Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

Abstract

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $d{X}_t-\text{div}\gamma (\nabla X_t)dt+\beta (X_t)dt\ni B(t,X_t)d{W}_t$, where $\gamma$ and $\beta$ are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on ${\mathbb{R}}^d$ and $\mathbb{R}$ respectively, and $W$ is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray-Lions conditions on $\gamma$ and with no restrictive smoothness or growth assumptions on $\beta$. The operator $B$ is assumed to be Hilbert-Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

Key words and phrases: doubly nonlinear stochastic equation, divergence, variational approach, existence of solutions, continuous dependence, multiplicative noise

Cited by in corpus (9)

• On the stochastic Cahn-Hilliard equation with a singular double-well potential
• Singular Stochastic Allen-Cahn equations with dynamic boundary conditions
• The stochastic viscous Cahn-Hilliard equation: well-posedness, regularity and vanishing viscosity limit
• Doubly nonlinear stochastic evolution equations
• A note on doubly nonlinear SPDEs with singular drift in divergence form
• Refined existence and regularity results for a class of semilinear dissipative SPDEs
• On the well-posedness of SPDEs with singular drift in divergence form
• The Energy-Dissipation Principle for stochastic parabolic equations
• Strong solutions to SPDEs with monotone drift in divergence form