Global classical solutions to a two-dimensional chemotaxis-fluid system involving signal-dependent degenerate diffusion

2025/09/21 by Yansheng Ma, Ma, Yansheng, Peter Y. H. Pang +2
Mathematics · Computer Science · #Mathematical Biology Tumor Growth #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.2509.17073

Abstract

This paper is concerned with the two-dimensional chemotaxis-fluid model \begincases n_t+u⋅∇ n=Δ(nϕ(v))+μn(1-n),
v_t+u⋅∇ v=Δv-nv,
u_t+ κ(u⋅∇) u=Δu+n∇Φ-∇ P, ∇⋅ u=0, \endcases accounting for signal-dependent motilities of microbial populations interacting with an incompressible liquid through transport and buoyancy, where the suitably smooth function ϕ satisfies ϕ>0 on (0,∞) with ϕ(0)=0 and ϕ'(0)>0, and the parameter μ≥ 0. For all reasonably regular initial data, if μ=0, the corresponding initial boundary value problem possesses global classical solutions with a smallness condition on ∫_Ωn₀; whereas if μ>0, this problem possesses global bounded classical solutions, which can converge toward (1,0,0) as time tends to infinity when a certain small mass is imposed on the initial data v₀. These results extend recent results for the fluid-free system to one in a Navier-Stokes fluid environment.

Citations

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• How far do chemotaxis-driven forces influence regularity in the\n Navier-Stokes system?
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• Global solutions for a 2D chemotaxis-fluid system with large measures as initial density and vorticity
• Global bounded and unbounded solutions to a chemotaxis system with\n indirect signal production