CPAA
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
$\mathbb{C}$
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.
keywords: Nernst-Planck-Poisson system Navier-Stokes system Gevrey regularity global solutions Besov spaces
DCDS-B
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
$C_{0}$
such that if the gravitational potential
$\phi \in \dot B_{p,1}^{3/p}({\mathbb{R}^3})$
and the initial data
$(u_{0}, n_{0}, c_{0}): = (u_{0}^{1}, u_{0}^{2}, u_{0}^{3}, n_{0}, c_{0}): = (u_{0}^{h}, u_{0}^{3}, n_{0}, c_{0})$
satisfies
$\begin{equation*}\begin{aligned} &\left(\left\|u_{0}^{h}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+\left\|\left(n_{0}, c_{0}\right)\right\|_{\dot{B}^{-2+3/q}_{q, 1}(\mathbb{R}^3) \times \dot{B}^{3/q}_{q, 1}(\mathbb{R}^3)}\right)\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left\{C_{0}\left(\left\|u_{0}^{3}\right\|_{\dot{B}^{-1+3/p}_{p, 1}(\mathbb{R}^3)}+1\right)^{2}\right\} \leq \sigma_{0}\end{aligned}\end{equation*}$
for some
$p, q$
with
$1<p, q<6,\frac{1}{p}+\frac{1}{q}>\frac{2}{3}$
and
$\frac{1}{\min\{p, q\}}-\frac{1}{\max\{p, q\}} \le \frac{1}{3}$
, 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
$u_{0}^{3}$
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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