Doubly nonlinear stochastic evolution equations

Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form dα(u)-div(β1(∇u))dt + β0(u)dt ∋ f(u)dt + G(u)dW. We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, α, β0, and β1 are allowed to be multivalued, as required by the modelization of solid-liquid phase transitions. In this regard, the equation corresponds to a nonlinear-diffusion version of the classical two-phase Stefan problem with stochastic perturbation. The existence of martingale solutions is proved via regularization and passage-to-the-limit. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô's formula. Under some more restrictive assumptions on the nonlinearities, existence and uniqueness of strong solutions follows. Besides the relation above, the theory covers equations with nonlocal terms as well as systems.