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Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations

Lili Du, Xinliang Li

TL;DR

This work proves sharp non-uniqueness of weak solutions for the 2D hyper-dissipative Navier–Stokes equations with $\alpha\in[1,\tfrac{3}{2})$ in super-critical spaces, at the two endpoints of the generalized LPS condition. It develops a space–time intermittent convex integration framework that concentrates Reynolds stress on a small temporal set using temporal building blocks $g_{(k)}$, spatial building blocks (2D accelerating jets or Mikado flows), and carefully designed amplitudes and correctors; this enables the construction of non-unique weak solutions with the same initial data and controlled regularity, including zero-measure singular sets. The results extend prior 2D NSE non-uniqueness at the endpoint $\alpha=1$ to the hyper-dissipative regime, and hold in Lebesgue, Besov, and Triebel–Lizorkin spaces, highlighting the sharpness of the endpoint gLPS conditions. Overall, the paper advances our understanding of ill-posedness in dissipative fluid dynamics by linking dissipation strength, intermittency, and endpoint regularity through a rigorous convex integration framework.

Abstract

In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^γW_{x}^{s,p}$ when $α\in[1,\frac{3}{2})$, and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Ladyženskaya-Prodi-Serrin condition with the triplet $(s,γ,p)=(s,\infty, \frac{2}{2α-1+s})$ and $(s, \frac{2α}{2α-1+s}, \infty)$. As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case $α\in(1,\frac{3}{2})$, and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity.

Sharp non-uniqueness for the 2D hyper-dissipative Navier-Stokes equations

TL;DR

This work proves sharp non-uniqueness of weak solutions for the 2D hyper-dissipative Navier–Stokes equations with in super-critical spaces, at the two endpoints of the generalized LPS condition. It develops a space–time intermittent convex integration framework that concentrates Reynolds stress on a small temporal set using temporal building blocks , spatial building blocks (2D accelerating jets or Mikado flows), and carefully designed amplitudes and correctors; this enables the construction of non-unique weak solutions with the same initial data and controlled regularity, including zero-measure singular sets. The results extend prior 2D NSE non-uniqueness at the endpoint to the hyper-dissipative regime, and hold in Lebesgue, Besov, and Triebel–Lizorkin spaces, highlighting the sharpness of the endpoint gLPS conditions. Overall, the paper advances our understanding of ill-posedness in dissipative fluid dynamics by linking dissipation strength, intermittency, and endpoint regularity through a rigorous convex integration framework.

Abstract

In this article, we study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces when , and obtain the conclusion that the non-uniqueness of the weak solutions at the two endpoints is sharp in view of the generalized Ladyženskaya-Prodi-Serrin condition with the triplet and . As a good observation, we use the intermittency of the temporal concentrated function in an almost optimal way. The research results extend the recent elegant works on 2D Navier-Stokes equations in [Cheskidov and Luo, Invent. Math., 229 (2022), pp. 987--1054; Cheskidov and Luo, Ann. PDE, 9:13 (2023)] to the hyper-dissipative case , and are also applicable in Lebesgue and Besov spaces. It is proved that even in the case of high viscosity, the behavior of the solution remains unpredictable and stochastic due to the lack of integrability and regularity.
Paper Structure (28 sections, 21 theorems, 257 equations, 2 figures)

This paper contains 28 sections, 21 theorems, 257 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Background.
  3. Summary of existence and uniqueness results.
  4. Statement of the problem.
  5. Convex integration technique.
  6. Notations.
  7. Main results.
  8. Comments on main results.
  9. Main iteration and concentrating Reynolds error
  10. Main iteration
  11. Concentrating the Reynolds error
  12. Velocity perturbations in the super-critical regime $\mathcal{A}_{1}$
  13. Temporal building blocks.
  14. Spatial building blocks.
  15. Velocity perturbations.
  16. ...and 13 more sections

Key Result

Theorem 1.2

(Main theorem) For $\alpha \in[1,\frac{3}{2})$, $(s, p, \gamma) \in \mathcal{A}_{1}\cup\mathcal{A}_{2}$. Given any smooth, divergence-free and mean-free vector field $\tilde{u}$ on $[0, T] \times \mathbb{T}^{2}$, the system 1.1 admits a velocity field $u$ and a set such that the following holds. $(i)$ Weak solution: $u$ is a mean-free and divergence-free weak solution to the system 1.1 with the i

Figures (2)

  • Figure 1: The case $\alpha \in\left[1, \frac{3}{2}\right), \ s\in[0,2).$
  • Figure 2: The case $\alpha \in[\frac{d+2}{4},\frac{d+1}{2}).$

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • ...and 17 more