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Optimal Regularity for the 2D Euler Equations in the Yudovich class

Nicola de Nitti, David Meyer, Christian Seis

Abstract

We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with BMO gradient. As an application, we study the 2D Euler equations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variation of Bahouri and Chemin's vortex patch example.

Optimal Regularity for the 2D Euler Equations in the Yudovich class

Abstract

We analyze the optimal regularity that is exactly propagated by a transport equation driven by a velocity field with BMO gradient. As an application, we study the 2D Euler equations in case the initial vorticity is bounded. The sharpness of our result for the Euler equations follows from a variation of Bahouri and Chemin's vortex patch example.
Paper Structure (14 sections, 15 theorems, 205 equations)

This paper contains 14 sections, 15 theorems, 205 equations.

Table of Contents

  1. Introduction and main results
  2. Outline
  3. Preliminaries
  4. Littlewood--Paley decomposition
  5. Orlicz and Zygmund spaces and their Luxemburg norms
  6. Besov spaces
  7. Calderon spaces
  8. Evolution of log2-Hölder norms by log-Lipschitz vector fields
  9. General setting
  10. Propagation of regularity for the transport equation
  11. Construction of the counterexample
  12. Construction of the perturbation
  13. A priori estimates
  14. Proof of the norm inflation on every Qnt

Key Result

Theorem 1.1

Let us assume that the velocity $u$ satisfies Then, the (unique) renormalized solution $\theta$ of the Cauchy problem eq:transport, with initial data $\theta_0$ such that $\theta_0 \in L^\infty(\mathbb{T}^d) \cap H^{\exp, a, s}(\mathbb{T}^d)$ and $\fint_{\mathbb{T}^d} \theta_0 = 0$, satisfies the estimate for all $a \le 1/2$ and all $s>0$, with a constant $C>0$ depending on $a$, $s$, and $d$.

Theorems & Definitions (34)

  • Theorem 1.1: Propagation of regularity for the transport equation
  • Remark 1.2: Advection-diffusion equation
  • Theorem 1.3: Sharpness of the regularity result
  • Remark 1.4: Loss of fractional Sobolev regularity
  • Lemma 2.1: Estimate of the Luxemburg norm
  • proof
  • Lemma 2.2: Besov-norm equivalence
  • proof
  • Lemma 2.3
  • proof
  • ...and 24 more