark here that, from the definition of \( {\mathcal {E}}(t) \) in (2.2) and (1.3), (3.2) automatically holds for \( \varepsilon \) small enough if \( \sup _{0\leqq t \leqq T} {\mathcal {E}}(t) < \infty \) and (3.1) holds.
Throughout the rest of this section, it is assumed that \( (\xi ,\psi ^h) \) with \( \psi ^z \) given by (1.10) is a solution to (1.8) which is smooth enough such that the estimates below can be established. To justify the arguments, one can employ the local well-posedness theory and the standard different quotient method to the corresponding lines below (replaced the differential operators by different quotients, for example); see, for instance, similar arguments in [49, 50].
We denote by \( {\mathfrak {G}}_p(t) \) a polynomial, with positive coefficients, quantity of the arguments
and
In particular, (3.1) of Proposition 1 is equivalent to
for some positive constant C. For the sake of convenience, we will shorten the notations \( {\mathfrak {G}}_p = {\mathfrak {G}}_p(t), \mathfrak H_p = {\mathfrak {H}}_p(t) \), below. We also remind the reader that we have assumed that (3.1) and (3.2) hold throughout this section.
3.1 Temporal Derivatives
We start by performing the time derivative estimate to the solutions to system (1.8). Applying \( \partial _t\) to system (1.8) we will have the following system:
where
We will show the following:
Lemma 1
In addition to the assumptions in Proposition 1, suppose that \( (\xi , \psi ^h) \), with \( \psi ^z \) given by (1.10), is a smooth solution to (1.8) in the time interval [0, T] . We have
Proof
Take the \( L^2 \)-inner product of (3.5)\(_{2}\) with \( \varepsilon ^2 \psi ^h_t \). After applying integration by parts and substituting (CPE)\(_{1}\) and (3.5)\(_{1}\), we have the following:
Next we estimate the right-hand side of (3.8). After substituting (3.6) into \( I_5 \), it can be written as
Notice that \( \zeta = \varepsilon ^2 \rho _1 + \xi \) is independent of the z variable. Then, for every \( \delta > 0 \) there exists a positive constant \( C_\delta \) such that
where we have applied the Minkowski, Hölder’s, the Sobolev embedding and Young’s inequalities. On the other hand, \( I_{5}'' \) can be estimated directly using Hölder’s, the Sobolev embedding and Young’s inequalities:
On the other hand, from (1.9) (or (1.10)), we have the identities
Therefore, \( I_{5}''' \) can be written as
Then, we plug in identity (1.11) and apply the Hölder, Minkowski and Young inequalities to infer
On the other hand, a straight forward estimate shows that
To estimate \( I_{5,3}''' \), we apply integration by parts as follows:
from which we infer
Therefore, we have shown
Next, after substituting (1.7) into \( I_2 \), it follows that
Similarly to before,
On the other hand, after substituting (3.9), \( I_2'' \) can be written as
Then we have the following estimate:
To estimate \( I_{2,2}'' \), we first apply integration by parts as follows:
which yields, similarly to the estimate of \( I_{5,3}''' \),
Therefore,
Now, we will estimate \( I_3 \), which reads as
Here we have employed the facts that \( \xi \) is independent of the z-variable and that \( \int _0^1 \mathrm {div}_h\,v_p \,\mathrm{d}z = \int _0^1 \mathrm {div}_h\,v_{p,t} \,\mathrm{d}z = 0 \). The rest is straightforward. For instance, substituting (1.7) in \( I_1 \) yields
We list estimates for \( I_4,I_6 \) as follows:
where we have used the fact that from (1.6)
Summing up inequalities (3.10), (3.11)–(3.15) and (3.8) completes the proof. \(\quad \square \)
The next lemma follows directly from system (1.8), and it shows the estimates of the temporal derivatives of \( \xi , \psi ^h \) in terms of the spatial derivatives.
Lemma 2
Under the same assumptions as in Lemma 1,
for some positive constant C independent of \( \varepsilon \).
Proof
Indeed, after integrating (1.8)\(_{1}\) in the z variable, we have, thanks to (1.5),
Then (3.16) follows easily after applying the Minkowski, Hölder and Sobolev embedding inequalities.
On the other hand, (3.17) follows from (1.8)\(_{2}\) after substituting
and applying the Minkowski, Hölder and Sobolev embedding inequalities. \(\quad \square \)
3.2 Horizontal Derivatives
We derive the required estimates for the horizontal derivatives in this subsection. After applying \( \partial _{hh} = \partial _h^2 \) to system (1.8), we obtain the following system:
where
Lemma 3
Under the same assumptions as in Lemma 1, we have
for some positive constant \( C_{\mu ,\lambda } \), which is independent of \( \varepsilon \).
Proof
Take the inner product of (3.18)\(_{2}\) with \( \psi ^h_{hh} \) and integrate the resultant over \( \Omega \). Similarly to before, we will have the following:
Then the lemma follows from careful estimates on the right-hand side of (3.21), which are similar to those in the proof of Lemma 1. Therefore, details are omitted here. \(\quad \square \)
Next, we will derive the required estimate of \( \xi _{hh} \). After integrating (1.8)\(_{2}\) over \( z \in (0,1) \), we have the following equation, thanks to (1.5):
After applying \( \partial _h \) to (3.22), one has
What we need is to estimate the \( L^2 \)-norm of the terms on the right-hand side of (3.23). In fact, after applying the Minkowski, Hölder and Sobolev embedding inequalities, one has
On the other hand, after substituting (3.33), and the identity
we have
Therefore, one has
After substituting (1.7) and (1.10) and applying integration by parts, we obtain
Hence, applying the Minkowski, Hölder and Sobolev embedding inequalities implies
Summing up these estimates, we get the following inequality from (3.23),
We summarize the result in the following:
Lemma 4
Under the same assumptions as in Lemma 1, the following holds:
for some positive constant C independent of \( \varepsilon \).
Proof
This is the direct consequence of (3.25) and the Poincaré inequality. \(\quad \square \)
3.3 Vertical Derivatives Estimates
Now we turn to the required estimates of vertical derivatives. To do so, we first apply \( \partial _z\) to system (1.8) and write down the resultant system as follows:
where
Then we apply \( \partial _z\) to system (3.27) again and obtain the following system:
where
Notice, here we have employed the fact that \( \rho , \xi , \rho _1, {\mathcal {R}} \) are independent of the z variable. Also (3.29)\(_{2}\) is a parabolic equation of \( \psi ^h_{zz} \). Now we perform standard \( L^2 \) estimate on system (3.29).
Lemma 5
Under the same assumptions as in Lemma 1, we have
for some positive constant \( C_{\mu ,\lambda } \) independent of \( \varepsilon \).
Proof
After taking the \( L^2 \)-inner product of (3.29)\(_{2}\) with \( \psi ^h_{zz} \), we have the following:
Again, we shall estimate the terms on the right-hand side of (3.32). We begin with the term \( I_{15} \). Notice first, after taking \( \partial _z, \partial _{zz} \) to (1.9), we have the following identities:
Consequently, after substituting (3.33) in \( I_{15} \), we have that:
The estimates of the terms \( I_{13}, I_{14} \) are as follows:
After summing the estimates for \( I_{13}, I_{14}, I_{15} \), above, and (3.32), we conclude (3.31). \(\quad \square \)
3.4 Mixed Horizontal and Vertical Derivatives Estimates
What is left is to estimate the \( L^2 \) norm of \( \partial _{hz} \psi ^h \). We apply \( \partial _h \) to (3.27) and write down the resultant system:
where
Lemma 6
Under the same assumptions as in Lemma 1, we have
for some positive constant \( C_{\mu ,\lambda } \), which is independent of \( \varepsilon \).
Proof
Take the \( L^2 \) inner produce of (3.34)\(_{2}\) with \( \psi ^h_{hz} \). It follows that
Then the lemma follows from careful estimates of the right-hand side of (3.37), which we omit here again, since they are similar to those we have done before. \(\quad \square \)
3.5 Proof of Proposition 1
After applying similar arguments as in the proofs of Lemmas 3, 5 and 6, one can easily check that the following inequalities hold:
Therefore, the above inequalities, together with (3.7), (3.16), (3.17), (3.20), (3.26), (3.31) and (3.36), imply that there exist positive constants \( c_i, ~ i \in \lbrace 1,2 \ldots 10 \rbrace \), such that
where we denote
Under the assumption (3.2), it is easy to check that
where \( {\mathcal {E}} = {\mathcal {E}}(t) \) and \( {\mathcal {D}} = \mathcal D(t) \) are defined in (2.2) and (2.3), respectively. Therefore (3.38) can be written, after choosing \( \delta \) small enough, as
Then after applying Grönwall’s inequality to (3.42), one concludes that
Under the assumptions of Proposition 1, this completes the proof of (3.3).
4 Low Mach Number Limit
In this section, we will establish the asymptotic behavior of \( (\xi , \psi ^h) = (\xi ^\varepsilon , \psi ^{\varepsilon ,h}) \) as \( \varepsilon \rightarrow 0^+ \). In particular, we prove Theorem 1.2 in this section.
First, as a consequence of Theorem 1.1, we have the following:
Corollary 1
Under the same assumptions of Theorem 1.1, consider any integer \( s \geqq 3 \). Then (3.1) holds true for \( v_{p,in} \in H^s(\Omega _h \times 2{\mathbb {T}}) \), with the compatibility conditions as in (1.15).
Proof
Directly from the conclusion of Theorem 1.1, for \( s \geqq 3 \) and \( v_{p,in} \in H^s(\Omega ) \) as stated in the theorem, one has
where we have used the Gagliardo–Nirenberg interpolation inequality
Moreover, applying the Minkowski and Hölder inequalities to the expression of \( w_p \) as in (1.11) yields
What is left is to estimate
To do this, we write down the elliptic problem for \( \rho _1 \), which is obtained by taking average over the z-variable and then taking \( \mathrm {div}_h\,\) in (PE)\(_{2}\), as follows
Then the \( L^p \) estimate of the Riesz transform implies, that together with the Minkowski, Hölder and Sobolev embedding inequalities,
Consequently, for \( s \geqq 2 \),
Furthermore, after taking time derivatives of (4.1), we have the following elliptic problems:
Thus similarly, one has, for \( s \geqq 3 \),
On the other hand, after taking time derivative of (PE)\(_{2}\), we have the identity
Therefore, directly one has,
where we have applied the Minkowski, Sobolev embedding and Hölder inequalities, and the following inequalities as the consequence of (1.11):
Consequently, one concludes that, for \( s \geqq 3 \),
where we have substituted inequality (4.5)\(_{2}\). Thus, (4.5)\(_{3}\) yields
This completes the proof. \(\quad \square \)
Now, given \( v_{p,in} \in H^s(\Omega _h \times 2{\mathbb {T}}) \), for any integer \( s \geqq 3 \), which is even in the z-variable and satisfies the compatibility conditions (1.15), one can apply the conclusion of Proposition 1 to establish the global bound of the perturbation energy \( {\mathcal {E}} \), provided it is initially small. This is done through a continuity argument. We state first the proposition concerning the local well-posedness of solutions to system (1.8) with \( {\mathcal {E}}_{in} \) small enough.
Proposition 2
Let \( v_p \) be the solution to system (PE), as stated in Theorem 1.1 with initial data \( v_{p,in} \in H^s(\Omega _h \times 2{\mathbb {T}}) \), for an integer \( s \geqq 3 \). Consider the initial data \( (\xi _{in}, \psi ^h_{in} ) \in H^2(\Omega _h\times 2 {\mathbb {T}}) \times H^2(\Omega _h \times 2\mathbb T) \) as in (1.12) and satisfying the compatibility condition (1.13). There is a positive constant \( {{\bar{\varepsilon }}} \in (0,1) \), small enough, and a positive time \( T_{{{\bar{\varepsilon }}}}\in (0,\infty ) \), such that if \( \varepsilon \in (0, {{\bar{\varepsilon }}}) \) and \( {\mathcal {E}}_{in} \leqq {{\bar{\varepsilon }}} \), there exists a unique strong solution \( (\xi ^\varepsilon , \psi ^{\varepsilon ,h}) \in L^\infty (0,T_{{{\bar{\varepsilon }}}};H^2(\Omega _h \times 2{\mathbb {T}})) \), with \( \psi ^{\varepsilon ,z} \) as in (1.10), to system (1.8) in the time interval \( [0,T_{{{\bar{\varepsilon }}}}] \). The existence time \( T_{{\bar{\varepsilon }}} \) depends only on \( {\bar{\varepsilon }} \) and \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{3}} \) and is independent of \( \varepsilon \). Here \( {\mathcal {E}}_{in} \) is as in (1.14). Moreover, \( \partial _t\xi ^\varepsilon , \partial _t\psi ^{\varepsilon ,h} \in L^\infty (0,T_{{{\bar{\varepsilon }}}};L^2(\Omega _h \times 2{\mathbb {T}})) \), \( \rho \in (\frac{1}{2}\rho _0, 2\rho _0) \) in \( \Omega \times [0,T_{{{\bar{\varepsilon }}}}] \), and there is a constant \( C'' > 0 \), independent of \( \varepsilon \) such that
where \( {\mathcal {E}}(t) \) is as in (2.2).
The proof of Proposition 2 can be done via a fixed point argument similar to that in our previous work [50] and it is omitted here.
Now we are ready to establish the proof of Theorem 1.2
Proof of Theorem 1.2
Consider \( s \geqq 3 \) and \( \varepsilon \in (0,{\bar{\varepsilon }}) \) with \( {{\bar{\varepsilon }}} \in (0,1) \) as given in Proposition 2. Let the initial data \( (\xi _{in}, \psi ^{h}_{in} ) \in H^2(\Omega _h) \times H^2(\Omega _h \times 2{\mathbb {T}}) \) satisfy (1.12), the compatibility conditions (1.13), and \( \mathcal E_{in} \leqq \varepsilon ^2 \), where \( {\mathcal {E}}_{in} \) is as in (1.14). Then \( {\mathcal {E}}_{in} \leqq {{\bar{\varepsilon }}} \), and there is a strong solution to system (1.8) as stated by Proposition 2 in the time interval \( [0,T_{{{\bar{\varepsilon }}}}] \), for some \( T_{{{\bar{\varepsilon }}}} \in (0,\infty ) \), independent of \( \varepsilon \). The strong solution satisfies
Such estimates, together with Theorem 1.1 and Corollary 1, imply that the assumptions in Proposition 1 hold true in the time interval \( [0,T_{{{\bar{\varepsilon }}}}] \). Therefore applying (3.3) yields
provided \( \varepsilon \in (0,\varepsilon _1) \subset (0,{{\bar{\varepsilon }}}) \), where \( \varepsilon _1 \) is small enough such that \( Q(C''\varepsilon _1^2) \leqq C' \) and \( C' e^{2C'} (\varepsilon _1^2 + (\varepsilon _1^2 + 1) Q(C''\varepsilon _1^2) ) \leqq 1/2 \). This inequality yields that
where \( C''' = 4 C' e^{2C'} \), and provided \( \varepsilon _1 \) is small such that \( C''' \varepsilon _1^2 < {{\bar{\varepsilon }}} \). In particular, \( {\mathcal {E}}(T_{{{\bar{\varepsilon }}}}) \leqq {{\bar{\varepsilon }}} \). We apply Proposition 2 again in the time interval \( [T_{{{\bar{\varepsilon }}}}, 2T_{{{\bar{\varepsilon }}}}] \), which states that there exists a strong solution satisfying
Together with (4.6), this implies that
Consequently, Proposition 1 applies. In particular, (3.3) yields
As above, this implies
provided that \( \varepsilon \in (0,\varepsilon _2) \subset (0,\varepsilon _1) \subset (0,{{\bar{\varepsilon }}}) \), for \( \varepsilon _2 \) small enough such that \( Q(C'''' \varepsilon _2^2) \leqq Q(C''\varepsilon _1^2) \). Then inductively, without needing to determine the smallness of \( \varepsilon \) again, the arguments from (4.6) to (4.7) hold true for \( T_{{{\bar{\varepsilon }}}}, 2T_{{{\bar{\varepsilon }}}} \) replaced by \( n T_{{{\bar{\varepsilon }}}}, (n+1)T_{{{\bar{\varepsilon }}}} \), \( n \geqq 2 \), respectively. In particular, (4.7) holds true for \( 2 T_{{{\bar{\varepsilon }}}} \) replaced by \( (n+1) T_{{{\bar{\varepsilon }}}} \). Recall that \( T_{{{\bar{\varepsilon }}}} \) is independent of \( \varepsilon \). This concludes the proof of (1.18). (1.19) is a direct consequence of (1.18), (1.3), (1.9), (1.11) and the fact that \( \bigl \Vert \rho _1 \bigr \Vert _{H^{2}} < \infty \) as in (4.2). Therefore let \( \varepsilon _0 = \varepsilon _2 \), and we complete the proof of Theorem 1.2. \(\quad \square \)
5 Global Regularity Estimates of the Solution to the Primitive Equations
Let us recall the primitive equations (PE) first. We shorten the notations \( v= v_p, w = w_p \) in this section. Recall that
where \( \mu> 0, \lambda> \frac{1}{3} \mu > 0 \). Here the symmetry (SYM-PE) and the side condition (1.1) are imposed. We will make further assumptions on the viscous coefficients later.
In this section, we will study the global regularity of the solution \( (\rho _1, v, w) \) to (PE) with initial data \( v_{p,in} \in H^s(\Omega _h\times 2{\mathbb {T}}) \) for arbitrary integer \( s \in \lbrace 1, 2,3 \ldots \rbrace \), with \( v_{p,in} \) being even in the z-variable and satisfying the compatible conditions as stated in Theorem 1.1.
We will show the following proposition:
Proposition 3
For \( 0< \lambda< 4\mu < 12 \lambda \), suppose (PE) is complemented with initial data \( v_{p,in} \in H^1(\Omega _h\times 2{\mathbb {T}}) \) as above. Then the unique solution \( v_p \) to the primitive equations (PE) satisfies the estimates as stated in Theorem 1.1.
As in [41], we focus on the a priori estimates below. In fact, the local-in-time regularity in Sobolev and analytic function spaces has been studied in [56], and therefore following a continuity argument, one can check the validity of the proof below.
5.1 Basic Energy Estimate
Take the \( L^2 \)-inner product of (PE)\(_{2}\) with v. We have
Integrating the above equation in the time variable yields
Moreover, under the assumption (1.2), after applying the Poincaré inequality in (5.1), we have the inequality
for some positive constant c. Thus one can derive from above that
for all \( t \in [0,\infty ) \).
5.2 \( H^1 \) Estimate
After applying \( \partial _z\) to (PE)\(_{2}\), we write down the following equation:
Then take the \( L^2 \) inner product of (5.4) with \( v_{z} \). It follows, after substituting (1.11), that
After applying integration by parts, one will have
Therefore, let p, q be some positive constants, to be determined later, satisfying
After applying Hölder’s, the Gagliardo–Nirenberg interpolation and Young’s inequalities, one has
Hence, after choosing \( \delta > 0 \), sufficiently small, in the above estimate, we have
Integrating the above inequality in the time variable yields
for \( q \in (3,\infty ) \), where we have substituted (5.2).
On the other hand, after applying \( \partial _h \) to (PE)\(_{2}\), one gets the equation
Then after taking \( L^2 \)-inner product of (5.8) with \( v_h \), we have
Now we estimate the terms on the right-hand side of the above equality. As before, let \( q > 3, \frac{1}{p} + \frac{1}{q} = \frac{1}{2} \). After applying integration by parts and the Hölder, Gagliardo–Nirenberg interpolation and Young inequalities, one has
On the other hand, after substituting (1.11) in the term \( L_4 \) and applying the Minkowski, Hölder’s, the Gagliardo–Nirenberg interpolation and Young’s inequalities, one has
Similarly, take the \( L^2 \) inner product of (PE)\(_{2}\) with \( v_t \). One has,
As before, applying the Hölder, Minkowski, Gagliardo–Nirenberg interpolation and Young inequalities yield, for \( q > 3, ~ \frac{1}{p} + \frac{1}{q} = \frac{1}{2} \),
After summing (5.9), (5.10) and the estimates of \( L_3, L_4, L_5, L_6 \) above with sufficiently small \( \delta \), one has
Then after applying the Grönwall’s inequality, it follows that
for some positive constant C and \( q \in (3,\infty ) \), where we have substituted (5.2) and (5.7).
5.3 \( L^q \) Estimate
We take the \( L^2 \)-inner product of (PE)\(_{2}\) with \( \bigl | v \bigr |^{q-2} v \). It follows that
By using the Cauchy–Schwarz inequality, it holds that
Therefore, (5.12) implies
provided
In order to estimate \( L_7 \), we first derive an estimate for the “pressure” \( \rho _1 \). Recall the elliptic problem (4.1):
Now we consider the \( L^{p_1} \) estimate of \( \rho _1 \). In fact, as a consequence of the \( L^p \) estimate of the Riesz transform, one has
provided
where we have applied the Minkowski, Hölder and Gagliardo–Nirenberg interpolation inequalities. Let \( q_1 > 0 \) be such that
Then we have, after applying the Minkowski, Hölder and Gagliardo–Nirenberg interpolation inequalities,
provided
where we have substituted (5.15) in the second but last inequality. Therefore after combining (5.14), (5.16), (5.18), for q satisfying
we conclude from (5.13) and (5.17) that
after choosing \( \delta \) sufficiently small above. Applying Grönwall’s inequality to the above inequality implies, for \( 2 < q \leqq \frac{4\mu }{\lambda } + 2 \),
for some positive constant C and \( C_q \) depending on q.
Therefore, after summing up the inequalities (5.2), (5.7), (5.11), (5.20), for \( \lambda< 4\mu < 12 \lambda \), one will have
for some positive constant \( C_{p,in} \) depending on \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{1}}, \bigl \Vert v_{p,in} \bigr \Vert _{L^{q}} \) with
In particular, it suffices to take
and therefore
depends only on \( \bigl \Vert v_{p,in} \bigr \Vert _{H^{1}} \) by noticing that \( \bigl \Vert v_{p,in} \bigr \Vert _{L^{q}} \lesssim \bigl \Vert v_{p,in} \bigr \Vert _{H^{1}} \) in this case. The estimate of \( \partial _tv_p \) follows directly from (5.21) and (PE)\(_{2}\).
5.4 \( H^s \) Estimates
Next, we will show the global regularity of the solution v to system (PE) with more regular initial data \( v_{p,in} \). That is, we complement (PE) with the initial data \( v_{p,in} \in H^s(\Omega ) \), with \( s \geqq 2 \). In fact, we will use the mathematical induction principle to show (1.17). Notice, the case when \( s = 1 \), i.e., (1.16), has been shown in (5.21).
First, for integer \( s \geqq 1 \), it is assumed that (1.17) holds true. Our goal is to show that the same estimate is also true for s replaced by \( s + 1 \). In order to do so, we apply \( \partial ^{s+1} \) to (PE)\(_{2}\) with \( \partial \in \lbrace \partial _{x}, \partial _y, \partial _z \rbrace \) and denote the k-order derivative by \( \cdot _{k} := \partial ^k \cdot \) for any \( k \in \lbrace 0,1,2 \ldots s+1 \rbrace \). Then we have the following equation:
Take the \( L^2 \)-inner product of (5.23) with \( v_{s+1} \). It follows,
We estimate \( K_1, K_2 \) on the right-hand side of (5.24). First, notice that \( K_1, K_2 \) can be written as
We consider the estimates of \( K_{j,i} \), for \( j\in \lbrace 1,2 \rbrace \) and \( i \in \lbrace 0,1,2 \ldots s \rbrace \) in three cases:
In the case when \( i \geqq 2 \), we have
Therefore, applying the Hölder, Sobolev embedding and Young inequalities implies
Similarly, we apply the Minkowski, Hölder, Sobolev embedding and Young inequalities to estimate \( K_{2,i} \). On the one hand, if \( w_{s+1-i} = \partial ^{s+1-i} w = \partial _h^{s+1-i} w \), we have, thanks to (1.11),
On the other hand, if \( w_{s+1-i} = \partial _zw_{s-i} \), we have
In the case when \( i = 1 \), direct application of the Hölder, Sobolev embedding and Young inequalities yields
Meanwhile, to estimate \( K_{2,1} \), we will again apply the Minkowski, Hölder, Sobolev embedding and Young inequalities. If \( w_s = \partial _h^s w \), we have, after substituting (1.11),
If \( w_s = \partial _zw_{s-1} \), we have
Finally, in the case when \( i = 0 \), we apply the Hölder, Sobolev embedding and Young inequalities to get
When \( w_{s+1} = \partial _h^{s+1} w \), applying the Minkowski, Hölder, Sobolev embedding and Young inequalities yields, after substituting (1.11),
When \( w_{s+1} = \partial _zw_{s} \), we have, after substituting (1.11),
From the above estimates, one can conclude from (5.24) that for any integer \( s \geqq 1 \),
Here we have used the notation \( \bigl \Vert v_{s+1} \bigr \Vert _{L^{2}}^2 \) to denote \( \sum _{\partial \in \lbrace \partial _x,\partial _y,\partial _z \rbrace } \bigl \Vert \partial ^{s+1} v \bigr \Vert _{L^{2}}^2 \). Then after taking \( \delta > 0 \) small enough and applying Grönwall’s inequality, together with the inequalities (1.17) and (5.21), we have
where \( C_{p,in,1} = C_{p,in} (\bigl \Vert v_{p,in} \bigr \Vert _{H^{1}}) \) is as in (5.22).
On the other hand, after replacing \( s+1 \) by s in (5.23), we have the identity
After taking the \( L^2 \)-inner product of (5.27) with \( \partial _tv_{s} \) and noticing the fact that
this implies
where we have applied that since \( s \geqq 1 \),
due to the fact that from (1.11),
Here we have applied the Minkowski, Hölder and Sobolev embedding inequalities. Similarly, taking \( s = s-1 \) in (5.28) yields
Integrating (5.26) in the time variable, together with (5.28), (5.29), implies (1.17) with s replaced by \( s + 1 \). This finishes the mathematical induction. Hence, this concludes the proof of Proposition 3.
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Open Access funding provided by Projekt DEAL. This work was supported in part by the Einstein Stiftung/Foundation—Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.
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Liu, X., Titi, E.S. Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data. Arch Rational Mech Anal 238, 705–747 (2020). https://doi.org/10.1007/s00205-020-01553-z
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DOI: https://doi.org/10.1007/s00205-020-01553-z