A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization

By Michael Winkler. Journal of Functional Analysis, 2019

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is considered in a bounded convex domain Ω ⊂ R with smooth boundary, where φ ∈ W (Ω) and f ∈ C1(Ω̄× [0,∞)), and where χ > 0, ρ ∈ R and μ > 0 are given parameters. It is proved that under the assumption that supt>0 ∫ t+1 t ‖f(·, s)‖ L 6 5 (Ω) ds be finite, for any sufficiently regular initial data (n0, c0, u0) satisfying n0 ≥ 0 and c0 ≥ 0, the initial-value problem for (⋆) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L1(Ω)× L6(Ω)× L(Ω;R). Moreover, under the explicit hypothesis that μ > χ √ ρ+ 4 , these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying

Subfields: Mathematical Physics, Modeling and Simulation, Public Health, Environmental and Occupational Health

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