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 $Q_T:=\mathrm{\Omega }\times (0,T)$, where $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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.