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On the well-posedness of the compressible Navier-Stokes equations

Zihua Guo, Minghua Yang, Zeng Zhang

TL;DR

The paper addresses Hadamard well-posedness for the barotropic compressible Navier–Stokes equations in critical Besov spaces ${\mathbb X}_p=\dot{B}_{p,1}^{d/p}\times\dot{B}_{p,1}^{-1+d/p}$ with $1\le p<2d$ and $d\ge 2$. It introduces a novel continuity mechanism for the solution map by combining a $L^1_tL^\infty_x$ difference estimate with Tao's frequency-envelope method and a Lagrangian transport formulation, proving that the map $S_T:(a_0,u_0)\mapsto (a,u)$ is continuous from ${\mathbb X}_p$ to $C([0,T];{\mathbb X}_p)$. The approach yields a new velocity-difference estimate and leverages a transport–parabolic framework to handle high- and low-frequency components uniformly. As a by-product, the continuous bijection property of the Lagrangian transform $(a,u)\mapsto (\bar a,\bar u)=(a\circ X, u\circ X)$ bridges Eulerian and Lagrangian methods and supports further extensions to density-dependent viscosities and full compressible systems.

Abstract

We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space $\mathbb{X}_p=\dot{B}_{p,1}^{\frac{d}{p}}\times \dot{B}_{p,1}^{-1+\frac{d}{p}}$ for $1\leq p<2d$ with $d\geq2$. The main new result is the continuity of the solution maps from $\mathbb{X}_p$ to $C([0,T]: \mathbb{X}_p)$, which was not proved in previous works \cite{D2001, D2005, D2014}. To prove our results, we derive a new difference estimate in $L_t^1L_x^\infty$. Then we combine the method of frequency envelope (see \cite{Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \cite{D2014}). As a by-product, the Lagrangian transform $(a,u)\to (\bar a, \bar u)=(a\circ X, u\circ X)$ used in \cite{D2014} is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.

On the well-posedness of the compressible Navier-Stokes equations

TL;DR

The paper addresses Hadamard well-posedness for the barotropic compressible Navier–Stokes equations in critical Besov spaces with and . It introduces a novel continuity mechanism for the solution map by combining a difference estimate with Tao's frequency-envelope method and a Lagrangian transport formulation, proving that the map is continuous from to . The approach yields a new velocity-difference estimate and leverages a transport–parabolic framework to handle high- and low-frequency components uniformly. As a by-product, the continuous bijection property of the Lagrangian transform bridges Eulerian and Lagrangian methods and supports further extensions to density-dependent viscosities and full compressible systems.

Abstract

We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space for with . The main new result is the continuity of the solution maps from to , which was not proved in previous works \cite{D2001, D2005, D2014}. To prove our results, we derive a new difference estimate in . Then we combine the method of frequency envelope (see \cite{Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \cite{D2014}). As a by-product, the Lagrangian transform used in \cite{D2014} is a continuous bijection and hence bridges the Eulerian and Lagrangian methods.
Paper Structure (4 sections, 11 theorems, 167 equations)

This paper contains 4 sections, 11 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Frequency envelope method: $1\leqslant p<2d$
  4. Bona-Smith method: $1\leqslant p<d$

Key Result

Theorem 1.2

Assume $d\geqslant 2$ and $1 \leqslant p <2d$. Let $c>0$ be small enough (depending on $p, d$). Then for any $(a_{0}, u_{0})\in {\mathbb X}_p$ with $\|a_{0}\|_{\dot{B}_{p,1}^{{d/p}}}\leqslant c$, there exists a neighbourhood $U$ of $(a_{0}, u_{0})$ in ${\mathbb X}_p$ and $T=T(U)>0$, such that for an Moreover, the solution map $S_T$ is continuous from $U$ to $Z_p(T)$.

Theorems & Definitions (27)

  • Definition 1.1: Well-posedness
  • Remark 1.1
  • Theorem 1.2: Local well-posedness
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2: see Lemma 2.1 in BaChDa11
  • Lemma 2.3
  • proof
  • Remark 2.1
  • ...and 17 more