The small-convection limit in a two-dimensional Keller-Segel-Navier-Stokes system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Wu, Chunyan¹  Xiang, Zhaoyin¹ 

[1] Univ Elect Sci & Technol China, Sch Math Sci, Chengdu 611731, Sichuan, Peoples R China

关键词: Keller-Segel;    Navier Stokes;    Small-convection limit;   

DOI  :  10.1016/j.jde.2019.01.027

来源: Elsevier

【 摘 要 】

In this paper, we will investigate the small-convection limit of solutions (n(kappa), c(kappa), u(kappa)) to the Keller Segel Navier-Stokes system {n(t)(kappa) + u(kappa) . del n(kappa) = Delta n(kappa) - del . (n(kappa)(1 + n(kappa)))(-alpha del C kappa)), c(t)(kappa) + u(kappa) . del c(kappa) = Delta c(kappa) - c(kappa) + n(kappa), u(t)(kappa) + kappa(u(kappa) . del)u(kappa) = Delta u(kappa) - del P-kappa + n(kappa)del phi, del . u(kappa) = 0 along with no-flux boundary conditions for n(kappa) and c(kappa)and a no-slip boundary condition for u(kappa), and with suitable regular initial data in a bounded convex domain Omega subset of R-2 with smooth boundary. Our first result asserts that for general large data, (n(kappa), c(kappa), u(kappa)) will stabilize to (n(0), c(0), u(0)) with an explicit rate and a time dependent coefficient as kappa -> 0+. Our second result further reveals that such a convergence is uniform with respect to kappa at an exponential time decay rate provided that the initial data is suitable small. To the best of our knowledge, this seems to be the first rigorous theoretical analysis on the convergence of solutions of the Keller-Segel-Naiver Stokes system with signal production to the corresponding Keller-Segel-Stokes system as kappa -> 0+. (C) 2019 Elsevier Inc. All rights reserved.

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