Global well-posedness and interior regularity of 2D Navier-Stokes equations with stochastic boundary conditions (Q6624799)

The authors consider two-dimensional Navier-Stokes equations on \(\mathbb T\times(0,a)\) with the boundary condition on \(\mathbb T\times\{a\}\) involving a noise (white in time, colored in space) which might be interpreted as idealization of the evolution of the velocity of the fluid in the ocean subject to a stochastic force. The analysis of this novel type of SDE problems begins with solving the linearized problem with that rough boundary condition, and then discussing the Navier-Stokes equations with random coefficients obtained as a perturbation of solution to this linear system. Results include global well-posedness of the initial-boundary value problem as well as smoothness in space (but not in time) of the obtained solution. A generalization to stochastic bulk forces is also mentioned.