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

2017 ◽  
Vol 16 (5) ◽  
pp. 1617-1639 ◽  
Author(s):  
Minghua Yang ◽  
◽  
Jinyi Sun ◽  
Keyword(s):  
Besov Spaces ◽  
Navier Stokes ◽  
Poisson System ◽  
Gevrey Regularity ◽  
Regularity And Existence ◽  
Critical Besov Spaces

scholarly journals On Gevrey regularity of the supercritical SQG equation in critical Besov spaces

2015 ◽  
Vol 269 (10) ◽  
pp. 3083-3119 ◽  
Author(s):  
Animikh Biswas ◽  
Vincent R. Martinez ◽  
Prabath Silva
Keyword(s):  
Besov Spaces ◽  
Gevrey Regularity ◽  
Critical Besov Spaces

Analyticity and Existence of the Keller–Segel–Navier–Stokes Equations in Critical Besov Spaces

2018 ◽  
Vol 18 (3) ◽  
pp. 517-535 ◽  
Author(s):  
Minghua Yang ◽  
Zunwei Fu ◽  
Suying Liu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Gevrey Regularity ◽  
Navier Stokes Equations ◽  
Global Existence Of Solutions ◽  
Regularity Estimates ◽  
Gevrey Estimates ◽  
The Cauchy Problem ◽  
Critical Besov Spaces ◽  
Derivatives Of

Abstract This paper deals with the Cauchy problem to the Keller–Segel model coupled with the incompressible 3-D Navier–Stokes equations. Based on so-called Gevrey regularity estimates, which are motivated by the works of Foias and Temam [20], we prove that the solutions are analytic for a small interval of time with values in a Gevrey class of functions. As a consequence of Gevrey estimates, we particularly imply higher-order derivatives of solutions in Besov and Lebesgue spaces. Moreover, we prove that the existence of a positive constant {\tilde{C}} such that the initial data {(u_{0},n_{0},c_{0}):=(u_{0}^{h},u_{0}^{3},n_{0},c_{0})} satisfy \tilde{C}\bigl{(}\lVert(n_{0},c_{0})\rVert_{\dot{B}^{-2+3/q}_{q,1}(\mathbb{R}^% {3})\times\dot{B}^{3/q}_{q,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}% ^{-1+3/p}_{p,1}(\mathbb{R}^{3})}+\lVert u_{0}^{h}\rVert_{\dot{B}^{-1+3/p}_{p,1% }(\mathbb{R}^{3})}^{\alpha}\lVert u_{0}^{3}\rVert_{\dot{B}^{-1+3/p}_{p,1}(% \mathbb{R}^{3})}^{1-\alpha}\bigr{)}\leq 1 for certain conditions on {p,q} and α implies the global existence of solutions with large initial vertical velocity component.

ILL-POSEDNESS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH THE VELOCITY IN FRAMEWORK

2017 ◽  
Vol 18 (4) ◽  
pp. 829-854 ◽  
Author(s):  
Jiecheng Chen ◽  
Renhui Wan
Keyword(s):  
Initial Data ◽  
Besov Space ◽  
Besov Spaces ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Coupling Term ◽  
Navier Stokes Equations ◽  
Critical Besov Space ◽  
Image Position ◽  
Critical Besov Spaces

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying $$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$ To accomplish this goal, we require a norm inflation coming from the coupling term $L(a)\unicode[STIX]{x1D6E5}u$ instead of $u\cdot \unicode[STIX]{x1D6FB}u$ and construct a new decomposition of the density.

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