Optimal decay rates of the solution for generalized Poisson–Nernst–Planck–Navier–Stokes equations in ${\mathbb {R}}^3$

Abstract

The Cauchy problem of a generalized Poisson–Nernst–Planck–Navier–Stokes system in ${\mathbb {R}}^3$ will be considered in this article. Based on the spectral analysis and the energy method, under some assumptions of the initial data, we obtain the lower bound and upper bound decay rates of the solution, which shows that the solution will converge to its constant equilibrium state at the same $L^2$-decay rates as the linearized one and the convergence rates are optimal.

Appendix A. Analytic tools

Lemma A.1

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

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

(3.43)

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

For $|\xi |\in [0, \eta ]$, we can derive from the conditions of Theorem 1.2 that

$$\begin{aligned} {\hat{U}}_0(0)\ne 0, \end{aligned}$$

(3.44)

and there exist some ${\bar{\xi }}_i\in (0,\xi )$, with $i=1,2,3$, such that

$$\begin{aligned} {\hat{U}}_0(\xi )={\hat{U}}_0(0)+\frac{\partial {\hat{U}}_0({\bar{\xi }}_i)}{\partial \xi }\xi . \end{aligned}$$

(3.45)

Proof

We can refer to [40] for the proof. $\square$