Stochastic Navier-Stokes Equations In Unbounded Channel Domains

In this talk, I will first describe the difficulty in using the standard L² energy methods for the solvability of the stochastic Navier-Stokes equations with non-zero, finite, random flux in two types of unbounded channel domains, namely,
(i) channels with outlets having constant width
(ii) channels with outlets that diverge at infinity
By considering an "admissible" channel domain, we will construct a unique basic vector field having the same flux as that of the original problem. A perturbed vector field with zero net flux is constructed using a suitable transformation involving the constructed basic vector field. Then a unique pathwise strong solution to this perturbed field will be proved by exploiting local monotonicity arguments. The perturbed pressure will be characterized using a generalization of the de Rham's Theorem to processes.