Existence and extinction in finite time for Stratonovich gradient noise porous media equations

Abstract

We study existence and uniqueness of distributional solutions to the stochastic partial differential equation $dX-(\nu \Delta X+\Delta \psi (X))dt={\sum }_{i=1}^N⟨b_i,\nabla X⟩\circ d{\beta }_i$ in $]0,T[\times \mathcal{O}$, with $X(0)=x(\xi )$ in $\mathcal{O}$ and $X=0$ on $]0,T[\times \partial \mathcal{O}$. Moreover, we prove extinction in finite time of the solutions in the special case of fast diffusion model and of self-organized criticality model.

Cited by in corpus (1)

• Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions