论傅立叶空间中具有流体效应的趋化系统的初始层和极限行为

Pub Date : 2024-11-06 DOI:10.1007/s10255-024-1134-3
Jian-xiang Wan, Hai-ping Zhong
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引用次数: 0

摘要

本文讨论了具有流体效应的趋化系统的考奇问题。\cdot \nabla {u^\varepsilon }- \Delta {u^\varepsilon }+ \nabla {{rm{P}}^\varepsilon }= {n^\varepsilon }\nabla {c^\varepsilon },}\hfill & {{\rm{in}} }\fill & {{\mathbb{R}^d}\times \left( {0,\infty } \right),} \hfill \cr {\nabla \cdot {u^\varepsilon } = 0,} \hfill \cr {\nabla \cdot {u^\varepsilon } = 0,}= 0,} \hfill & {{\rm{in}}*times *left }\times \left( {0,\infty } \right),} \hfill \cr {n_t^\varepsilon + {u^\varepsilon }\cdot \nabla {n^\varepsilon }- \Delta {n^\varepsilon }= - \nabla \cdot \left( {{n^\varepsilon }\nabla {c^\varepsilon }} \right),} \hfill & {{{\rm{in}}}*fill & {{mathbb{R}^d}\times \left( {0,\infty } \right),} \hfill \cr {{1 \over \varepsilon }c_t^\varepsilon - \Delta {c^\varepsilon }= {n^\varepsilon },} \hfill & {{\rm{in}}}\fill & {{\mathbb{R}^d}\times \left( {0,\infty } \right),} \hfill \cr {\left( {{u^\varepsilon },{n^\varepsilon },{c^\varepsilon }} \right){|_{t = 0}} = \left( {{u_0},{n_0},{c_0}} \right),} \hfill & {{\rm{in}}\fill & {{\mathbb{R}^d},} \fill \cr }}\right.$$ 其中 d ≥ 2。众所周知,对于每个 ϵ >0和所有属于特定傅立叶空间的足够小的初始数据(u0, n0, c0),问题在傅立叶空间具有唯一的全局解(uϵ, nϵ, cϵ)。此外,我们还证明了 cϵ(t)在ϵ-1→0时具有初始层。
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On the Initial Layer and the Limit Behavior for Chemotaxis System with the Effect of Fluid in Fourier Space

The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid

$$\left\{ {\matrix{ {u_t^\varepsilon + {u^\varepsilon } \cdot \nabla {u^\varepsilon } - \Delta {u^\varepsilon } + \nabla {{\rm{P}}^\varepsilon } = {n^\varepsilon }\nabla {c^\varepsilon },} \hfill & {{\rm{in}}} \hfill & {{\mathbb{R}^d} \times \left( {0,\infty } \right),} \hfill \cr {\nabla \cdot {u^\varepsilon } = 0,} \hfill & {{\rm{in}}} \hfill & {{\mathbb{R}^d} \times \left( {0,\infty } \right),} \hfill \cr {n_t^\varepsilon + {u^\varepsilon } \cdot \nabla {n^\varepsilon } - \Delta {n^\varepsilon } = - \nabla \cdot \left( {{n^\varepsilon }\nabla {c^\varepsilon }} \right),} \hfill & {{\rm{in}}} \hfill & {{\mathbb{R}^d} \times \left( {0,\infty } \right),} \hfill \cr {{1 \over \varepsilon }c_t^\varepsilon - \Delta {c^\varepsilon } = {n^\varepsilon },} \hfill & {{\rm{in}}} \hfill & {{\mathbb{R}^d} \times \left( {0,\infty } \right),} \hfill \cr {\left( {{u^\varepsilon },{n^\varepsilon },{c^\varepsilon }} \right){|_{t = 0}} = \left( {{u_0},{n_0},{c_0}} \right),} \hfill & {{\rm{in}}} \hfill & {{\mathbb{R}^d},} \hfill \cr } } \right.$$

where d ≥ 2. It is known that for each ϵ > 0 and all sufficiently small initial data (u0, n0, c0) belongs to certain Fourier space, the problem possesses a unique global solution (uϵ, nϵ, cϵ) in Fourier space. The present work asserts that these solutions stabilize to (u, n, c) as ϵ−1 → 0. Moreover, we show that cϵ(t) has the initial layer as ϵ−1 → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.

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