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- Title
The fast signal diffusion limit in Keller–Segel(-fluid) systems.
- Authors
Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin
- Abstract
This paper deals with convergence of solutions to a class of parabolic Keller–Segel systems, possibly coupled to the (Navier–)Stokes equations in the framework of the full model ∂ t n ε + u ε · ∇ n ε = Δ n ε - ∇ · (n ε S (x , n ε , c ε) · ∇ c ε) + f (x , n ε , c ε) , ε ∂ t c ε + u ε · ∇ c ε = Δ c ε - c ε + n ε , ∂ t u ε + κ (u ε · ∇) u ε = Δ u ε + ∇ P ε + n ε ∇ ϕ , ∇ · u ε = 0 to solutions of the parabolic–elliptic counterpart formally obtained on taking ε ↘ 0 . In smoothly bounded physical domains Ω ⊂ R N with N ≥ 1 , and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on ∇ c ε and u ε are bounded in L λ ((0 , T) ; L q (Ω)) and in L ∞ ((0 , T) ; L r (Ω)) , respectively, for some λ ∈ (2 , ∞ ] , q > N and r > max { 2 , N } such that 1 λ + N 2 q < 1 2 . To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system. This general result will thereafter be concretized in the context of two examples: firstly, for an unforced Keller–Segel–Navier–Stokes system we shall establish a statement on global classical solutions under suitable smallness conditions on the initial data, and show that these solutions approach a global classical solution to the respective parabolic–elliptic simplification. We shall secondly derive a corresponding convergence property for arbitrary solutions to fluid-free Keller–Segel systems with logistic source terms, which in spatially one-dimensional settings turn out to allow for a priori estimates compatible with our general theory. Building on the latter in conjunction with a known result on emergence of large densities in the associated parabolic–elliptic limit system, we will finally discover some quasi-blowup phenomenon for the fully parabolic Keller–Segel system with logistic source and suitably small parameter ε > 0 .
- Subjects
STOKES equations; MATHEMATICAL logic; DIFFUSION
- Publication
Calculus of Variations & Partial Differential Equations, 2019, Vol 58, Issue 6, pN.PAG
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-019-1656-3