Decay estimates of solutions to the bipolar compressible Euler–Poisson system in $\pmb {\mathbb {R}^3}$

Abstract

This work is concerned with the bipolar compressible Euler–Poisson equations with damping in three-dimensional space. We consider the optimal decay rates of the solution to the Cauchy problem, provided that the initial perturbation at the constant equilibrium state is sufficiently small. Under some assumptions of the initial data, we show that the solution to the Cauchy problem converges to its constant equilibrium state at the exact same $L^2$ decay rates as the linearized equations, which shows the convergence rates are sharp. The proof is based on the spectral analysis of the semigroup generated by the linearized equations and the nonlinear energy estimates.

Appendix A. Analytic tools

Lemma A.1

Let $r_1>1$, $0\le r_2\le r_1$, then it holds that

$$\begin{aligned} \int \limits _0^t(1+t-s)^{-r_1}(1+s)^{-r_2}\mathrm{d}s\le C(r_1,r_2)(1+t)^{-r_2}, \end{aligned}$$

(A.1)

where $C(r_1,r_2)$ is defined as

$$\begin{aligned} C(r_1,r_2)=\displaystyle {\frac{2^{r_2+1}}{r_1-1}}. \end{aligned}$$

Proof

The proof can be seen in [8]. $\square$

Lemma A.2

Let $l\ge 0$ be an integer, it holds

$$\begin{aligned} \left\| \nabla ^l(gh)\right\| _{L^{p_0}}\lesssim \left\| g\right\| _{L^{p_1}} \left\| \nabla ^{l}h\right\| _{L^{p_2}} +\left\| \nabla ^l g\right\| _{L^{p_3}} \left\| h\right\| _{L^{p_4}}. \end{aligned}$$

(A.2)

In the above, $p_0,p_1,p_2,p_3,p_4\in [1,+\infty ]$ such that

$$\begin{aligned} \frac{1}{p}_0=\frac{1}{p}_1+\frac{1}{p}_2=\frac{1}{p}_3+\frac{1}{p}_4. \end{aligned}$$

Proof

The proof can be seen in [7]. $\square$