Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions. (March 2019)

• Record Type: Journal Article
• Authors: Black, Tobias
• Is Part Of: Nonlinear analysis. Volume 180(2019)
• Page Start: 129
• Page End: 153
• Publication Date: 2019-03
• Subjects: primary 35D30 35D99 -- secondary 35K55 35A01 35Q92 35Q35 92C17; Chemotaxis–fluid -- Porous medium type diffusion -- Global weak and generalized solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.10.003
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this work we extend a recent result to chemotaxis–fluid systems which include matrix-valued sensitivity functions S ( x, n, c ) : Ω × [ 0, ∞ ) 2 → R 3 × 3 in addition to the porous medium type diffusion, which were discussed in the previous work. Namely, we will consider the system n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ ( n S ( x, n, c ) ∇ c ), x ∈ Ω, t > 0, c t + u ⋅ ∇ c = Δ c − c + n, x ∈ Ω, t > 0, u t + ( u ⋅ ∇ ) u = Δ u + ∇ P + n ∇ ϕ, x ∈ Ω, t > 0, ∇ ⋅ u = 0, x ∈ Ω, t > 0, in a bounded domain Ω ⊂ R 3 with smooth boundary. Assuming that m ≥ 1, α ≥ 0 satisfy m + α > 4 3, that the matrix-valued function S ( x, n, c ) : Ω × [ 0, ∞ ) 2 → R 3 × 3 satisfies | S ( x, n, c ) | ≤ S 0 ( 1 + n ) α for some S 0 > 0 and suitably regular nonnegative initial data, we show that the corresponding no-flux-Dirichlet boundary value problem emits at least one global very weak solution. Upon comparison with results for the fluid-free system this condition appears to be optimal. Moreover, imposing a stronger condition for the exponents m and α, i.e. m + 2 α > 5 3, we will establish the existence of at least one global weak solution in the standard sense.