Stability of steady-state solutions to Navier-Stokes-Poisson systems
The authors approach the well-known Navier-Stokes-Poisson system which describes the dynamics of electrons in semiconductors with a viscosity term in momentum equations. In fact, they consider a Cauchy problem because the system is supplemented by an initial condition. In this context, the stability problem is to investigate the global existence of smooth solutions when the initial data are close to the steady-state. In previous studies, this problem is solved in the Sobolev spaces or Besov spaces framework by using the classical energy method and linearized method. In the situation of the present study, these methods no longer work due to the appearance of lower order terms which will make essential difficulties in the energy estimates. Then, the authors solve this problem by using an anti-symmetric matrix technique and employing an induction argument on the order of the derivatives of solutions in energy estimates. So, they obtain refined estimates by use of classical Sobolev embedding theorems and integrations by parts. This allows to use a slightly different induction argument. The two important results of this study are: 1) demonstration of global existence of solutions to the above mentioned Cauchy problem; 2) the solution tends to the steady-state for large time.
scientific article; zbMATH DE number 6864632
scientific article