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Stability of steady states of the 3-D Navier-Stokes-Poisson equations with non-flat doping profile in exterior domains

Yingzhi Du, Hairong Liu

Abstract

This paper concerns an initial boundary value problem of compressible Navier-Stokes-Poisson equations with the non-flat doping profile in a 3-D exterior domain.The global existence of strong solutions near a steady state for compressible Navier-Stokes-Poisson equations with the general Navier-slip boundary conditions is established. For our setting, not only a steady state should be constructed in the exterior domain by the sub and super solution method, but also some new techniques should be adopted to obtain a priori estimates, especially some refined elliptic estimates and the estimates on the boundary.

Stability of steady states of the 3-D Navier-Stokes-Poisson equations with non-flat doping profile in exterior domains

Abstract

This paper concerns an initial boundary value problem of compressible Navier-Stokes-Poisson equations with the non-flat doping profile in a 3-D exterior domain.The global existence of strong solutions near a steady state for compressible Navier-Stokes-Poisson equations with the general Navier-slip boundary conditions is established. For our setting, not only a steady state should be constructed in the exterior domain by the sub and super solution method, but also some new techniques should be adopted to obtain a priori estimates, especially some refined elliptic estimates and the estimates on the boundary.

Paper Structure

This paper contains 6 sections, 22 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Notation and basic lemmas
  3. Steady state
  4. Global existence of strong solutions
  5. Estimates for lower-order derivatives
  6. Estimates for the second-order derivatives

Key Result

Theorem 1.1

Let $\Omega=\mathbb{R}^3 \setminus B_R$ be a smooth exterior domain in $\mathbb{R}^3$, where $B_R$ is a ball with radius $R$. Assume $b(x)>0$ and $\nabla b \in H^1(\Omega)$. Let $\tilde{ \rho}>0$, $\tilde{u} \equiv 0$ and $\tilde{\Phi}$ be a smooth steady state solution of (origin) such that $\frac{ and then there exists a unique global strong solution $(\rho, u ,\Phi)(t,x)$ to the initial bounda

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • ...and 35 more