Global solutions for a 2D chemotaxis-fluid system with large measures as initial density and vorticity

Abstract

We consider a chemotaxis-fluid system in the whole plane $\mathbb{R}^{2}$ which describes the motion of bacteria suspended in a Navier-Stokes fluid and attracted by a chemical (oxygen). Employing the vorticity formulation for the fluid equations, we obtain local and global solutions with large (Radon) measures as initial data for the bacterial density and vorticity. The gravitational/centrifugal potential is taken with finite $L^{2}$-gradient that can be large. The uniqueness property is also discussed. For the global result, we need to assume a smallness condition only on the $L^{\infty}$-norm of the initial oxygen concentration. In comparison with previous works, our results provide a new class for the initial density and vorticity, as well as for the potential, covering particularly singular measures such as Dirac delta, measure concentrated on smooth curves (filaments and rings), among others. For that, we approach the system via critical functional spaces, Kato-type norms, and suitable $L^{p}$-estimates uniformly in time.