Narrow Results

In this work, we investigate the question of existence of weak solutions to the density dependent stochastic Navier–Stokes equations. The noise considered contains functions which depend nonlinearly on the velocity and which do not satisfy the Lipschitz condition. Furthermore, the initial density is allowed to vanish. We introduce a suitable notion of probabilistic weak solution for the problem and prove its existence.

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.