Convergence of boundary layers of chemotaxis models with physical boundary conditions I: degenerate initial data
arxiv（2024）
摘要
The celebrated experiment of Tuval et al. showed
that the bacteria living a water drop can form a thin layer near the air-water
interface, where a so-called chemotaxis-fluid system with physical boundary
conditions was proposed to interpret the mechanism underlying the pattern
formation alongside numerical simulations. However, the rigorous proof for the
existence and convergence of the boundary layer solutions to the proposed model
still remains open. This paper shows that the model with physical boundary
conditions proposed in in one dimension can generate
boundary layer solution as the oxygen diffusion rate ε>0 is small.
Specifically, we show that the solution of the model with ε>0 will
converge to the solution with ε=0 (outer-layer solution) plus the
boundary layer profiles (inner-layer solution) with a sharp transition near the
boundary as ε→ 0. There are two major difficulties in
our analysis. First, the global well-posedness of the model is hard to prove
since the Dirichlet boundary condition can not contribute to the gradient
estimates needed for the cross-diffusion structure in the model. Resorting to
the technique of taking anti-derivative, we remove the cross-diffusion
structure such that the Dirichlet boundary condition can facilitate the needed
estimates. Second, the outer-layer profile of bacterial density is required to
be degenerate at the boundary as t → 0 ^+, which makes the
traditional cancellation technique incapable. Here we employ the Hardy
inequality and delicate weighted energy estimates to overcome this obstacle and
derive the requisite uniform-in-ε estimates allowing us to pass the
limit ε→ 0 to achieve our results.
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