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Global boundedness of weak solutions for a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and rotation.
- Source :
-
Journal of Differential Equations . May2020, Vol. 268 Issue 11, p7047-7091. 45p. - Publication Year :
- 2020
-
Abstract
- This paper deals with the chemotaxis-Stokes system with nonlinear diffusion and rotation: n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n S (x , n , c) ⋅ ∇ c) , c t + u ⋅ ∇ c = Δ c − n c , u t + ∇ P = Δ u + n ∇ ϕ + f (x , t) and ∇ ⋅ u = 0 , in a bounded domain Ω ⊂ R 3 , where m > 0 , and ϕ : Ω ¯ → R , f : Ω ¯ × [ 0 , ∞) → R 3 and S : Ω ¯ × [ 0 , ∞) 2 → R 3 × 3 are given sufficiently smooth functions such that f is bounded in Ω × (0 , ∞) and S satisfies | S (x , n , c) | ≤ S 0 (c) (1 + n) − α for all (x , n , c) ∈ Ω ¯ × [ 0 , ∞) 2 with α > 0 and some nondecreasing function S 0 : [ 0 , ∞) → [ 0 , ∞). It is shown that if m + α > 10 9 and m + 5 4 α > 9 8 , then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Dirichlet initial-boundary problem possesses a globally bounded weak solution. This extends the previous global boundedness result for m > 9 8 and α = 0 [43] , and improves that for m ≥ 1 and m + α > 7 6 [34] , or for m + α > 7 6 in the associated fluid-free system [31]. Our proof consists at its core in using, inter alia , the maximal Sobolev regularity theory to elaborately derive some spatio-temporal estimates for the signal and the fluid equations so as to decouple the system. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 268
- Issue :
- 11
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 142165480
- Full Text :
- https://doi.org/10.1016/j.jde.2019.11.072