The initial value problem of the fractional compressible Navier-Stokes-Poisson system

Abstract

We consider the initial value problem to the fractional generalized compressible Navier-Stokes-Poisson equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations and the diffusion term for the internal electrostatic potential are described respectively by the nonlocal fractional Laplace operators. The global-in-time existence of the smooth solution is proven under the assumption that the initial data are given in a small neighborhood of a constant state in the sense of Sobolev's space. The optimal decay rates depending upon the orders of two fractional Laplace operators are established, and that the momentum of the fractional Navier-Stokes-Poisson system exhibits a slower convergence rate in time to the constant state compared to that of the fractional compressible Navier-Stokes system is also shown.