Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation

Advances in Nonlinear Analysis

article

Yulan Wang¹  Michael Winkler²  Zhaoyin Xiang³ 

[1] School of Science, Xihua University;Institut für Mathematik, Universität Paderborn;School of Mathematical Sciences, University of Electronic Science and Technology of China

关键词: chemotaxis;    Stokes;    logistic source;    generalized solution;   

DOI  :  10.1515/anona-2020-0158

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

The Keller-Segel-Stokes system (*) n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n ∇ c ) + ρ n − μ n α , c t + u ⋅ ∇ c = Δ c − c + n , u t = Δ u + ∇ P − n ∇ Λ , ∇ ⋅ u = 0 , $$\begin{eqnarray*} \left\{ \begin{array}{lcll} n_t + u\cdot\nabla n &=& \it\Delta n - \nabla \cdot (n\nabla c) + \rho n - \mu n^\alpha, \\[1mm] c_t + u\cdot\nabla c &=& \it\Delta c-c+n, \\[1mm] u_t &=& \it\Delta u + \nabla P - n\nabla \it\Lambda, \qquad \nabla\cdot u =0, \end{array} \right. \end{eqnarray*}$$ is considered in a bounded domain Ω ⊂ ℝ 3 with smooth boundary, with parameters ρ ≥ 0, μ > 0 and α > 1, and with a given gravitational potential Λ ∈ W 2,∞ ( Ω ). It is shown that in this general setting, when posed under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u , and for any suitably regular initial data, an associated initial value problem possesses at least one globally defined solution in an appropriate generalized sense. Since it is well-known that in the absence of absorption, already the corresponding fluid-free subsystem with u ≡ 0 and μ = 0 admits some solutions blowing up in finite time, this particularly indicates that any power-type superlinear degradation of the form in (*) goes along with some significant regularizing effect.

【 授权许可】

CC BY