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Volume 18, Issue 3

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

Minghua Yang
/ Zunwei Fu
/ Suying Liu
Published Online: 2018-01-20 | DOI: https://doi.org/10.1515/ans-2017-6046

## 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 $\stackrel{~}{C}$ such that the initial data $\left({u}_{0},{n}_{0},{c}_{0}\right):=\left({u}_{0}^{h},{u}_{0}^{3},{n}_{0},{c}_{0}\right)$ satisfy

$\stackrel{~}{C}\left({\parallel \left({n}_{0},{c}_{0}\right)\parallel }_{{\stackrel{˙}{B}}_{q,1}^{-2+3/q}\left({ℝ}^{3}\right)×{\stackrel{˙}{B}}_{q,1}^{3/q}\left({ℝ}^{3}\right)}+{\parallel {u}_{0}^{h}\parallel }_{{\stackrel{˙}{B}}_{p,1}^{-1+3/p}\left({ℝ}^{3}\right)}+{\parallel {u}_{0}^{h}\parallel }_{{\stackrel{˙}{B}}_{p,1}^{-1+3/p}\left({ℝ}^{3}\right)}^{\alpha }{\parallel {u}_{0}^{3}\parallel }_{{\stackrel{˙}{B}}_{p,1}^{-1+3/p}\left({ℝ}^{3}\right)}^{1-\alpha }\right)\le 1$

for certain conditions on $p,q$ and α implies the global existence of solutions with large initial vertical velocity component.

MSC 2010: 35Q30; 35Q335; 76D03; 35E15; 42B37

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Revised: 2018-01-04

Accepted: 2018-01-07

Published Online: 2018-01-20

Published in Print: 2018-08-01

Funding Source: National Science Foundation

Award identifier / Grant number: 11671185

Award identifier / Grant number: 11771195

Funding Source: Natural Science Foundation of Shandong Province

Award identifier / Grant number: ZR2017MA041

This work was partially supported by NSF of China (grant nos. 11671185, 11771195) and NSF Shandong Province (grant no. ZR2017MA041).

Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 3, Pages 517–535, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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