Gevrey regularity and existence of Navier-Stokes-Nernst-Planck-Poisson system in critical Besov spaces
Minghua Yang Jinyi Sun
Communications on Pure & Applied Analysis 2017, 16(5): 1617-1639 doi: 10.3934/cpaa.2017078
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
such that if the initial data
$(u_{0}, n_{0}, c_{0})=(u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$
$\begin{aligned}&\|(n_{0}, c_{0},u_{0}^{h})\|_{\dot{B}^{-2+3/q}_{q, 1}× \dot{B}^{-2+3/q}_{q, 1}×\dot{B}^{-1+3/p}_{p, 1}}+\|u_{0}^{h}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{α}\|u_{0}^{3}\|_{\dot{B}^{-1+3/p}_{p, 1}}^{1-α}≤q1/\mathbb{C}\end{aligned}$
$p, q, α$
$1<p<q≤ 2p<\infty, \frac{1}{p}+\frac{1}{q}>\frac{1}{3}, 1< q<6, \frac{1}{p}-\frac{1}{q}≤\frac{1}{3}$
, then global existence of solutions with large initial vertical velocity component is established.
keywords: Nernst-Planck-Poisson system Navier-Stokes system Gevrey regularity global solutions Besov spaces
Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces
Minghua Yang Zunwei Fu Jinyi Sun
Discrete & Continuous Dynamical Systems - B 2018, 23(8): 3427-3460 doi: 10.3934/dcdsb.2018228
In this article, we consider the Cauchy problem to chemotaxis model coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish the global-in-time existence of the solution when the gravitational potential ϕ and the small initial data
$(u_{0}, n_{0}, c_{0})$
in critical Besov spaces under certain conditions. Moreover, we prove that there exist two positive constants σ0 and
such that if the gravitational potential
$\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$
and the initial data
$(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$
$\begin{equation*}\begin{aligned} &\left(\left\|u_{0}^{h}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+\left\|\left(n_{0}, c_{0}\right)\right\|_{\dot{B}^{-2+3/q}_{q, 1}(\mathbb{R}^3) \times \dot{B}^{3/q}_{q, 1}(\mathbb{R}^3)}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left\{C_{0}\left(\left\|u_{0}^{3}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+1\right)^{2}\right\} \leq \sigma_{0}\end{aligned}\end{equation*}$
for some
$p, q$
$1<p, q<6,\frac{1}{p}+\frac{1}{q}>\frac{2}{3}$
$\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$
, then the global existence results can be extended to the global solutions without any small conditions imposed on the third component of the initial velocity field
in critical Besov spaces with the aid of continuity argument. Our initial data class is larger than that of some known results. Our results are completely new even for three-dimensional chemotaxis-Navier-Stokes system.
keywords: Chemotaxis-Navier-Stokes equation global solution Besov space Littlewood-Paley theory

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