Local existence of the strong solutions for the full Navier-Stokes-Poisson equations
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. \(\rho_0\geq 0\). The proof is based on two main steps: firstly a linearized problem is considered, deriving some local estimates for the solutions independent of the lower bound of the initial density. Secondly, a classical iteration argument based on the uniform estimates is applied.
scientific article; zbMATH DE number 5586777