Deterministic and Stochastic Partial Differential Equations in Fluid and Solid Mechanic

The dissertation is divided into three parts. In the first part, the object of study is to establish the existence and uniqueness of weak solutions of variational parabolic inequalities with pseudo-monotonicity and nonzero initial data using finite differences in time with an implicit Euler scheme.;The second part is to devoted to the study of the single layer shallow water equations on a bounded domain in space dimension two forced by a multiplicative white noise. We are able to obtain the existence and uniqueness of a maximal pathwise solution for a short period of time. The third part is concerned about the existence of global pathwise solutions for the Stochastic non-Newtonian incompressible fluid equations in two space dimensions. Moreover, we show that said solutions converge in probability to solutions of the stochastic Navier Stokes equations in the appropriate limit. Our approach is based on Galerkin approximations and the theory of martingale solutions.