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On the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes system with density-dependent viscosity

Dongjuan Niu, Lu Wang

Abstract

In this paper, we are concerned with the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity when the initial velocity is sufficiently small in the critical Besov space $\dot{B}^{\frac 12}$. Compared with the previous result of Abidi and Zhang (Science China Mathematics 58 (6) (2015) 1129-1150), we remove the smallness assumption of the viscosity $μ(ρ_0)-1$ in $L^{\infty}$-norm.

On the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes system with density-dependent viscosity

Abstract

In this paper, we are concerned with the global well-posedness of 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity when the initial velocity is sufficiently small in the critical Besov space . Compared with the previous result of Abidi and Zhang (Science China Mathematics 58 (6) (2015) 1129-1150), we remove the smallness assumption of the viscosity in -norm.
Paper Structure (12 sections, 22 theorems, 248 equations)

This paper contains 12 sections, 22 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Littlewood-Paley theory
  4. Regularity results
  5. Local well-posedness of (\ref{['a1']})
  6. Global well-posedness of (\ref{['a1']})
  7. $L^2$ estimate of $u$
  8. $\dot{H}^1$ estimate of $u$
  9. The a priori time-weighted estimates to (\ref{['a1']})
  10. The $L^{1}([t_0,T];L^\infty(\mathbb{R}^{3}))$ estimate for $\nabla u$
  11. Proof of Theorem \ref{['Theorem-1']}
  12. The large time decay-estimate of $u$

Key Result

Theorem 1.1

Let $q\in(3,6)$ and $\delta\in(\frac{1}{2},\frac{3}{4})$. Assume that the initial data $(\rho_{0},u_{0})$ satisfies the regularity condition Then there exists some small positive constant $\varepsilon$ depending on $\|\rho_{0}-1\|_{B^{\frac{3}{2}}_{2,1}},$$\|\nabla\mu(\rho_0)\|_{L^{q}}$ and $\|u_0\|_{\dot{H}^{-2\delta}}$ such that if the Cauchy problem $(a1)$ admits a unique global strong soluti

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • ...and 28 more