On the local pressure of the Navier-Stokes equations and related systems

TL;DR

This paper develops a method to construct local pressure distributions for weak solutions of Navier-Stokes and related systems, addressing the challenge of the absence of global pressure in complex domains.

Contribution

It introduces a novel approach to represent distributions related to fluid velocity and forces as local pressure gradients, enhancing analysis of local regularity.

Findings

Constructed local pressure distributions from weak solutions.

Provided a representation of distributions as pressure gradients.

Facilitated local regularity analysis in complex domains.

Abstract

In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain informations on the pressure. This fact has been used by V. Scheffer in the notion of a suitable weak to the Navier-Stokes equation, and in the proof of the partial regularity due to Caffarelli. Kohn and Nirenberg. In general domains, or in case of complex viscous fluid models a global pressure doesn't necessarily exist. To overcome this problem, in the present paper we construct a local pressure distribution by showing that every distribution $ \partial _t \bu +\bF $, which vanishs on the set of smooth solenoidal vector fields can be represented by a distribution $ \partial _t \nabla p_h +\nabla p_0 $, where $\nabla p_h \sim \bu $ and $ \nabla p_0 \sim \bF$.