A Smallness Condition Ensuring Boundedness in a Two-dimensional Chemotaxis-Navier—Stokes System involving Dirichlet Boundary Conditions for the Signal

Abstract

The chemotaxis-Navier—Stokes system

$$\left\{{\matrix{{{n_t} + u \cdot \nabla n = \Delta n - \nabla \cdot \left({n\nabla c} \right),} \hfill \cr {{c_t} + u \cdot \nabla c = \Delta c - nc,} \hfill \cr {{u_t} + \left({u \cdot \nabla} \right)u = \Delta u + \nabla P + n\nabla \phi ,\,\,\,\,\nabla \cdot u = 0} \hfill \cr}} \right.$$

is considered in a smoothly bounded planar domain Ω under the boundary conditions

$$\left({\nabla n - n\nabla c} \right) \cdot \nu = 0,\,\,\,\,\,\,c = {c_ *},\,\,\,\,\,u = 0,\,\,\,\,\,x \in \partial \Omega ,t > 0,$$

with a given nonnegative constant C_✭. It is shown that if (n₀, c₀, u₀) is sufficiently regular and such that the product \({\left\| {{n_0}} \right\|_{{L^1}\left(\Omega \right)}}\left\| {{c_0}} \right\|_{{L^\infty}\left(\Omega \right)}^2\) is suitably small, an associated initial value problem possesses a bounded classical solution with (n, c, u)|_t=0 = (n₀, c₀, u₀).