Global Regularity for the Chemotaxis-Stokes System with Signal Consumption in $\mathbb{R}^3$

Abstract

In 2005, Tuval et al. introduced a continuum model for the interaction of aerobic bacteria, oxygen, and an incompressible fluid. For the associated three-dimensional chemotaxis-consumption systems, global classical regularity for large data has remained a longstanding question, even after omitting the nonlinear convection term from the fluid equation. We answer this question and prove global classical solvability of the chemotaxis--Stokes Cauchy problem in \(\mathbb{R}^3\), with linear cell diffusion and bilinear signal consumption, under the following data assumptions. The initial density and signal are smooth, bounded, and nonnegative, with finite density mass and second moment and finite signal Fisher energy; the divergence-free initial velocity belongs to \(H^4\). No smallness assumption is imposed, and a positive constant signal background is allowed. The solution remains bounded on every finite time interval. The main new ingredient is a localized entropy estimate uniform under density normalization, even when the resulting consumption coefficient becomes arbitrarily large. A terminal representation of the signal separates two complementary mechanisms: consumption excludes normalized density concentration at positive-signal points, while a localized weighted estimate improves density integrability at zero-signal points. Together, these mechanisms yield decay of a scale-invariant density quantity and allow a local $\varepsilon$-regularity criterion to be applied without a separate smallness assumption on the signal gradient.