Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:261

Article

Wang, Yulan¹  Xiang, Zhaoyin² 

[1] Xihua Univ, Sch Sci, Chengdu 610039, Peoples R China

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

关键词: Keller-Segel-Stokes system;    Tensor-valued sensitivity;    Global existence;    Boundedness;   

DOI  :  10.1016/j.jde.2016.07.010

来源: Elsevier

【 摘 要 】

In this paper we continue to deal with the initial boundary value problem for the coupled Keller-Segel- Stokes system {n(t) + u center dot del n = Delta n - del center dot(nS(x,n,c)center dot del c), (x,t) is an element of Omega x (0,T), c(t) + u center dot del c = Delta c - c+n, (x,t) is an element of Omega x (0,T), u(t) + del P = Delta u + n del phi, (x,t) is an element of Omega x (0,T), del center dot u = 0, (x,t) is an element of Omega x (0,T), where Omega subset of R-d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R-dxd, and satisfies vertical bar S(x,n,c)vertical bar <= C-s(1+ n)(-alpha) with some C-s > 0 and alpha > 0. When d = 2, our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption alpha > 0, which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3, we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption alpha > 1/2, which is slightly stronger than the corresponding subcritical assumption alpha > 1/3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L-P regularity of u from a rather low L-P regularity of n, which will be obtained by a new combinational functional. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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