Chemotaxis-consumption system with Robin boundary conditions coupled to the (Navier-)Stokes equations

초록

In this paper, we consider the chemotaxis-consumption system on a bounded smooth domain ω C Rn, n = 2,3, with fluid coupling ρt + u δρδ (D(ρ)δρ) = δ (ρS(x, ρ, c) δc),u δc-δc = ρc,ut + (u δ)uδu+δφ = ρδω, δ u = 0, subject to the boundary conditions ν × (D(ρ)ρ + ρS(x, ρ, c)c)|ω = 0, (ν × c + c)|ω = γ, and u|ω = 0. When (n, κ) = (2, 1), we establish the global existence and uniform boundedness of classical solutions for all suitably regular initial data, under general structural conditions on the tensor-valued sensitivity S and a strictly positive lower bound on the diffusivity D. In case (n, κ) = (3, 0), we show that the same result holds provided that D meets a certain diffusion enhancement condition depending on γ. Moreover, we construct finite-Time blow-up solutions for the radially symmetric, fluid-free system when n = 2, 3, D(ξ) ≲ (1 + ξ) m-1 with 0 < m < 2 n 0 m 2 n and S I n × n S I-n ×n. We prove that, for any prescribed initial mass, blow-up occurs when γ is sufficiently large. © 2026 the author(s), published by De Gruyter, Berlin/Boston 2026.

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