Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp nonuniqueness for the forced 2D Navier-Stokes and dissipative SQG equations

Francisco Mengual, Marcos Solera

TL;DR

This work establishes sharp nonuniqueness results for the forced $(\alpha,\beta)$-SQG equation, covering the diffusive and generalized regimes, by combining Vishik’s spectral method with Golovkin’s trick and a self-similar instability framework. The authors show nonuniqueness across multiple solution notions, including $\dot{H}^s$-energy, Leray–Hopf, Resnick, and Marchand-type solutions, under precise thresholds like $s+\beta-\alpha<1$ (and $s<2-\beta$ when $\alpha=1$). They further derive sharp nonuniqueness below classical $L_t^pL^q$ criteria (LPS for NS, Constantin–Wu for SQG, and DCZL for generalized SQG), offering forcing data that yield two distinct solutions from zero initial data. The analysis hinges on self-similar coordinates, unstable vortices, and spectral continuity in the diffusion parameter, providing deep insights into the limits of uniqueness in 2D fluid models and sharp thresholds aligned with classical stability criteria.

Abstract

We prove a sharp nonuniqueness result for the forced generalized SQG equation. First, this yields nonunique $\dot{H}^s$- energy solutions below the Miura-Ju class. In particular, this shows that the solutions constructed by Resnick and Marchand for the dissipative SQG equation are not necessarily unique. Second, this establishes nonuniqueness below the Ladyzhenskaya-Prodi-Serrin class for the 2D Navier-Stokes equation, as well as below the Constantin-Wu and Dong-Chen-Zhao-Liu classes for the dissipative SQG equation.

Sharp nonuniqueness for the forced 2D Navier-Stokes and dissipative SQG equations

TL;DR

This work establishes sharp nonuniqueness results for the forced -SQG equation, covering the diffusive and generalized regimes, by combining Vishik’s spectral method with Golovkin’s trick and a self-similar instability framework. The authors show nonuniqueness across multiple solution notions, including -energy, Leray–Hopf, Resnick, and Marchand-type solutions, under precise thresholds like (and when ). They further derive sharp nonuniqueness below classical criteria (LPS for NS, Constantin–Wu for SQG, and DCZL for generalized SQG), offering forcing data that yield two distinct solutions from zero initial data. The analysis hinges on self-similar coordinates, unstable vortices, and spectral continuity in the diffusion parameter, providing deep insights into the limits of uniqueness in 2D fluid models and sharp thresholds aligned with classical stability criteria.

Abstract

We prove a sharp nonuniqueness result for the forced generalized SQG equation. First, this yields nonunique - energy solutions below the Miura-Ju class. In particular, this shows that the solutions constructed by Resnick and Marchand for the dissipative SQG equation are not necessarily unique. Second, this establishes nonuniqueness below the Ladyzhenskaya-Prodi-Serrin class for the 2D Navier-Stokes equation, as well as below the Constantin-Wu and Dong-Chen-Zhao-Liu classes for the dissipative SQG equation.
Paper Structure (14 sections, 21 theorems, 113 equations)

This paper contains 14 sections, 21 theorems, 113 equations.

Table of Contents

  1. Introduction and main results
  2. Main result
  3. Nonuniqueness of energy solutions
  4. $\dot{H}^{\frac{\alpha-2}{2}}$--energy solutions
  5. $L^2$--energy solutions
  6. Nonuniqueness in $L_t^pL^q$ spaces
  7. The Navier-Stokes case
  8. The generalized SQG case
  9. Other uniqueness criteria
  10. Self-similar coordinates
  11. Proof of Theorem \ref{['thm:main']}
  12. Self-similar instability
  13. Case $0\leq\alpha<1$
  14. Case $\alpha=1$

Key Result

Theorem 1.1

Let $0\leq\alpha\leq 1$ and $0<\beta< 3+\alpha$. There exists a force $f$ for which there are two distinct solutions $\theta_1$ and $\theta_2$ to the $(\alpha,\beta)$-SQG equation eq:SQG with $\theta^\circ=0$. Moreover, for all $r,s\geq -1$ and $1\leq a,b,p,q\leq\infty$ in the regimes Moreover, for $p=\infty$ the solutions are continuous in time and belong to the critical space

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Nonuniqueness of $\dot H^s$-energy solutions
  • proof
  • Theorem 1.4: Nonuniqueness of Leray--Hopf and Marchand solutions
  • proof
  • Theorem 1.5: Nonuniqueness of Resnick solutions
  • proof
  • Theorem 1.6: Nonuniqueness below the Ladyzhenskaya–Prodi–Serrin class
  • proof
  • ...and 30 more