Eventual smoothness of generalized solutions to a singular chemotaxis-Stokes system in 2D

By Tobias Black. Journal of Differential Equations, 2017

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Abstract We study the chemotaxis–fluid system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n c ∇ c ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + n ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u, where Ω ⊂ R 2 is a bounded domain with smooth boundary and ϕ ∈ C 2 ( Ω ¯ ) . From recent results it is known that for suitable regular initial data, the corresponding initial–boundary value problem possesses a global generalized solution. We will show that for small initial mass ∫ Ω n 0 these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties. Moreover, from the analysis of certain energy-type inequalities arising during the investigation of the eventual regularity, we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of n 0 in L 1 ( Ω ) and in L log ⁡ L ( Ω ) , u 0 in L 4 ( Ω ) , and of ∇ c 0 in L 2 ( Ω ) .

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/e5e5f8ab21df834f377ac2dbdfd61090e933a11a

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Citations in the corpus

Depleting the signal: Analysis of chemotaxis‐consumption models—A survey (2023, background): “3 ([11]) Suppose that Ω ⊂ R2 is a bounded domain with smooth boundary, and assume (3.”