# Stochastic Porous Media Equations and Self-Organized Criticality

@article{Barbu2009StochasticPM, title={Stochastic Porous Media Equations and Self-Organized Criticality}, author={Viorel Barbu and Giuseppe Da Prato and Michael R{\"o}ckner}, journal={Communications in Mathematical Physics}, year={2009}, volume={285}, pages={901-923} }

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 extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized criticality behavior of stochastic nonlinear diffusion equations with critical states.

## 79 Citations

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The long time behaviour of solutions to generalised stochastic porous media equations on bounded domains with Dirichlet boundary data is studied. We focus on a degenerate form of nonlinearity arising…

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Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are…

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- 2011

This work addresses the stochastic porous media equation with multiplicative noise and diffusivity function depending on the space variable. The first part of the paper proves an existence and…

## References

SHOWING 1-10 OF 37 REFERENCES

Existence and uniqueness of nonnegative solutions to the stochastic porous media equation

- Mathematics
- 2007

It is proved that the stochastic porous media equation in a bounded domain of R 3 , with multiplicative noise, with a monotone nonlin- earity of polynomial growth has a unique nonnegative solution in…

Existence of strong solutions for stochastic porous media equation under general monotonicity conditions

- Mathematics
- 2007

This paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dX ( X)dt = B(X)dW (t) in bounded domains of R d with Dirichlet boundary conditions. Here is a…

The two phase stochastic Stefan problem

- Mathematics
- 2002

Abstract. This work is concerned with existence for a stochastic free boundary problem arising in phase transition (the Stefan two phase problem). The existence of an invariant ergodic measure…

Dissipative transport in open systems: An investigation of self-organized criticality.

- PhysicsPhysical review letters
- 1989

A dynamic renormalization-group calculation allows us to determine various critical exponents exactly in all dimensions of dissipative transport in open systems.

Self-orgainzed criticality and singular diffusion.

- MathematicsPhysical review letters
- 1990

A continuum limit is rigorously established for a one-dimensional automaton that has this property, and it is shown that certain exponents and the distribution of events are simply related to the order of the diffusion singularity.

Conservation laws, anisotropy, and "self-organized criticality" in noisy nonequilibrium systems.

- PhysicsPhysical review letters
- 1990

It is argued in the context of noisy, nonequilibrium Langevin models that systems with conserving deterministic dynamics and noise which violates the conservation law always exhibit self-organized…

Smoothness of weak solutions to a nonlinear fluid-structure interaction model

- Mathematics
- 2008

The nonlinear fluid-structure interaction coupling the Navier-Stokes equations with a dynamic system of elasticity is consid- ered. The coupling takes place on the boundary (interface) via the con-…

A Concise Course on Stochastic Partial Differential Equations

- Mathematics
- 2007

Motivation, Aims and Examples.- Stochastic Integral in Hilbert Spaces.- Stochastic Differential Equations in Finite Dimensions.- A Class of Stochastic Differential Equations.