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Viorel Barbu Giuseppe Da Prato Michael Röckner

Stochastic Porous Media Equations

Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zürich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and New York Anna Wienhard, Heidelberg

2163

More information about this series at http://www.springer.com/series/304

Viorel Barbu • Giuseppe Da Prato • Michael RRockner

Stochastic Porous Media Equations

123

Viorel Barbu Department of Mathematics Al. I. Cuza University & Octav Mayer Institute of Mathematics of the Romanian Academy Iasi, Romania

Giuseppe Da Prato Classe di Scienze Scuola Normale Superiore di Pisa Pisa, Italy

Michael RRockner Department of Mathematics University of Bielefeld Bielefeld, Germany

ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-41068-5 DOI 10.1007/978-3-319-41069-2

ISSN 1617-9692 (electronic) ISBN 978-3-319-41069-2 (eBook)

Library of Congress Control Number: 2016954369 Mathematics Subject Classification (2010): 60H15, 35K55, 76S99, 76M30, 76M35 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

This book is concerned with stochastic porous media equations with main emphasis on existence theory, asymptotic behaviour and ergodic properties of the associated transition semigroup. The general form of the porous media equation is dX  ˇ.X/dt D .X/dW;

(1)

where ˇ W R ! R is a monotonically increasing function (possibly multivalued) and W is a cylindrical Wiener process. P in stochastic porous media equation Stochastic perturbations of the form .X/W were already considered by physicists but until recently no rigorous mathemati