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

Xiaomeng Chen; Shuai Li; Wendong Wang

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arXiv · 2207.04870

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

Xiaomeng Chen·Shuai Li·Wendong Wang

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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}.

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Xiaomeng Chen, Shuai Li, Wendong Wang. 2022-07-11. Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time. https://arxiv.org/abs/2207.04870

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