Finite Speed of Propagation for Stochastic Porous Media Equations

  title={Finite Speed of Propagation for Stochastic Porous Media Equations},
  author={Benjamin Gess},
  journal={SIAM J. Math. Anal.},
  • B. Gess
  • Published 8 October 2012
  • Mathematics
  • SIAM J. Math. Anal.
We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding random attractor has infinite fractal dimension. 

Finite Speed of Propagation and Waiting Times for the Stochastic Porous Medium Equation: A Unifying Approach

An energy method to study finite speed of propagation and waiting time phenomena for the stochastic porous media equation with linear multiplicative noise in up to three spatial dimensions is developed and a sufficient criterion on the growth of initial data is formulated which locally guarantees a waiting time phenomenon to occur almost surely.

On stochastic porous-medium equations with critical-growth conservative multiplicative noise

First, we prove existence, nonnegativity, and pathwise uniqueness of martingale solutions to stochastic porous-medium equations driven by conservative multiplicative power-law noise in the Ito-sense.

Nonnegativity preserving convergent schemes for stochastic porous-medium equations

It is shown that linear, multiplicative noise in the sense of Ito, which enters the equation as a source-term, has a decelerating effect on the average propagation speed of the boundary of the support of solutions.

Numerical approximation of singular-degenerate parabolic stochastic PDEs

A fully discrete numerical approximation of the considered SPDEs based on the very weak formulation is proposed and it is proved the convergence of the numerical approximation towards the unique solution is convergence.

Zero-contact angle solutions to stochastic thin-film equations

We establish existence of nonnegative martingale solutions to stochastic thin-film equations with quadratic mobility for compactly supported initial data under Stratonovich noise. Based on so-called

Equations with Maximal Monotone Nonlinearities

We shall study here Eq. (1.1) for general (multivalued) maximal monotone graphs \(\beta: \mathbb{R} \rightarrow 2^{\mathbb{R}}\) with polynomial growth. The principal motivation for the study of

Hölder regularity for a non-linear parabolic equation driven by space-time white noise

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$ driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

This paper considers exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system and applies the Leray-Schauder fixed point theorem to obtain controllability for arbitrarily large initial data.



Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Abstract Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in L 1 ( O ) . The generation of a

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L 2-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates


A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to

Stochastic Porous Media Equations and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

Explicit conditions are presented for the existence, uniqueness, and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence

Rough evolution equations

We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular

On a random scaled porous media equation

Weak solutions to stochastic porous media equations

Abstract A stochastic version of the porous medium equation is studied. The corresponding Kolmogorov equation is solved in a space $L^{2}(H, \nu)$ where ν is an invariant measure. Then a weak