Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time

TL;DR

This paper studies the partial regularity of weak solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time, showing the singular set has vanishing Hausdorff measure, extending partial regularity theory to this complex fluid model.

Contribution

It introduces a new approach to analyze the singular set of weak solutions in the chemotaxis-fluid model, including a local energy inequality and handling non-scaling invariant quantities.

Findings

The Hausdorff measure of the singular set vanishes at the first blow-up time.

A new local energy inequality is established for the model.

First description of the singular set for weak solutions of the chemotaxis-fluid system.

Abstract

In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first blow-up time. The new ingredients are to establish certain type of local energy inequality and deal with the non-scaling invariant quantity of $n\ln n$, which seems to be the first description for the singular set of weak solutions of the chemotaxis-fluid model, which is motivated by Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}.