Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

Abstract

A class of chemotaxis-Stokes systems generalizing the prototype \[\left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \nabla \cdot \big(n^{m-1}\nabla n\big) - \nabla \cdot \big(n\nabla c\big), c_t + u\cdot\nabla c &=& \Delta c-nc, u_t +\nabla P &=& \Delta u + n \nabla \phi, \qquad \nabla\cdot u =0, \end{array} \right. \] is considered in bounded convex three-dimensional domains, where $\phi\in W^{2,\infty}(\Omega)$ is given. The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that \[ m>\frac{9}{8}. \qquad (\star) \] Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{|\Omega|} \int_\Omega n_0,0,0)$ in the large time limit. This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than ($\star$).