Doubly nonlinear stochastic evolution equations II

Completing the analysis in Scarpa (Math Models Methods Appl Sci 30(5): 991–1031 2020), we investigate the well-posedness of SPDEs problems of doubly nonlinear type. These arise ubiquitously in the modeling of dissipative media and correspond to generalized balance laws between conservative and nonconservative dynamics. We extend the reach of the classical deterministic case by allowing for stochasticity. The existence of martingale solutions is proved via a regularization technique, hinging on the validity of an Itô formula in a minimal regularity setting. Under additional assumptions, the well-posedness of stochastically strong solutions is also shown.