Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?

Abstract

In a bounded planar domain varOmega with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system \t\t\t{nt+∇⋅(nu)=Δn−∇⋅(n∇c),x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t} + \\nabla \\cdot (nu) = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in \\varOmega , \\ t>0, \\\\ 0 = \\Delta c -c+n, & x\\in \\varOmega , \\ t>0, \\end{array}\\displaystyle \\right . \\end{aligned}$$ \\end{document} is considered, where u is a given sufficiently smooth velocity field on overline {varOmega }times [0,infty ) that is tangential on partial varOmega but not necessarily solenoidal.It is firstly shown that for any choice of n_{0}in C^{0}(overline {varOmega }) with int _{varOmega}n_{0}<4pi , this problem admits a global classical solution with n(cdot ,0)=n_{0}, and that this solution is even bounded whenever u is bounded and int _{varOmega}n_{0}<2pi . Secondly, it is seen that for each m>4pi one can find a classical solution with int _{varOmega}n(cdot ,0)=m which blows up in finite time, provided that varOmega satisfies a technical assumption requiring partial varOmega to contain a line segment.In particular, this indicates that the value 4pi of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.