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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 18, Issue 3

Issues

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

Minghua YangORCID iD: http://orcid.org/0000-0001-7704-2442 / Zunwei FuORCID iD: http://orcid.org/0000-0001-9109-4142 / Suying LiuORCID iD: http://orcid.org/0000-0001-8581-7551
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 C~ such that the initial data (u0,n0,c0):=(u0h,u03,n0,c0) satisfy

C~((n0,c0)B˙q,1-2+3/q(3)×B˙q,13/q(3)+u0hB˙p,1-1+3/p(3)+u0hB˙p,1-1+3/p(3)αu03B˙p,1-1+3/p(3)1-α)1

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

Keywords: Keller–Segel System; Navier–Stokes Equation; Gevrey Regularity; Global Solution; Besov Space

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

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About the article


Received: 2017-03-24

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, DOI: https://doi.org/10.1515/ans-2017-6046.

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