The Stochastic Porous Media Equations in $\R^d$
april, 2015
Publication type:
Paper in peer-reviewed journals
Journal:
Journal de Mathematiques Pures et Appliquees., vol. 103 (4), pp. 1024-1052
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Keywords :
Stochastic; porous media;
Wiener process; maximal monotone graph; distributions.
Abstract:
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.
BibTeX:
@article{Bar-Roc-Rus-2015, author={Viorel Barbu and Michael Röckner and Francesco Russo }, title={The Stochastic Porous Media Equations in $\R^d$ }, doi={10.1016/j.matpur.2014.10.004 }, journal={Journal de Mathematiques Pures et Appliquees. }, year={2015 }, month={4}, volume={103 (4) }, pages={1024--1052}, }