SCIENCE@home

Global solvability of chemotaxis–fluid systems with nonlinear diffusion and matrix-valued sensitivities in three dimensions

Tobias Black

Published 2018 in Nonlinear Analysis

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.

PUBLICATION RECORD

CITATION MAP

EXTRACTION MAP

CLAIMS

  • No claims are published for this paper.

CONCEPTS

  • No concepts are published for this paper.

REFERENCES

Showing 1-30 of 30 references · Page 1 of 1

CITED BY

Showing 1-29 of 29 citing papers · Page 1 of 1