Partial regularity for Navier-Stokes and liquid crystals inequalities without maximum principle

TL;DR

This paper extends partial regularity results to Navier-Stokes and liquid crystal inequalities without maximum principles, showing the singular set has zero Lebesgue measure under certain conditions, generalizing classical results.

Contribution

It establishes partial regularity for solutions to Navier-Stokes and liquid crystal inequalities lacking maximum principles, broadening understanding of solution regularity in these systems.

Findings

The singular set has zero Lebesgue measure for these inequalities.

Partial regularity results match classical theorems under additional assumptions.

The results recover and extend Caffarelli-Kohn-Nirenberg's partial regularity for Navier-Stokes.

Abstract

In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the "Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. We extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the "director field" $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$ which they use to establish partial regularity (specifically, $\mathcal{P}^{1}(\mathcal{S})=0$) of solutions. For the analogous "inequality" one loses this maximum principle, but here we nonetheless establish certain partial regularity results (namely $\mathcal{P}^{\frac 92 + \delta}(\mathcal{S})=0$, so that in particular the putative singular set $\mathcal{S}$ has space-time Lebesgue measure zero). Under an additional assumption on $d$ for any fixed value of a certain parameter $\sigma \in (5,6)$ (which for $\sigma =6$ reduces precisely to the boundedness of $d$ used by Lin and Liu), we obtain the same partial regularity ($\mathcal{P}^{1}(\mathcal{S})=0$) as do Lin and Liu. In particular, we recover the partial regularity result ($\mathcal{P}^{1}(\mathcal{S})=0$) of Caffarelli-Kohn-Nirenberg (1982) for "suitable weak solutions" of the Navier-Stokes system, and we verify Scheffer's assertion that the same hold for solutions of the weaker "inequality" as well.