Weak Solutions of a Stochastic Model for Two-Dimensional Second Grade Fluids

Abstract

We initiate the investigation of a stochastic system of evolution partial differential equations modelling the turbulent flows of a second grade fluid filling a bounded domain of R 2 . We establish the global existence of a probabilistic weak solution.

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

“…, .)) which is equivalent to the usual H 1 (D)-scalar product and the scalar product [11] or [13] for the definitions of these spaces. We denote by C any unessential positive constant independent of α, which may change from one line to the next.…”

Strong solution for a stochastic model of two-dimensional second grade fluids: Existence, uniqueness and asymptotic behavior

“…Similar investigations for stochastic models of Non-Newtonian fluids have almost not been undertaken except in very few works. We refer, for instance, to [7], [29], [30], [31], [41], [42], [43] for some relevant results in that direction. It should be noted that the models investigated in the first four papers occur very naturally from the kinetic theory of polymer dynamics.…”

“…To the best of our knowledge the system (1) was only analyzed in [41] and [43] where the authors focused on proving the existence of weak probabilistic solutions and the behaviour of the solution as α → 0, respectively. The existence in [41] was achieved by using Galerkin method combined with some deep compactness results due to Prokhorov and Skorokhod.…”

“…The existence in [41] was achieved by using Galerkin method combined with some deep compactness results due to Prokhorov and Skorokhod. The main goals of the present work are the following:…”

“…• Show that, in contrast to [41] again, the whole sequence of Galerkin approximation converges to the strong probabilistic solution in mean square (see Theorem 4.2).…”