Existence and regularity of the pressure for the stochastic Navier-Stokes equations

Abstract

We prove, on one hand, that for a convenient body force with values in the distribution space (H−1(D))d, where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier-Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V0 of the divergence free subspace V of (H1 0(D))d, in general it is not possible to solve the stochastic Navier-Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier-Stokes equations could be meaningful for them.