Local well-posedness of the three dimensional compressible Euler-Poisson equations with physical vacuum (Q267065)

The motion of a self-gravitational inviscid gaseous star is studied in the paper. The mathematical model of the motion is the free boundary problem to the Euler-Poisson equations NEWLINE\[NEWLINE\begin{cases} \frac{\partial \rho}{\partial t}+\text{div}\,(\rho u)=0 &\text{in }\Omega(t), \\ \rho\left[\frac{\partial u}{\partial t}+u\cdot\nabla u\right]+\nabla p=\rho\nabla \phi &\text{in }\Omega(t),\\ -\Delta\phi=4\pi g\rho &\text{in }\Omega(t), \\ \nu(\Gamma(t))=u\cdot n(t) &\text{on }\Gamma(t), \\ (\rho,u)=(\rho_0,u_0) &\text{in }\Omega(0),\\ p=C\rho^\gamma,\quad \gamma>1,\quad C=\text{const}. \end{cases}\tag{1}NEWLINE\]NEWLINE Here \(\Omega(t)\subset\mathbb{R}^3\) is the time depended open, bounded domain occupied by a gas, \(\Gamma(t)=\partial\Omega(t)\), \(\nu(\Gamma(t))\) is the velocity of the boundary \(\Gamma(t)\), \(\rho(x,t)\) is the gas density, \(\rho>0\) in \(\Omega(t)\) and \(\rho=0\) in \(\mathbb{R}^3\setminus \Omega(t)\), \(u(x,t)\) is the velocity field, \(p(x,t)\) is the pressure, \(n(t)\) is the exterior unit normal to \(\Gamma(t)\), \(\phi\) is the potential, \(g\) is the gravity constant. The system (1) is added by the ``physical vacuum'' boundary condition NEWLINE\[NEWLINE-\infty<\frac{\partial }{\partial n_0}\,\frac{\partial p}{\partial \rho}(x,0)<0\quad \text{on}\;\Gamma, \tag{2}NEWLINE\]NEWLINE where \(\Gamma\equiv\Gamma(0)=\partial\Omega(0)\) is the initial boundary and \(n_0\) is the unit outward normal to \(\Gamma\).NEWLINENEWLINEIt is proved that the problem (1), (2) has a local in time solution for the initial data \((\rho_0,u_0)\) from certain weighted Sobolev spaces and for adiabatic exponent \(1<\gamma<3\).