Decay of the Navier-Stokes-Poisson equations

TL;DR

This paper establishes optimal time decay rates for solutions to the compressible Navier-Stokes-Poisson system, highlighting the roles of electric field dispersion and viscous dissipation in enhancing decay.

Contribution

It introduces a refined energy method to derive optimal decay rates, including higher-order derivatives, for the Navier-Stokes-Poisson system, improving understanding of its long-term behavior.

Findings

Optimal decay rates for solutions and derivatives are obtained.

Negative Sobolev norms are preserved and enhance decay.

Electric field dispersion and viscosity jointly improve decay rates.

Abstract

We establish the time decay rates of the solution to the Cauchy problem for the compressible Navier-Stokes-Poisson system via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The $\Dot{H}^{-s}$($0\le s<3/2$) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. As a corollary, we also obtain the usual $L^p$--$L^2$($1<p\le 2$) type of the optimal decay rates. Compared to the compressible Navier-Stokes system and the compressible irrotational Euler-Poisson system, our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density. Our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.