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

Jiri Neustupa; Sarka Necasova; Petr Kucera

doi:10.1007/s00021-020-00500-y

paper

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

arXiv:1911.04007 · doi:10.1007/s00021-020-00500-y

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 $Q_{T} := Ω \times (0, T)$ , where $Ω$ 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 $Ω$ 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.