Global mass-preserving solutions in a two-dimensional chemotaxis-Stokes system with rotational flux components

Abstract

In a bounded domain $\Omega \subset \mathbb {R}^2$, we consider the the chemotaxis-Stokes system

$$\begin{aligned} \left\{ \begin{array}{ll} n_t + u\cdot \nabla n = \Delta n - \nabla \cdot \Big (nS(x,n,c) \cdot \nabla c \Big ), \qquad &{} x\in \Omega , \ t>0,\\ c_t + u\cdot \nabla c = \Delta c - nf(c), \qquad &{} x\in \Omega , \ t>0,\\ u_t = \Delta u + \nabla P + n\nabla \phi , \qquad \nabla \cdot u=0, \qquad &{} x\in \Omega , \ t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$

which arises as a model for populations of aerobic bacteria swimming in a sessile water drop. In accordance with refined modeling approaches, we do not necessarily assume the chemotactic sensitivity S herein to be a scalar function, but rather allow S to attain values in $\mathbb {R}^{2\times 2}$. As compared to the well-studied case of scalar-valued sensitivities in which an analysis can be based on favorable energy-type inequalities, this modification brings about significant new challenges which require to adequately cope with only little a priori information on regularity of solutions of ($\star$). The present work creates a functional setup which despite this allows for the construction of certain global mass-preserving generalized solutions to an associated initial-boundary value problem in planar convex domains with smooth boundary, provided that the initial data and the parameter functions S, f and $\phi$ are sufficiently smooth, and that S is bounded and f is nonnegative with $f(0)=0$.