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

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.

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