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Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

Jinlu Li, Xing Wu, Yanghai Yu

TL;DR

This paper analyzes ill-posedness and inviscid-limit behavior for the Euler, Navier-Stokes, and surface quasi-geostrophic equations in Besov spaces on the torus. By constructing explicit high-frequency traveling-wave data via lacunary Fourier series, it proves ill-posedness of the data-to-solution map in $B^s_{p,\infty}$ for the Euler and QG equations and shows non-Hölder time regularity of the Euler flow in these spaces. It also demonstrates non-convergence of the vanishing viscosity limit from Navier–Stokes to Euler in the same Besov scale, and establishes analogous ill-posedness and non-convergence results for the dissipative and inviscid QG equations, using dyadic analysis and precise frequency-localized constructions. Overall, the work extends endpoint ill-posedness results to broader Besov settings and clarifies the delicate inviscid-limit behavior in critical Besov spaces.

Abstract

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of $B^s_{p,\infty}$ solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \cite{CS} and Misiołek-Yoneda \cite{MY}. Secondly, we prove the failure of the $B^s_{p,\infty}$-convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.

Ill-posedness and inviscid limit of basic equations of fluid dynamics in Besov spaces

TL;DR

This paper analyzes ill-posedness and inviscid-limit behavior for the Euler, Navier-Stokes, and surface quasi-geostrophic equations in Besov spaces on the torus. By constructing explicit high-frequency traveling-wave data via lacunary Fourier series, it proves ill-posedness of the data-to-solution map in for the Euler and QG equations and shows non-Hölder time regularity of the Euler flow in these spaces. It also demonstrates non-convergence of the vanishing viscosity limit from Navier–Stokes to Euler in the same Besov scale, and establishes analogous ill-posedness and non-convergence results for the dissipative and inviscid QG equations, using dyadic analysis and precise frequency-localized constructions. Overall, the work extends endpoint ill-posedness results to broader Besov settings and clarifies the delicate inviscid-limit behavior in critical Besov spaces.

Abstract

In this paper, we consider the Cauchy problem to the basic equations of fluid dynamics on the torus. Firstly, we construct a new initial data and provide a simple proof on the ill-posedness of solution of the Euler equations and the surface quasi-geostrophic equation, which covers the results obtained by Cheskidov-Shvydkoy \cite{CS} and Misiołek-Yoneda \cite{MY}. Secondly, we prove the failure of the -convergence in the inviscid limit for both the Navier-Stokes equations and the surface quasi-geostrophic equation.

Paper Structure

This paper contains 7 sections, 12 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1']}
  4. Proof of Theorem \ref{['th2']}
  5. Proof of Theorem \ref{['th4']}
  6. Proof of Theorems \ref{['Qth1']}-\ref{['Qth2']}
  7. Proof of Theorem \ref{['Qth3']}

Key Result

Theorem 1.1

Let $d\geq2$ and $s>0$ with $1\leq p\leq \infty$. There exists a divergence-free vector field $u_0\in B^s_{p,\infty}(\mathbb{T}^d)$ such that the data-to-solution map $u_0\mapsto u^{\rm E}(t,u_0)\in B^s_{p,\infty}(\mathbb{T}^d)$ of the Euler equations e satisfies where $\eta_0$ is some positive constant only dependent on $p$ and $d$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: MY
  • Theorem 1.5: $\kappa>0$
  • Remark 1.3
  • Theorem 1.6
  • Theorem 1.7: $\kappa=0$
  • ...and 11 more