Chemotaxis-Stokes system with $p$-Laplacian slow diffusion: global boundedness across the full $p>2$ range

Abstract

The global boundedness problem for three-dimensional chemotaxis-fluid systems with nonlinear diffusion presents substantial analytical challenges, particularly in the near-linear diffusion regime. Indeed, the porous-medium and p-Laplacian diffusion operators approach the classical Laplacian as $m\searrow1$ and $p\searrow2$, and the extra a priori control afforded by nonlinear structures becomes increasingly difficult to exploit. This difficulty is reflected in the porous-medium setting, where establishing boundedness throughout $m>1$ required a sequence of substantial developments spanning more than a decade (2010-2024). For the p-Laplacian case, despite significant progress, the problem remained open in the near-linear range $2<p\le p^*(\approx2.01)$, due to the saturation of classical Moser-type iterations as $p\searrow2$. In this paper, we establish global bounded strong solutions for a three-dimensional chemotaxis-Stokes system with p-Laplacian slow diffusion and signal consumption. Our approach departs from the conventional $L^q$-iteration framework and develops a weighted energy method that directly exploits the coercive structure of the nonlinear diffusion operator. A weighted Rellich identity yields density estimates. To control the resulting boundary contribution on general smooth domains, we first use tangential differentiation along the boundary to lower its order, and then apply a trace-embedding estimate. Combined with a high-gradient/low-gradient decomposition, these estimates allow us to control the mixed chemotactic terms and close the coupled estimates for the density, signal, and fluid. The resulting estimates hold throughout $2<p\le 3$, thereby resolving the previously open range $2<p\le p^*$. Together with existing results for $p>p^*$, our work establishes boundedness throughout the entire slow-diffusion range $p>2$.