Non-uniqueness of weak solutions to hyperviscous Navier–Stokes equations: on sharpness of J.-L. Lions exponent

Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity (-Δ)θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\theta }$$\end{document}, whenever the exponent θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is less than Lions’ exponent 5/4, i.e., when θ<5/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta < 5/4$$\end{document}.


Introduction
In this paper we consider the question of non-uniquness of weak solutions to the 3D Navier-Stokes equations with fractional viscosity (FVNSE) on where θ ∈ R is a fixed constant, and for u ∈ C ∞ (T 3 ) with´T 3 u(x)dx 0, the fractional Laplacian is defined via the Fourier transform as Definition (weak solutions) A vector field v ∈ C 0 weak (R; L 2 (T 3 )) is called a weak solution to the FVNSE if it solves (1) in the sense of distribution.
In the recent breakthrough work [5], Buckmaster and Vicol obtained non-uniqueness of weak solutions to the three-dimensional Navier-Stokes equations. They developed a new convex integration scheme in Sobolev spaces using intermittent Beltrami flows which combined concentrations and oscillations. Later, the idea of using intermittent flows was used to study non-uniqueness for transport equations in [23][24][25] employing scaled Mikado waves, and for stationary Navier-Stokes equations in [7,22] employing viscous eddies.
The schemes in [5,24] are based on the convex integration framework in Hölder spaces for the Euler equations, introduced by De Lellis and Székelyhidi [12], subsequently refined in [2,3,10,15], and culminated in the proof of the second half of the Onsager conjecture by Isett in [16]; also see [4] for a shorter proof. For the first half of the Onsager conjecture, see, e.g., [1,9], and the references therein.
The main contribution of this note is to show that the results in Buckmaster-Vicol's paper hold for FVNSE (1) for θ < 5/4: Theorem 1 Assume that θ ∈ [1, 5/4). Suppose u is a smooth divergence-free vector field, define on R + × T 3 , with compact support in time and satisfies the condition Then for any given ε 0 > 0, there exists a weak solution v to the FVNSE (1), with compact support in time, satisfying As a consequence there are infinitely many weak solutions of the FVNSE (1) which are compactly supported in time; in particular, there are infinitely many weak solutions with initial values zero.

Remark 1
In the above theorem we assume that θ ∈ [1, 5/4). However, using the constructions in [5] with a slightly different choice of parameters, one can actually show that Theorem 1.2 and Theorem 1.3 in [5] hold for the 3D FVNSE, i.e., there exist non-unique weak solutions v ∈ C 0 t W β,2 x , with a different β > 0, depending on θ . However, in this paper we choose to prove a weaker result, Theorem 1, in order to simplify the presentation while retaining the main idea.

Remark 2
For the case θ ∈ (−∞, 1), the same construction also yields weak x with a suitable choice of parameters.
We now make some comments on the analysis in this paper. Using the technique in [5], we adapt a convex integration scheme with intermittent Beltrami flows as the building blocks. The main difficulty in a convex integration scheme for (FVNSE), is the error induced by the frictional viscosity ν(− ) θ v, which is greater for a larger exponent θ . This error is controlled by making full use of the concentration effect of intermittent flows introduced in [5]. As it is shown in the crucial estimate (36), the error is controllable only for θ < 5/4. Compared with [5], since our goal is to construct weak x , we adapt a slightly simpler cut-off function and prove only estimates that are sufficient for this purpose.

Iteration lemma
Following [5], we consider the approximate system where R is a symmetric 3 × 3 matrix.
for some δ q+1 > 0. Then for any given δ q+2 > 0, there exists a smooth solution (v q+1 , R q+1 ) of (2) with Here for a given set A ⊂ R, the δ-neighborhood of A is denoted by where the positive constant C depends only on θ .

Proof of Theorem 1
where R is the symmetric anti-divergence operator established in Lemma 5, below. Clearly (v 0 , R 0 ) solves (2). Set Apply Lemma 1 iteratively to obtain smooth solution (v q , R q ) to (2). It follows from (6) Thus v q converge strongly to some Furthermore, it follows from (5) that . Now we show the existence of infinitely many weak solutions with initial values zero. Let . Thus ∇ · u 0 satisfies the conditions of the theorem. Hence there exists a weak solution v to (1) close enough to u so that v ≡ 0.

Notations and parameters
For a complex number ζ ∈ C, we denote by ζ * its complex conjugate. Let us normalize the volume For M, N ∈ [0, +∞], denote the Fourier projection of u by We also denote P ≤k P [0,k) and P ≥k P [k,+∞) for k > 0. Following the notation in [5], we introduce here several parameters σ , r , λ, with where λ λ q+1 ∈ 5N is the 'frequency' parameter; σ with 1/σ ∈ N is a small parameter such that λσ ∈ N parameterizes the spacing between frequencies; r ∈ N denotes the number of frequencies along edges of a cube; μ measures the amount of temporal oscillation. Later σ , r , μ will be chosen to be suitable powers of λ q+1 . We also fix a constant p > 1 which will be chosen later to be close to 1. The constants implicitly in the notation ' ' may depend on p but are independent of the parameters σ , r , λ.

Intermittent Beltrami flows
We use intermittent Beltrami flows introduced in [5] as the building blocks. Recall some basic facts of Beltrami waves.
Let be a given finite subset of S 2 such that − , and λ ∈ Z be such that λ ⊂ Z 3 . Then for any choice of coefficients a ξ ∈ C with a * ξ a −ξ the vector field is real-valued, divergence-free and satisfies Furthermore, Clearly we have 5 ∈ Z 3 , and min ξ ,ξ ∈ ,ξ +ξ 0 Also it is direct to check that In fact, representations of this form exist for symmetric matrices close to the identity. We have the following simple variant of [5, Proposition 3.2].
Proposition 2 Let B ε (Id) denote the ball of symmetric matrices, centered at the identity, of radius ε. Then there exist a constant ε γ > 0 and smooth positive functions Define the Dirichlet kernel It has the property that, for 1 < p ≤ ∞, Following [5], for ξ ∈ + , define a directed and rescaled Dirichlet kernel by and for ξ ∈ − , define Since the map is the composition of a rotation by a rational orthogonal matrix mapping {e 1 , e 2 , e 3 } to {ξ, A ξ , ξ × A ξ }, a translation, and a rescaling by integers, for 1 < p ≤ ∞, we have Let W (ξ ) be the Beltrami plane wave at frequency λ, Define the intermittent Beltrami wave W (ξ ) as It follows from the definitions and (9) that The following properties are immediate from the definitions.

Perturbations
Let ψ(t) be a smooth cut-off function such that Take a smooth increasing function χ such that and set where ε γ is the constant in Proposition 2. Then clearly It follows from the above definition that Therefore, the amplitude functions are well-defined and smooth. Define the velocity perturbation to be w w q+1 : where P L H Id − ∇ −1 div is the Leray-Helmholtz projection into divergence-free vector field, and P 0 f f − ffl T 3 f dx. It is well-known that P L H is bounded on L p , 1 < p < ∞ (see, e.g., [14]). It follows from Proposition 3 that

Lemma 2
The following bounds hold: Proof It follows from (3) that It is direct to verify (21) and (23), while (22) and (24) follow from (17) and (21). Now we can estimate the time support of w q+1 : We need the following Lemma, which is a variant of [5, Lemma 3.6].

Lemma 4 Suppose the parameters satisfy (8) and
Then the following estimates for the perturbations hold: Proof Since W (ξ ) is (T/λσ ) 3 periodic, it follows from (15), (23), and Lemma 3 that In view of (8), (15) and (16) yield that where the boundedness of P L H and P 0 on L p , for 1 < p < ∞, is used in the first inequality of the estimate for w In the same way, we can estimate For N ≥ 1, using (15) and (16), we obtain that where we use (8) and (27).

Estimates for the stress
Let us recall the following operator in [12].
Lemma 5 (symmetric anti-divergence) There exists a linear operator R, of order −1, mapping vector fields to symmetric matrices such that with standard Calderon-Zygmund estimates, for 1 < p < ∞, It is direct to verify that R(u) is a symmetric matrix field depending linearly on u and satisfies (33). Note that R is a constant coefficient ellitpic operator of order −1. We refer to [14] for the Calderon-Zygmund estimates R L p →W 1, p 1 and RP 0 u L p |∇| −1 P 0 u L p . Combining these with Sobolev embeddings, we have Ru C α We have the following variant of [5, Lemma B.1] in [5].
It remains to estimate R oscillation , which can be handled in the same way as in [5]. It follows from (19) that Since E (ξ,ξ ) has zero mean, we can split it as : E (ξ,ξ ,1) + E (ξ,ξ ,2) .
Finally, we estimate the time support of R q+1 . Using (25) we obtain supp t R q+1 ⊂ supp t w q+1 ∪ supp t R q ⊂ N δ q+1 (supp t R q ).