Navier-Stokes equations with Navier boundary conditions and stochastic Lie transport: Well-posedness and inviscid limit

We prove the existence and uniqueness of global, probabilistically strong, PDE-strong solutions of the 2D Stochastic Navier-Stokes Equation under Navier boundary conditions with a transport and stretching noise. We emphasise that the Navier boundary conditions enable energy estimates which appear to be prohibited for the usual no-slip condition. The importance of the Stochastic Advection by Lie Transport (SALT) structure, in comparison to a purely transport Stratonovich noise, is also highlighted in these estimates. In the particular cases of the free boundary condition and a convex domain, the inviscid limit exists and is a global, probabilistically weak, PDE-weak solution of the corresponding Stochastic Euler Equation with impermeable boundary condition.