Gluing Non-unique Navier–Stokes Solutions

We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-uniqueness discovered by the authors in [1].


Introduction
In the recent work [1], we constructed non-unique Leray solutions of the Navier-Stokes equations in the whole space with forcing: ∂ t u + u · ∇u − ∆u + ∇p = f div u = 0 .
The non-unique solutions are driven by the extreme instability of a "background" solutionū, which has a self-similar structure: u(x, t) = 1 √ tŪ (1.1) In particular, the non-uniqueness "emerges" from the irregularity at the spacetime origin and is expected to be local. However, whileū is compactly supported, the non-uniqueness in [1] involves another solution whose support is R 3 × [0, T ]. Below, we demonstrate a certain locality and robustness of the non-uniqueness discovered in [1] by gluing it into any smooth, bounded domain Ω ⊂ R 3 with noslip boundary condition u| ∂Ω = 0 and into the torus T 3 := R 3 /(2πZ) 3 , i.e., the fundamental domain [−π, π] 3 with periodic boundary conditions. Theorem 1.1 (Non-uniqueness in bounded domains). Let Ω be a smooth, bounded domain in R 3 or the torus T 3 . There exist T > 0, f ∈ L 1 t L 2 x (Ω × (0, T )), and two distinct suitable Leray-Hopf solutions u,ū to the Navier-Stokes equations on Ω × (0, T ) with body force f , initial condition u 0 ≡ 0, and no-slip boundary condition. 1 We assume a certain familiarity with the conventions of [1], although it will be convenient to recall the basics below. For x ∈ R 3 and t ∈ (0, +∞), define the similarity variables ξ = x √ t , τ = log t . (

1.2)
A velocity field u and its similarity profile U are related via the transformation The pressure p, force f , and their respective profiles P , F transform according to (1.4) The Navier-Stokes equations in similarity variables are (1.5) ThenŪ ∈ C ∞ 0 (B 1 ) constructed in [1] (see (1.1) above) is an unstable steady state of (1.5) with suitable smooth, compactly supported forcing termF , and the nonunique solutions are trajectories on the unstable manifold associated toŪ .
In this paper, we take the following perspective. The force f and one solution u are exactly the ones from [1]. They are self-similar, smooth for positive times, and compactly supported inside the domain Ω, which we assume contains the ball of radius 1/2 centered at the origin. Each non-unique solution in [1] constitutes then an "inner solution" which lives at the self-similar scaling |x| ∼ t 1/2 , and this solution can be glued to an "outer solution" (namely, u ≡ 0), which lives at the scaling |x| ∼ 1. The boundary conditions are satisfied by the outer solution. The solutions are glued by truncating on an intermediate scale |x| ∼ 1/10. Let η(x) be a suitable cut-off function with η ≡ 1 on B 1/9 and η ≡ 0 on R 3 \ B 1/7 . Our main ansatz is u =ū + φη + ψ , (1.6) whereū is the compactly supported self-similar solution of the previous work, φ is the inner correction defined on the whole R 3 (although only the values in supp η matter for the definition of u), and ψ is the outer correction defined on the torus. Since φ is the inner correction, it will be natural to track its similarity profile Φ (we keep the lower and uppercase convention). We likewise decompose the pressure althoughp = 0 from the construction in [1]. The PDE to be satisfied in Ω by φ and ψ is together with div(φη + ψ) = 0. We distribute the terms into an "inner equation", which we think of as an equation for φ involving some terms in ψ, localized around the origin, and an "outer equation", thought of as an equation for ψ. The inner and outer equations, when satisfied separately, imply that (1.8) is satisfied.
1.1. Inner equation. The inner equation has to be satisfied on the support of η, which is contained in B 1/7 : and it is coupled to the divergence-free condition div φ = 0 . (1.10) We introduce the operator L ss , i.e., the linearized operator of (1.5) aroundŪ : In self-similar variables, we rewrite the cut-off η(x) = N(ξ, τ ). We rewrite the inner equation (1.9) as 12) where N (ξ, τ ) = N(ξ/3, τ ). We now require that it is satisfied in the whole R 3 , not merely on the support of N.

Outer equation.
Using that (ū · ∇η)φ = 0 and ∂ t η = 0, as a consequence of our choice of η, we deduce the following system for the outer equation: The problem (1.13) is to be solved in Ω with the boundary condition ψ| ∂Ω = 0.
We now consider the PDEs (1.9) and (1.13) as a system for (Φ, ψ). The two components will be controlled using two different linear operators, L ss and P∆.
In dividing the terms of (1.8) into the inner and outer equations, we put the "boundary terms", i.e., terms involving derivatives of η, into the outer equation, whereas the we put the termsŪ · ∇Ψ and Ψ · ∇Ū into the inner equation.
Crucially, we expect that the boundary terms are small because solutions of the inner equation are well localized. Consequently, ψ decouples from φ as t → 0 + , and therefore the linear part of the system should be invertible. 1 For this to work, it is necessary to show that the boundary terms are negligible, which requires knowledge of the inner correction Φ in weighted spaces.
With this knowledge, we solve the full nonlinear system via a fixed point argument. The details of the scheme will be discussed in Section 3.
1 One can compare this to the matrix a b ε d where ε represents the boundary terms, b represents theŪ · ∇Ψ + Ψ · ∇Ū terms, and the diagonal elements a and d are O(1). In fact, eventually we will see that ψ decays faster than φ as t → 0 + , so the terms corresponding to b are small, and the whole system decouples.
Our method is inspired by the parabolic "inner-outer" gluing technique exploited in [3] to analyze bubbling and reverse bubbling in the two-dimensional harmonic map heat flow into S 2 . The reverse bubbling in [3] is also an example of gluing techniques applied to non-uniqueness, although its mechanism is quite different. It is worth noting that, in that setting, the harmonic map heat flow actually has a natural uniqueness class [9].
We expect that Theorem 1.1 may be extended in a number of ways. Our techniques extend with minimal effort to non-uniqueness centered at k points. We expect that the conditionally non-unique solutions of Jia andŠverák [7] can also be glued. 2 Finally, it would be interesting to glue the two-dimensional Euler constructions of [11,12] (see also [2]) into the torus or bounded domains. This is likely to be more challenging than the present work, since the Euler equations are quasilinear and the construction of the unstable manifold more involved. We leave these and other extensions to future work.

Preliminaries
Consider p ∈ (1, +∞) and Ω = R 3 , T 3 , or a smooth, bounded domain in R 3 . We define which can be understood as the space of L p velocity fields with div φ = 0 on Ω and φ · ν = 0 on ∂Ω, where ν is the exterior normal to Ω. See [5,Chapter III] or [10,Lemma 1.4]. Notice that the boundary condition is vacuous when Ω = R 3 , T 3 . There exists a bounded projection P : where ∆ N is the Neumann Laplacian. This is the Leray projection. By density of divergence-free test fields, it agrees across L p spaces and, in particular, with the extension of the L 2 -orthogonal projection onto divergence-free fields; see [5,Chapter III] or [10, Theorem 1.5].
2.1. Linear instability. The following theorem provides an unstable background for the 3D Navier-Stokes equations. We refer the reader to [1] for its proof.
The following lemma, borrowed from [1,Lemma 4.4], provides sharp growth estimates on the semigroup e τ Lss . Lemma 2.2. LetŪ be as in Theorem 2.1. Then, for any where ξ = (1 + |ξ| 2 ) 1/2 is the Japanese bracket notation. We further define ) and p ∈ [1, +∞], the solution operator e τ Lss P div M is easily shown to be well defined by standard arguments. Namely, consider the solution u to the following PDE: The mild solution theory of the above PDE can be developed using properties of the semigroup e t∆ P div (whose kernel consists of derivatives of the Oseen kernel, see (2.14)-(2.15) below) by considering P div(ū ⊗ u + u ⊗ū) as a perturbation in Duhamel's formula. In particular, it is standard to demonstrate that, for all T > 1 and t ∈ (1, T ], we have With this in mind, we focus below on growth estimates for the semigroup.
Proof of Lemma 2.3. To begin, we establish weighted estimates for the semigroup e τ A P div, where We have the representation formula where g is tensor-valued and consists of derivatives of the Oseen kernel (see, e.g., [10, p. 80]), satisfying the pointwise estimate Using the representation formula and elementary estimates for convolution (see Lemma 7.1 and Remark 7.2), we have two estimates. First, we have the short-time estimate Moreover, we have the long-time estimate This completes the semigroup estimates for e τ A P div. We now turn our attention to the growth estimate for e τ Lss P div. First, we prove We already have this estimate for τ ∈ (0, 2], see (2.10) in Remark 2.4, so we focus on τ ≥ 2. This is done by splitting e τ Lss P div = e (τ −1)Lss P • e Lss P div, using estimate (2.10) (with p = q = 2) for the operator e Lss P div, and using the growth estimate We will combine the semigroup estimates (2.19) and (2.20) for A with (2.21) and the fact thatŪ is compactly supported. We end up with 24) where we used that p > 3. This holds for all δ > 0, completing the proof.
where −F =Ū ⊗ρ+ρ⊗Ū. Notably, local elliptic regularity implies that ρ is smooth on the support ofŪ . Hence, F ∈ L ∞ w . Next, we 'undo' the similarity variables by defining 27) and we have the representation formula This completes the proof.  Then the Stokes operator A generates an analytic semigroup (e tA ) t≥0 , and we have, for all p ∈ (1, +∞) and q ∈ [p, +∞], the smoothing estimates The function u(x, t) = (e tA u 0 )(x) solves the Stokes equations with no-slip boundary conditions ∂ t u − ∆u + ∇π = 0 , u(·, t) = 0 on ∂Ω , Hence, the associated Stokes semigroup (e tA ) t≥0 coincides with the heat semigroup and enjoys the smoothing estimates provided h satisfies the compatibility condition T 3 h(x, t) dx = 0 for a.e. t ∈ (0, T ). The solution is in the very weak sense, that is, div u = h in the sense of distributions, and, for all As in Remark 2.8, the initial condition is only "modulo gradients". It is immediate to check that for any r ∈ [1, ∞] and p ∈ (1, ∞). Moreover, there is uniqueness when u ∈ L r t L p x (T 3 × (0, T )). That is, necessarily u is given by (2.40). Indeed, if div u = 0, then u = Pu, and (2.41) simply asserts that u solves the heat equation with zero initial condition.
For F ∈ L 1 (R 3 ) compactly supported in B R with R > 0, we evidently have For F ∈ L p w (R 3 ) with p ∈ (1, +∞) and R ≥ 2, we require the estimate Then, in the near field, we have whereas, whenever ξ ∈ B 10R \ B R , we have the contribution as in Remark 7.2, from the far field. Hence,

The integral equations
In what follows Ω is either a smooth, bounded domain or the periodic box T 3 . Forτ ∈ R,t > 0, and α, β > 0, we define the norms where r, p ≫ 1 will be fixed later. The function spaces X ᾱ τ and Y β t consist of C((−∞,τ ]; L ∞ w (R 3 )) and measurable functions, respectively, with finite norm. Let We drop the dependence onτ from Z α,β t since we always assume thatτ = logt. We use the decomposition The integral equations will be a reformulation of the inner and outer equations introduced in Section 1. We want to show that, for an appropriate choice of the parameters α and β, defined in (4.3), and r, p ≫ 1, there existst > 0 such that the integral equations admit a unique solution (Φ per , ψ) ∈ Z α,β t . In what follows, we allow the implied constants to depend on r, p, and a.
We now determine the above operators, beginning with the inner integral equation.

Inner integral equation. Recall that the inner PDE is
which must be satisfied on the support of N, and which we seek to solve in the whole space. With the decomposition (3.5), we can derive an equation for Φ per . The equation is where L is a linear operator in (Φ per , ψ) given by . (3.10) . (3.11) We finally have −G = NΦ lin ⊗ Φ lin .
(3.12) The associated integral operators are (3.17) It will be convenient to rewrite, for each component φ i of the vector field φ, to keep everything in divergence form: The PDE is supplemented with the boundary condition ψ| ∂Ω = 0. The inner pressure π, which appears in the boundary term π∇η, is given by its similarity profile Hence, we can rewrite it in physical variables as where ℓ, b, and g will represent L, B, and G in physical variables as opposed to similarity variables.
The integral equation for ψ is We rewrite it as where L o acts linearly on (Φ per , ψ) according to  and finally, where the implied constants depend on β, β ′ , r. Hence, (3.28) where the 1/r arises from the change of measure e τ dτ = dt. (3.30)

Outer estimates
We begin with the outer estimates. Crucially, we will see that the boundary terms from Φ lin will limit the decay rate β of ψ.

Inner estimates
We now turn to the inner estimates, for which our main tool is Lemma 2.3.
That the solution is indeed mild is a technical point, which we now justify. Initially, we know that, for all divergence-free w ∈ C 1 c ((0, T ); C 2 ∩ C 0 (Ω)), we have and u(·, t) ∈ L p σ (Ω) for a.e. t ∈ (0,t). In particular, u = Pu, and it is weakly continuous in (0,t) due to (6.4). Consider ε ∈ (0,t) such that u(·, ε) ∈ L p σ (Ω). Let v be the mild solution to the Stokes equations on Ω × (ε,t) with initial data v(·, ε) = u(·, ε) and right-hand side − div u ⊗ u + f . Then u − v is a very weak solution in the sense of Lemma 2.7 with zero initial data, zero right-hand side, and zero divergence. By uniqueness, u ≡ v on Ω × (ε,t).