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Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

Bing Yuan, Rong Zhang, Peng Zhou

TL;DR

The paper proves global existence of strong solutions for the 2D Cauchy problem of nonhomogeneous incompressible Navier–Stokes with density‑dependent viscosity and vacuum, covering both vacuum and nonvacuum far fields. A central innovation is a robust $L^q$ bound for $\|\nabla\rho\|_{L^q}$, $q>2$, obtained from the system’s structure and the material derivative, which enables global extension without smallness assumptions. The analysis leverages a Beale–Kato–Majda‑type inequality, a crucial substitute estimate $\|\nabla(\mu(\rho)\omega)\|_{L^2}+\|\nabla P\|_{L^2} \le C\|\sqrt{\rho}\dot u\|_{L^2}$, and careful handling of density–diffusion coupling to obtain higher‑order and time‑weighted estimates. The results hold for general density‑dependent viscosity with $\mu'\in L^{\infty}$ and include the constant viscosity limit, significantly advancing the understanding of large, vacuum‑bearing data in 2D.

Abstract

This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key estimates of $\|\nabla ρ\|_{L_t^\infty L_x^q},q>2$ without any smallness asuumption on the initial data, and thus establish the global existence of the strong solutions with the far-field density being either vacuum or nonvacuum. Notably, the initial data can be arbitrarily large and the initial density is allowed to vanish. Furthermore, the large-time asymptotic behavior of the gradients of the velocity and the pressure is also established.

Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

TL;DR

The paper proves global existence of strong solutions for the 2D Cauchy problem of nonhomogeneous incompressible Navier–Stokes with density‑dependent viscosity and vacuum, covering both vacuum and nonvacuum far fields. A central innovation is a robust bound for , , obtained from the system’s structure and the material derivative, which enables global extension without smallness assumptions. The analysis leverages a Beale–Kato–Majda‑type inequality, a crucial substitute estimate , and careful handling of density–diffusion coupling to obtain higher‑order and time‑weighted estimates. The results hold for general density‑dependent viscosity with and include the constant viscosity limit, significantly advancing the understanding of large, vacuum‑bearing data in 2D.

Abstract

This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key estimates of without any smallness asuumption on the initial data, and thus establish the global existence of the strong solutions with the far-field density being either vacuum or nonvacuum. Notably, the initial data can be arbitrarily large and the initial density is allowed to vanish. Furthermore, the large-time asymptotic behavior of the gradients of the velocity and the pressure is also established.
Paper Structure (9 sections, 23 theorems, 140 equations)

This paper contains 9 sections, 23 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vacuum at infinity ($\boldsymbol{\rho_\infty=0}$)
  4. A priori estimates (I): upper bound of the $\boldsymbol{\|\nabla\rho\|_{L^q}}$
  5. A priori estimates (II): higher order estimates
  6. Nonvacuum at infinity ($\boldsymbol{\rho_\infty>0}$)
  7. A priori estimates (I): upper bound of the $\boldsymbol{\|\nabla\rho\|_{L^q}}$
  8. A priori estimates (II): higher order estimates
  9. Proof of Theorems \ref{['T1']} and \ref{['T2']}

Key Result

Theorem 1.1

Let $\rho_\infty=0$, assume the initial data $(\rho_0\ge 0, u _0)$ satisfy that for some $a\in(1, 2)$ and $q>2$, where Then the Cauchy problem NS-ydxw has a unique global strong solution $(\rho, u, P)$ satisfying that for any $0<T<\infty$, and for some positive constant $N_1$. Moreover, $(\rho,u,P)$ has the following decay rates, that is, for any $p\in[2,\infty)$ and $t\geq1$, we have where $

Theorems & Definitions (40)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2: Gagliardo-Nirenberg
  • Lemma 2.3
  • ...and 30 more