Existence and large time behavior to coupled chemotaxis-fluid equations in Besov–Morrey spaces

By Minghua Yang, Zunwei Fu, Jinyi Sun. Journal of Differential Equations, 2019

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Abstract The paper deals with the Cauchy problem of coupled chemotaxis-fluid equations. By taking advantage of a coupling structure of the equations and using the semigroup approach, we show existence and asymptotic stability with small initial data and external force ( u 0 , n 0 , ∇ c 0 , c 0 ) ∈ N ˙ r 1 , λ , ∞ − β 1 × N ˙ r 2 , λ , ∞ − β 2 × N ˙ r 3 , λ , ∞ − β 3 × L ∞ , ∇ ϕ ∈ M N − λ , λ for certain technical assumptions. For the embedded relationship between function spaces, we show that our initial data class is larger than that of Kozono et al. (2016) [11] and covers physical cases of initial aggregation at points (Diracs) and on filaments. As an application, we obtain a class of asymptotically existence of a basin of attraction for each self-similar solutions with homogeneous initial data.

Subfields: Finance, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/8375a2a831fdc6cd97d8cc68199ab72e6dc148a9