Weak and Strong Solutions to Nonlinear SPDEs with Unbounded Noise

We introduce an extended variational framework for nonlinear SPDEs with unbounded noise, defining three different solution types of increasing strength along with criteria to establish their existence. The three notions can be understood as probabilistically and analytically weak, probabilistically strong and analytically weak, as well as probabilistically and analytically strong. Our framework facilitates several well-posedness results for the Navier-Stokes Equation with transport noise, equipped with the no-slip and Navier boundary conditions.