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# Communications in Mathematical Sciences

## Volume 21 (2023)

### Number 2

### Optimal large time behavior of the compressible bipolar Navier–Stokes–Poisson system

Pages: 323 – 349

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n2.a2

#### Authors

#### Abstract

This paper is concerned with the Cauchy problem of the 3D compressible bipolar Navier–Stokes–Poisson (BNSP) system. Our main purpose is three-fold: First, under the assumption that $H^l \cap L^1 (l \leq 3)$-norm of the initial data is small, we prove the time decay rates of the solution as well as its spatial derivatives from the first-order to the highest-order. Similar to the results on the heat equation and the compressible Navier–Stokes equations, these decay rates for the BNSP system are optimal. Second, for well–chosen initial data, we also show the lower bounds on the decay rates. Third, we give the explicit influences of the electric field on the qualitative behaviors of solutions, which are totally new as compared to the results for the compressible unipolar Navier–Stokes–Poisson (UNSP) system [H.L. Li et al., *Arch. Ration. Mech. Anal.*, 196:681–713, 2010; Y.J. Wang, *J. Differ. Equ.*, 253:273–297, 2012]. More precisely, we show that the densities of the BNSP system converge to their corresponding equilibriums at the same $L^2$-rate $(1+t)^{-\frac{3}{4}}$ as the compressible Navier–Stokes equations, but the momentums of the BNSP system decay at the $L^2$-rate $(1+t)^{-\frac{3}{4} (\frac{1}{p}-\frac{1}{2})}$ with $1 \leq p \leq \frac{3}{2}$, which depend directly on the initial low frequency assumption of electric field, namely, the smallness of ${\lVert \nabla \phi_0 \rVert}_{L^p} \;$. This phenomenon is the most important difference from the compressible Navier–Stokes equations.

#### Keywords

bipolar compressible Navier–Stokes–Poisson system, unequal viscosities, optimal time decay rates

#### 2010 Mathematics Subject Classification

35B40, 35D35

Qing Chen’s research is supported in part by National Natural Science Foundation of China (Nos. 12271114, 12171401) and National Science Foundation of Fujian Province, China (No. 2022J011241).

Guochun Wu’s research is partially supported by National Natural Science Foundation of China (No. 12271114) and National Science Foundation of Fujian Province, China (No. 2022J01304).

Yinghui Zhang’s research is partially supported by National Natural Science Foundation of China (No. 12271114) and Guangxi Natural Science Foundation (No. 2019JJG110003, 2019AC20214).

Received 19 September 2021

Received revised 8 May 2022

Accepted 9 May 2022

Published 1 February 2023