Local existence of the strong solutions for the full Navier-Stokes-Poisson equations
type: Item
instance of: scholarly article
title: Local existence of the strong solutions for the full Navier-Stokes-Poisson equations
zbMATH Open document ID: 1173.35103
author: Zhong Tan
author: Junping Yin
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-07-24T00:00:00Z
review text: The paper is devoted to the study of full system of compressible Navier-Stokes-Poisson equations, considering the following continuity, momentum and Poisson equation: \[ \partial_t\rho+ \text{div}(\rho u)=0, \] \[ \partial_t(\rho u)+ \text{div}(\rho u u)+\rho\nabla \Phi-\mu\Delta u- (\mu+\lambda) \nabla \operatorname{div}u +\nabla P=\rho f, \] \[ \Delta \Phi = 4\pi g \left(\rho - \frac{m_0}{|\Omega|}\right). \] When the thermal conductivity coefficient is a positive constant \(\kappa > 0\), it is considered the internal energy equation \[ \partial_t(\rho e)+ \text{div}(\rho ue)- \kappa \Delta e + P \operatorname{div} u=\frac{\mu}{2}|\nabla u+\nabla u^T|^2 + \lambda |\operatorname{div} u|^2 + \rho h, \] and the state law \[ P=(\gamma -1)\rho e \quad (\gamma>1). \] When \(\kappa=0\) and \(h=0\), these last two equations are changed by the pressure equation \[ \partial_t P+ u\cdot\nabla P +\gamma P \operatorname{div} u=(\gamma-1)\left(\frac{\mu}{2}|\nabla u+\nabla u^T|^2 + \lambda |\operatorname{div} u|^2 \right). \] Here \(\rho\geq 0\), \(u\), \(e\) and \(\Phi\) denotes respectively the density, internal energy and velocity field of the fluid and the Newtonian gravitational potential, with \(\mu,\lambda\) the Lamé coefficients (constants). These two PDE systems are ended with Dirichlet boundary conditions. Under reasonable assumptions on the Lamé coefficients \[ \mu>0,\quad 2\mu+3\lambda\geq 0, \] and the following compatibility conditions on the initial data \[ \begin{aligned} -\mu\Delta u_0 - (\mu+\lambda)\nabla \operatorname{div} u_0 + \nabla P_0 &= \rho_0^{1/2} g_2,\\ -\kappa \Delta e_0 - \frac{\mu}{2}|\nabla u_0 + \nabla u_0^T|^2 - \lambda |\operatorname{div} u_0|^2 &= \rho_0^{1/2} g_1, \end{aligned} \] for some \((g_1,g_2)\) (in the case of the \((\rho,u,\Phi,P)\)-system, only the first condition must be assumed), the paper proves the existence and uniqueness of local in time strong solutions. It is important that the initial density may allow vacuum, i.e. \(\rh…
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 76D03
zbMATH DE Number: 5586777
zbMATH Keywords: full Navier-Stokes-Poisson equations
zbMATH Keywords: viscous polytropic fluids
zbMATH Keywords: local strong solution
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
reviewed by: Francisco Guillén-González
MaRDI profile type: Publication
full work available at URL: j.na.2009.01.074
OpenAlex ID: W2068869970
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DOI: 10.1016/J.NA.2009.01.074
description: scientific article; zbMATH DE number 5586777