Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces

Minghua Yang; Jinyi Sun

doi:10.3934/cpaa.2017078

Article Contents

2017, Volume 16, Issue 5: 1617-1639. Doi: 10.3934/cpaa.2017078

This issue Previous Article Positive solutions for quasilinear Schrödinger equations in $\mathbb{R}^N$ Next Article Existence and concentration for Kirchhoff type equations around topologically critical points of the potential

Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces

Minghua Yang^1, and
Jinyi Sun^2, ,

1.
School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

^* Corresponding author
^* Corresponding author

Received: July 31, 2016

Revised: February 28, 2017

Published: October 2017

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Abstract

The paper deals with the Cauchy problem of Navier-Stokes-Nernst-Planck-Poisson system (NSNPP). First of all, based on so-called Gevrey regularity estimates, which is motivated by the works of Foias and Temam [J. Funct. Anal., 87 (1989), 359-369], we prove that the solutions are analytic in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly obtain higher-order derivatives of solutions in Besov and Lebesgue spaces. Finally, we prove that there exists a positive constant $\mathbb{C}$ such that if the initial data $(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$ satisfies

$\begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}$

for $p, q, α$ with $1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$, then global existence of solutions with large initial vertical velocity component is established.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 35E15.

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