Global large-data generalized solutions in a two-dimensional chemotaxis-Stokes system with singular sensitivity

Boundary value problems

Yilong Wang¹ 

[1] School of Sciences, Southwest Petroleum University, Chengdu, China

关键词: chemotaxis;    Stokes;    global existence;    generalized solutions;    35Q30;    35Q35;    35K55;    35Q92;    92C17;   

DOI  :  10.1186/s13661-016-0687-3

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

This paper considers the following chemotaxis-Stokes system:{nt+u⋅∇n=Δn−∇⋅(nc∇c),ct+u⋅∇c=Δc−nc,ut=Δu+∇P+n∇ϕ,∇⋅u=0,$$ \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot\nabla n=\Delta n-\nabla\cdot(\frac{n}{c}\nabla c), \\ c_{t}+u\cdot\nabla c=\Delta c-nc, \\ u_{t}=\Delta u+\nabla P+n\nabla\phi, \\ \nabla\cdot u=0, \end{array}\displaystyle \right . $$in two-dimensional smoothly bounded domains, which can be seen as a model to describe the migration of aerobic bacteria swimming in an incompressible fluid. It is proved that the corresponding initial-boundary value problem possesses a global generalized solution for any sufficiently regular initial data(n0,c0,u0)$(n_{0}, c_{0}, u_{0})$satisfyingn0≥0$n_{0}\geq0$andc0>0$c_{0}>0$. Moreover, the solution component c satisfiesc(⋅,t)⇀⋆0$c(\cdot,t)\overset{\star}{\rightharpoonup}0$inL∞(Ω)$L^{\infty}(\Omega )$ast→∞$t\rightarrow\infty$andc(⋅,t)→0$c(\cdot,t)\rightarrow0$inLp(Ω)$L^{p}(\Omega)$ast→∞$t\rightarrow\infty$for anyp∈[1,∞)$p\in[1,\infty)$. To the best of our knowledge, this is the first result on global solvability in a chemotaxis-Stokes system with singular sensitivity and signal absorption.

【 授权许可】

CC BY