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On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition

Quansen Jiu, Jie Tan, Zhihong Yan

Abstract

In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition \(\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty\). If \(\frac{9}{2} < p < 6\), Liouville-type theorem holds under the additional oscillation condition for momentum \(ρu \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(ρu \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(ρu \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for \(2 \le s \le 6\) and \(\frac{3}{2} < q \le 3\).

On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition

Abstract

In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), , with bounded density, one obtains . This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition . If , Liouville-type theorem holds under the additional oscillation condition for momentum \(ρu \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(ρu \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(ρu \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for and .
Paper Structure (3 sections, 3 theorems, 67 equations)

This paper contains 3 sections, 3 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Proofs of Main Theorems
  3. Declaration of competing interest

Key Result

Theorem 1

Let $(u,\rho)$ is a smooth solution of eqNS-dfqt which satisfies $u\in L^p(\mathbb{R}^3)$ and $\rho\in L^{\infty}(\mathbb{R}^3)$ with $3\leq p\leq\frac{9}{2}$, then $u=0$ and $\rho=$constant.

Theorems & Definitions (9)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • proof : Proof of Theorem \ref{['thNS1']}
  • proof : Proof of Theorem \ref{['thNS2']}
  • proof : Proof of Theorem \ref{['thNS3']}