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 (u^ϵ, n^ϵ, c^ϵ) in Fourier space. The present work asserts that these solutions stabilize to (u^∞, n^∞, c^∞) as ϵ⁻¹ → 0. Moreover, we show that c^ϵ(t) has the initial layer as ϵ⁻¹ → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.