Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three

TL;DR

This paper derives pointwise asymptotic estimates for solutions to the three-dimensional bipolar Navier-Stokes-Poisson system, revealing a generalized Huygens' principle and extending decay rate results in various Lebesgue spaces.

Contribution

It provides the first detailed pointwise estimates for the bipolar system, highlighting differences from the unipolar case and extending decay rate results.

Findings

Solution exhibits generalized Huygens' principle.

Extended decay rates from L^2 to L^p for p>1.

Improved decay estimates compared to previous results.

Abstract

The Cauchy problem of the bipolar Navier-Stokes-Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit generalized Huygens' principle as the Navier-Stokes system. This phenomenon is the the most important difference from the unipolar Navier-Stokes-Poisson system. Due to non-conservative structure of the system (1.1) and interplay of two carriers which counteracts the influence of electric field (a nonlocal term), some new observations are essential for the proof. We make full use of the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier-Stokes-Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual $L^2(\mathbb{R}^3)$-decay rate to $L^p(\mathbb{R}^3)$-decay rate with $p>1$ and also improve former decay rates in part.