A Pressure Associated with a Weak Solution to the Navier-Stokes Equations with Navier's Boundary Condition

TL;DR

This paper establishes the existence and regularity properties of the pressure associated with weak solutions to the Navier-Stokes equations under Navier's boundary conditions, including in smooth domains and under Serrin's integrability criteria.

Contribution

It demonstrates the existence of a structured pressure distribution for weak solutions with Navier boundary conditions and analyzes its regularity in smooth domains and sub-domains.

Findings

Pressure exists as a distribution with a specific structure.

In smooth domains, pressure is represented by an integrable function.

Pressure regularity improves under Serrin's integrability conditions.

Abstract

We show that if u is a weak solution to the Navier-Stokes initial-boundary value problem with Navier's slip boundary conditions in $Q_T:=\Omega\times(0,T)$, where $\Omega$ is a domain in $R^3$, then an associated pressure $p$ exists as a distribution with a certain structure. Furthermore, we also show that if $\Omega$ is a "smooth" domain in $R^3$ then the pressure is represented by a function in $Q_T$ with a certain rate of integrability. Finally, we study the regularity of the pressure in sub-domains of $Q_T$, where $u$ satisfies Serrin's integrability conditions.