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Stochastic porous media equations in \(\mathbb R^d\). (English. French summary) Zbl 1405.76047

Summary: Existence and uniqueness of solutions to the stochastic porous media equation \(dX-\Delta\psi(X)dt=XdW\) in \(\mathbb R^d\) are studied. Here, \(W\) is a Wiener process, \(\psi\) is a maximal monotone graph in \(\mathbb R\times\mathbb R\) such that \(\psi(r)\leq C| r|^m\), \(\forall r\in\mathbb R\). In this general case, the dimension is restricted to \(d\geq 3\), the main reason being the absence of a convenient multiplier result in the space \(\mathcal H=\{\varphi\in\mathcal S'(\mathbb R^d);|\xi|(\mathcal F\varphi)(\xi) \in L^2(\mathbb R^d)\}\), for \(d\leq 2\). When \(\psi\) is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space \(H^{-1}(\mathbb R^d)\). If \(\psi(r)r\geq\rho| r|^{m+1}\) and \(m=\frac{d-2}{d+2}\), we prove the finite time extinction with strictly positive probability.

MSC:

76S05 Flows in porous media; filtration; seepage
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K55 Nonlinear parabolic equations
35K67 Singular parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
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References:

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