Local‐in‐Time Solutions for a 2D Chemotaxis‐Fluid System With Singular Measure Data

Abstract

ABSTRACT In this work, we study an whole‐plane chemotaxis‐fluid system that describes the motion of bacteria attracted by a chemical (oxygen) and suspended in a Navier‐Stokes fluid. By employing the vorticity formulation for the fluid equations, we derive local solutions. The centrifugal or gravitational potential is assumed to have a finite gradient, which may be large. Furthermore, we address the uniqueness of these solutions. Compared to previous studies, our results introduce a new class for the initial density, vorticity, and potential. This class includes particularly singular measures such as the Dirac delta, measures concentrated on smooth curves, and others. To achieve this, we utilize critical functional spaces, Kato‐type norms, and suitable estimates, ensuring uniform bounds over time.