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Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces
Author(s) -
Yang Minghua
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4538
Subject(s) - mathematics , navier–stokes equations , compressibility , initial value problem , mathematical analysis , class (philosophy) , fourier transform , besov space , decomposition , non dimensionalization and scaling of the navier–stokes equations , hagen–poiseuille flow from the navier–stokes equations , cauchy distribution , mathematical physics , pure mathematics , functional analysis , physics , interpolation space , ecology , biochemistry , chemistry , artificial intelligence , biology , computer science , gene , thermodynamics
In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u F := e t Δ u 0 ; we prove that there exist 2 positive constants σ 0 and C 0 such that if the gravitational potential ϕ ∈B ˙p , 1 3 / p ( R 3 ) and the initial data ( u 0 , n 0 , c 0 ) satisfyu F · ∇ u FL 1 ( R + ;B ˙p , 1 − 1 + 3 / p ( R 3 ) ) +n 0 , c 0B ˙q , 1 − 2 + 3 / q ( R 3 ) ×B ˙q , 1 3 / q ( R 3 )× expC 0u 0B ˙p , 1 − 1 + 3 / p ( R 3 ) + 12⩽ σ 0for some p , q with 1 ⩽ p , q < ∞ , 1 p + 1 q > 1 3 , 1 ⩽ q < 6 and 1 min { p , q } − 1 max { p , q } ⩽ 1 3 , then the global solutions can be established in critical Besov spaces.