一类具有低正则初值的趋化–流体耦合模型解的整体存在性

Global Existence of Solutions for a Chemotaxis-Fluid Coupling Model with Low Regular Initial Data

摘要: 考虑在三维有界区域上带有logistic源的具有信号消耗机制的趋化–流体耦合方程组：的初边值问题，其中Ω⊂ℝ³是一个具有光滑边界的有界区域；n和c满足齐次Neumann边界条件，u满足Dirichlet边界条件；Φ∈W^2,∞(Ω)；r>0，μ>0，α>1是给定的参数。此前的结果表明：当初值满足n₀∈C⁰(Ω)时，该模型在三维有界凸区域上存在整体弱解。本文进一步研究了当初值条件正则性更低时，该模型弱解的整体存在性。具体而言，初值n₀满足n₀∈L¹(Ω)，该模型在三维有界凸区域上存在整体弱解。

Abstract: This paper mainly studies the initial-boundary value problem of chemotaxis-fluid coupling equations with signal consumption mechanism on a three-dimensional bounded domain with logistic source where Ω⊂ℝ³ is bounded domain with smooth boundary; n and c satisfy the homogeneous Neu-mann boundary condition and u satisfy the Dirichlet boundary condition; where Φ∈W^2,∞(Ω); where r>0, μ>0, and α>1 are given parameters. Previous results show that the initial value satisfies n₀∈C⁰(Ω), the model has a globally weak solution on the three-dimensional bounded convex region. This paper further examines when the initial value condition regularity is lower, the globally existence of weak solutions in this model. Specifically, the initial value n₀ satisfies n₀∈L¹(Ω), the model has a global weak solution on a three-dimensional bounded convex region.