The Stochastic Porous Media Equations in $\R^d$

Viorel Barbu, Michael Röckner and Francesco Russo
april, 2015
Publication type:
Paper in peer-reviewed journals
Journal de Mathematiques Pures et Appliquees., vol. 103 (4), pp. 1024-1052
images/icons/icon_arxiv.png 1312.6234
Keywords :
Stochastic; porous media; Wiener process; maximal monotone graph; distributions.
Existence and uniqueness of solutions to the stochastic porous media equation $dX-\D\psi(X) dt=XdW$ in $\R^d$ are studied. Here, $W$ is a Wiener process, $\psi$ is a maximal monotone graph in $\R\times\R$ such that $\psi(r)\le C|r|^m$, $\ff r\in\R$, $W$ is a coloured Wiener process. In this general case the dimension is restricted to $d\ge 3$, the main reason being the absence of a convenient multiplier result in the space ${\mathcal H}=\{\varphi\in\mathcal{S}'(\R^d);\ |\xi|({\mathcal F}\varphi)(\xi)\in L^2(\R^d)\}$, for $d\le2$. When $\psi$ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space $H^{-1}(\R^d)$. If $\psi(r)r\ge\rho|r|^{m+1}$ and $m=\frac{d-2}{d+2}$, we prove the finite time extinction with strictly positive probability.
    author={Viorel Barbu and Michael Röckner and Francesco Russo },
    title={The Stochastic Porous Media Equations in $\R^d$ },
    journal={Journal de Mathematiques Pures et Appliquees. },
    year={2015 },
    volume={103 (4) },