In this paper we provide an elementary proof of the classical result of J.L. Lions and G. Prodi on the global unique solvability of two-dimensional Navier-Stokes equations that avoids compact embedding and strong convergence. The method applies to unbounded domains without special treatment. The essential idea is to utilize the local monotonicity of the sum of the Stokes operator and the inertia term. This method was first discovered in the context of stochastic Navier-Stokes equations by J.L. Menaldi and S.S. Sritharan.
"A simple proof of global solvability of 2-D Navier-Stokes equations in unbounded domains." Differential Integral Equations 23 (3/4) 223 - 235, March/April 2010. https://doi.org/10.57262/die/1356019315