Local smooth solutions to the Euler-Poisson equations for semiconductor in vacuum

Abstract

In this paper, we study the initial-boundary value problem to the Euler-Poisson equations for semiconductors, which involves the vacuum for the electronic density, a challenging case because of its degeneracy and singularity. The main issue is to investigate the local well-posedness of smooth solutions to the isentropic system with an adiabatic exponent $\gamma >1$, a degenerate hyperbolic-elliptic system on the free boundary. By setting the system to the Lagrangian coordinates, we reduce it to the quasi-linear wave equation coupling the Poisson equations, where the initial degeneracy can be explicitly expressed by the function ${\rho }_0^{\gamma -1}$ of the initial density ${\rho }_0$, which equals to the distance function near boundaries. By applying the Hardy inequality and weighted Sobolev spaces depending on the distance function, we can overcome the degeneracy and singularity of the system caused by the vacuum, and we technically establish some crucial priori estimates and then prove the existence and uniqueness of the local smooth solution. This is the first result on the smooth solution to the Euler-Poisson equations for semiconductors in vacuum.