Local kinetic energy and singularities of the incompressible Navier--Stokes Equations

TL;DR

This paper investigates the partial regularity of solutions to the incompressible Navier--Stokes equations, establishing a reverse H"older inequality for velocity gradients under bounded local kinetic energy, and providing bounds on the singular set's dimensions.

Contribution

It introduces a new reverse H"older inequality for velocity gradients under bounded local kinetic energy and derives bounds on the singular set's dimensions for weak solutions.

Findings

Reverse H"older inequality for velocity gradient with increasing support

Bound on Hausdorff and Minkowski dimensions of singular set

New bounds for singularities in weak solutions in $L^ abla(0,T;L^{3,w})$

Abstract

We study the partial regularity problem of the incompressible Navier--Stokes equations. In this paper, we show that a reverse H\"older inequality of velocity gradient with increasing support holds under the condition that a scaled functional corresponding the local kinetic energy is uniformly bounded. As an application, we give a new bound for the Hausdorff dimension and the Minkowski dimension of singular set when weak solutions $v$ belong to $L^\infty(0,T;L^{3,w}(\mathbb{R}^3))$ where $L^{3,w}(\mathbb{R}^3)$ denotes the standard weak Lebesgue space.