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Global well-posedness of 3D inhomogenous incompressible Navier-Stokes equations with density-dependent viscosity

Dongjuan Niu, Lu Wang

Abstract

The issue of global well-posedness for the 3D inhomogenous incompressible Navier-Stokes equations was first addressed by Kazhikov in 1974. In this manuscript, we obtain its global well-posedness for the system with density-dependent viscosity under the smallness assumption of initial velocity in the critical space $\dot{B}_{p,1}^{-1+\frac 3p}$ with $p\in ]1, 9/2]$. To the best of our knowledge, this is the first result about the global well-posedness for which one does not assume any smallness condition on the density when the initial density is far away from vacuum.

Global well-posedness of 3D inhomogenous incompressible Navier-Stokes equations with density-dependent viscosity

Abstract

The issue of global well-posedness for the 3D inhomogenous incompressible Navier-Stokes equations was first addressed by Kazhikov in 1974. In this manuscript, we obtain its global well-posedness for the system with density-dependent viscosity under the smallness assumption of initial velocity in the critical space with . To the best of our knowledge, this is the first result about the global well-posedness for which one does not assume any smallness condition on the density when the initial density is far away from vacuum.
Paper Structure (12 sections, 21 theorems, 251 equations)

This paper contains 12 sections, 21 theorems, 251 equations.

Table of Contents

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

Key Result

Theorem 1.1

Let $q\in[1,2]$ and $p\in]1,9/2]$ with $\max\{\frac{1}{p},1-\frac{2}{p}\}\leq\frac{1}{q}\leq\frac{1}{p}+\frac{1}{3}.$ Assume that the initial data $(\rho_{0},u_{0})$ satisfying where $r\in ]3,9[$ and $\delta\in]1/2,3/4[.$ Then there exists a small positive constant $\varepsilon$ depending on $\|\rho_{0}-1\|_{B^{\frac{3}{q}}_{q,1}},$$\|\nabla\mu(\rho_0)\|_{L^{r}}$ and $\|u_0\|_{\dot{H}^{-2\delta}}

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 31 more