An Application of BMO-type Space to Chemotaxis-fluid Equations

Abstract

We consider a Keller–Segel model coupled to the incompressible Navier–Stokes system in 3-dimensional case. We prove that the system has a unique local solution when (u₀, n₀, c₀) ∈ Φ ¹₀₁ × Φ ²₀₁ × Φ ³₀₁ , where Φ ¹₀₁ × Φ ²₀₁ × Φ ³₀₁ is a subspace of $\text{bmo}^{-1}(\mathbb{R}^{3})\times\dot{B}_{p,\infty}^{-2+3/p}(\mathbb{R}^{3})\times(\dot{B}_{q,\infty}^{3/q}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^3))$. Furthermore, we obtain that the system exists a unique global solution for any small initial data $(u_{0},n_{0},c_{0})\in\text{BMO}^{-1}(\mathbb{R}^{3})\times\dot{B}_{p,\infty}^{-2+3/p}(\mathbb{R}^{3})\times(\dot{B}_{q,\infty}^{3/q}(\mathbb{R}^{3})\cap L^{\infty}(\mathbb{R}^3))$. For the difference between these spaces and known ones, our results may be regarded as a new existence theorem on the system.