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Stochastic Porous Media Equations

Authors: Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found.
The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model".
The book will be of interest to PhD students and researchers in mathematics, physics and biology.

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Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract

This is an introductory chapter mainly devoted to the formulation of problems, models and some preliminaries on convex and infinite dimensional analysis, indispensable for understanding the sequel.

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 2. Equations with Lipschitz Nonlinearities

Abstract

We start here by studying the porous media equation problem (1.1) when \(\beta: \mathbb{R} \rightarrow \mathbb{R}\) is monotonically increasing and Lipschitz continuous. The main reason is that general maximal monotone graphs β can be approximated by their Yosida approximations β _ε which are Lipschitz continuous and monotonically increasing. So, several estimates proved in this chapter will be exploited later for studying problems with more general β.

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 3. Equations with Maximal Monotone Nonlinearities

Abstract

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 these equations comes from nonlinear diffusion models presented in Sect. 1.1

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 4. Variational Approach to Stochastic Porous Media Equations

Abstract

We shall briefly present here a different approach to stochastic porous media equations which in analogy to the variational formulation of parabolic boundary value problems will be called variational approach. It is based on a general existence result for infinite dimensional stochastic equations of the form

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 5. L 1-Based Approach to Existence Theory for Stochastic Porous Media Equations

Abstract

The existence theory developed in the previous chapter was based on energy estimates in the space H ⁻¹ obtained via Itô’s formula in approximating equations. This energetic approach leads to sharp existence results, but requires polynomial growth assumptions or strong coercivity for the nonlinear function β. The case of general maximal monotone functions β of arbitrary growth and in particular with exponential growth was beyond the limit of the previous theory. Here we develop a different approach based on sharp L ¹-estimates for the corresponding approximating equations which allows to treat these general situations.

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 6. The Stochastic Porous Media Equations in

Abstract

Here we shall treat Eq. (3.1) in the domain \(\mathcal{O} = \mathbb{R}^{d}\). Though the methods are similar to those used for bounded domains, there are, however, some notable differences and as seen below the dimension d of the space plays a crucial role.

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Chapter 7. Transition Semigroup

Abstract

This chapter is devoted to existence of invariant measures for transition semigroups associated with stochastic porous media equations with additive noise studied in previous chapters.

Viorel Barbu, Giuseppe Da Prato, Michael Röckner

Backmatter

Title: Stochastic Porous Media Equations
Authors: Viorel Barbu
Giuseppe Da Prato
Michael Röckner
Copyright Year: 2016
Publisher: Springer International Publishing
Electronic ISBN: 978-3-319-41069-2
Print ISBN: 978-3-319-41068-5
DOI: https://doi.org/10.1007/978-3-319-41069-2