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Global well-posedness for the 2D incompressible heat conducting Navier-Stokes equations with temperature-dependent coefficients and vacuum

Wenchao Dong, Qingyan Li

Abstract

We consider the initial boundary problem of 2D non-homogeneous incompressible heat conducting Navier-Stokes equations with vacuum, where the viscosity and heat conductivity depend on temperature in a power law of Chapman-Enskog. We derive the global existence of strong solution to the initial-boundary value problem, which is not trivial, especially for the nonisentropic system with vacuum. Significantly, our existence result holds for the cases that the viscosity and heat conductivity depend on $θ$ with possibly different power laws (i.e., $μ=θ^α, κ=θ^β$ with constants $ α,β\geq0$) with smallness assumptions only on $α$ and the measure of initial vacuum domain. In particular, the initial data can be arbitrarily large. Moreover, it is obtained that both velocity and temperature decay exponentially as time tends to infinity.

Global well-posedness for the 2D incompressible heat conducting Navier-Stokes equations with temperature-dependent coefficients and vacuum

Abstract

We consider the initial boundary problem of 2D non-homogeneous incompressible heat conducting Navier-Stokes equations with vacuum, where the viscosity and heat conductivity depend on temperature in a power law of Chapman-Enskog. We derive the global existence of strong solution to the initial-boundary value problem, which is not trivial, especially for the nonisentropic system with vacuum. Significantly, our existence result holds for the cases that the viscosity and heat conductivity depend on with possibly different power laws (i.e., with constants ) with smallness assumptions only on and the measure of initial vacuum domain. In particular, the initial data can be arbitrarily large. Moreover, it is obtained that both velocity and temperature decay exponentially as time tends to infinity.
Paper Structure (6 sections, 20 theorems, 147 equations)

This paper contains 6 sections, 20 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Priori Estimates
  4. Bootstrap Argument
  5. Time-Weighted Estimates
  6. Global Well-Posedness

Key Result

Theorem 1.1

For some positive constants $q\in(2,\infty)$ and $\underline{\theta}$, suppose that the initial data $(\rho_0, u_0, \theta_0)$ satisfies and the compatibility conditions for some $P_0\in H^1$ and $g_1,g_2\in L^2$. Then there exist two positive constants $\epsilon_0$ and $c_0$ depending only on $\tilde{\rho}$ and $\Vert(u_0,\theta_0)\Vert_{H^2}$, such that if and where $|V|$ is the measure of t

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 26 more