Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains?
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Can Fluid Interaction Influence the Critical Mass for Taxis-Driven Blow-up in Bounded Planar Domains? Michael Winkler1
Received: 26 July 2019 / Accepted: 16 January 2020 © Springer Nature B.V. 2020
Abstract In a bounded planar domain Ω with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system nt + ∇ · (nu) = n − ∇ · (n∇c), x ∈ Ω, t > 0, 0 = c − c + n,
x ∈ Ω, t > 0,
is considered, where u is a given sufficiently smooth velocity field on Ω × [0, ∞) that is tangential on ∂Ω but not necessarily solenoidal. It is firstly shown that for any choice of n0 ∈ C 0 (Ω) with Ω n0 < 4π, this problem admits a global classical solution with n(·, 0) = n0 , and that this solution is even bounded whenever u is bounded and Ω n0 < 2π. Secondly, it is seen that for each m > 4π one can find a classical solution with Ω n(·, 0) = m which blows up in finite time, provided that Ω satisfies a technical assumption requiring ∂Ω to contain a line segment. In particular, this indicates that the value 4π 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. Keywords Chemotaxis · Critical mass · Blow-up Mathematics Subject Classification (2010) Primary 35B44 · 35K55 · 92C17 · Secondary 35Q92
1 Introduction Understanding possible effects of fluid interaction on chemotaxis systems has been the objective of considerable efforts in the mathematical literature during the past decade. Motivated by experimentally obtained results reporting significant influences of corresponding
B M. Winkler
[email protected]
1
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
M. Winkler
transport mechanisms on the structure-enhancing potential of aggregation due to attractive cross-diffusion ([23]), a noticeable literature has been concerned with various types of associated chemotaxis-fluid systems. Even in the most complex case in which, according to the modeling approach presented in [23], the fluid velocity itself is an unknown system variable according to buoyancy-induced feedback effects of cells on the fluid flow, beyond establishing basic solution theories ([3, 4, 7, 8, 12, 21, 24, 25, 27, 29, 31]) it has been possible to address aspects related to qualitative solution behavior in some situations ([4, 5, 14, 28, 30, 32, 33, 35]). While the latter class of findings seems yet limited to results identifying conditions under which spatial homogeneity ultimately prevails due to dominance of various dissipative mechanisms, somewhat deeper insight with regard to genuine structure formation could be gained upon renouncing any feedback of the considered population on the fluid evolution, thus considering the fluid flow as an externally given system ingredient. Examples in this direction address cor
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