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Nonlocal Diffusion Problems Mathematical Surveys and Monographs Volume 165 Nonlocal Diffusion Problems Fuensanta Andreu-Vaillo José M. Mazón Julio D. Rossi J. Julián Toledo-Melero American Mathematical Society Real Sociedad Matemática Española surv-165-mazon-cov.indd 1 8/4/10 2:33 PM http://dx.doi.org/10.1090/surv/165 Nonlocal Nonlocal Diffusion Problems Diffusion Problems Fuensanta Andreu-Vaillo José M. Mazón Julio D. Rossi J. Julián Toledo-Melero Mathematical Surveys and Monographs Volume 165 Nonlocal Diffusion Problems Fuensanta Andreu-Vaillo José M. Mazón Julio D. Rossi J. Julián Toledo-Melero American Mathematical Society Providence, Rhode Island Real Sociedad Matemática Española Madrid, Spain Editorial Committee of Mathematical Surveys and Monographs Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein Editorial Committee of the Real Sociedad Matem´atica Espa˜nola Pedro J. Pa´ul, Director Luis Al´ıas Alberto Elduque Emilio Carrizosa Rosa Mar´ıa Mir´o Bernardo Cascales Pablo Pedregal Javier Duoandikoetxea Juan Soler 2010 Mathematics Subject Classification. Primary 45E10, 45A05, 45G10, 47H20, 45M05, 35K05, 35K55, 35K57, 35K92. For additional information and updates on this book, visit www.ams.org/bookpages/surv-165 Library of Congress Cataloging-in-Publication Data Nonlocal diffusion problems / Fuensanta Andreu-Vaillo ...[et al.]. p. cm. — (Mathematical surveys and monographs ; v. 165) Includes bibliographical references and index. ISBN 978-0-8218-5230-9 (alk. paper) 1. Integral equations. 2. Semigroups of operators. 3. Parabolic operators. I. Andreu-Vaillo, Fuensanta, 1955– . QA431.N585 2010 515.45—dc22 2010020473 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2010 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 151413121110 To Mayte Nati, Alba and Hugo and, of course, especially to Fuensanta Contents Preface xi Chapter 1. The Cauchy problem for linear nonlocal diffusion 1 1.1. The Cauchy problem 1 1.1.1. Existence and uniqueness 5 1.1.2. Asymptotic behaviour 6 1.2. Refined asymptotics 10 1.2.1. Estimates on the regular part of the fundamental solution 12 1.2.2. Asymptotics for the higher order terms 17 1.2.3. A different approach 20 1.3. Rescaling the kernel. A nonlocal approximation of the heat equation 22 1.4. Higher order problems 23 1.4.1. Existence and uniqueness 24 1.4.2. Asymptotic behaviour 25 1.4.3. Rescaling the kernel in a higher order problem 28 Bibliographical notes 29 Chapter 2. The Dirichlet problem for linear nonlocal diffusion 31 2.1. The homogeneous Dirichlet problem 31 2.1.1. Asymptotic behaviour 32 2.2. The nonhomogeneous Dirichlet problem 36 2.2.1. Existence, uniqueness and a comparison principle 36 2.2.2. Convergence to the heat equation when rescaling the kernel 38 Bibliographical notes 40 Chapter 3. The Neumann problem for linear nonlocal diffusion 41 3.1. The homogeneous Neumann problem 41 3.1.1. Asymptotic behaviour 42 3.2. The nonhomogeneous Neumann problem 45 3.2.1. Existence and uniqueness 46 3.2.2. Rescaling the kernels. Convergence to the heat equation 48 3.2.3. Uniform convergence in the homogeneous case 54 3.2.4. An L1-convergence result in the nonhomogeneous case 56 3.2.5. A weak convergence result in the nonhomogeneous case 57 Bibliographical notes 63 vii viii CONTENTS Chapter 4. A nonlocal convection diffusion problem 65 4.1. A nonlocal model with a nonsymmetric kernel 65 4.2. The linear semigroup revisited 69 4.3. Existence and uniqueness of the convection problem 76 4.4. Rescaling the kernels. Convergence to the local convection-diffusion problem 82 4.5. Long time behaviour of the solutions 90 4.6. Weakly nonlinear behaviour 96 Bibliographical notes 98 Chapter 5. The Neumann problem for a nonlocal nonlinear diffusion equation 99 5.1. Existence and uniqueness of solutions 100 5.1.1. Notation and preliminaries 100 5.1.2. Mild solutions and contraction principle 104 5.1.3. Existence of solutions 112 5.2. Rescaling the kernel. Convergence to the local problem 115 5.3. Asymptotic behaviour 118 Bibliographical notes 122 Chapter 6. Nonlocal p-Laplacian evolution problems 123 6.1. The Neumann problem 124 6.1.1. Existence and uniqueness 125 6.1.2. A precompactness result 128 6.1.3. Rescaling the kernel. Convergence to the local p-Laplacian 131 6.1.4. A Poincar´etypeinequality 137 6.1.5. Asymptotic behaviour 141 6.2. The Dirichlet problem 142 6.2.1. A Poincar´etypeinequality 144 6.2.2. Existence and uniqueness of solutions 146 6.2.3. Convergence to the local p-Laplacian 149 6.2.4. Asymptotic behaviour 153 6.3. The Cauchy problem 154 6.3.1. Existence and uniqueness 154 6.3.2. Convergence to the Cauchy problem for the local p-Laplacian 157 6.4. Nonhomogeneous problems 160 Bibliographical notes 161 Chapter 7. The nonlocal total variation flow 163 7.1. Notation and preliminaries 164 7.2. The Neumann problem 165 7.2.1. Existence and uniqueness 166 7.2.2. Rescaling the kernel. Convergence to the total variation flow 169 7.2.3. Asymptotic behaviour 174 7.3. The Dirichlet problem 175 7.3.1. Existence and uniqueness 176 7.3.2. Convergence to the total variation flow 180 7.3.3. Asymptotic behaviour 188 Bibliographical notes 189 Chapter 8. Nonlocal models for sandpiles 191 CONTENTS ix 8.1. A nonlocal version of the Aronsson-Evans-Wu model for sandpiles 191 8.1.1. The Aronsson-Evans-Wu model for sandpiles 191 8.1.2. Limit as p →∞in the nonlocal p-Laplacian Cauchy problem 193 8.1.3. Rescaling the kernel. Convergence to the local problem 195 8.1.4. Collapse of the initial condition 197 8.1.5. Explicit solutions 200 8.1.6. A mass transport interpretation 210 8.1.7. Neumann boundary conditions 212 8.2. A nonlocal version of the Prigozhin model for sandpiles 213 8.2.1. The Prigozhin model for sandpiles 214 8.2.2. Limit as p → +∞ in the nonlocal p-Laplacian Dirichlet problem 214 8.2.3. Convergence to the Prigozhin model 217 8.2.4. Explicit solutions 219 Bibliographical notes 222 Appendix A. Nonlinear semigroups 223 A.1. Introduction 223 A.2. Abstract Cauchy problems 224 A.3. Mild solutions 227 A.4. Accretive operators 229 A.5. Existence and uniqueness theorem 235 A.6. Regularity of the mild solution 239 A.7. Convergence of operators 241 A.8. Completely accretive operators 242 Bibliography 249 Index 255 Preface The goal in this monograph is to present recent results concerning nonlocal evolution equations with different boundary conditions. We deal with existence and uniqueness of solutions and their asymptotic behaviour. We also give some results concerning limits of solutions to nonlocal equations when a rescaling parameter goes to zero. We recover in these limits some of the most frequently used diffusion models such as the heat equation, the p-Laplacian evolution equation, the porous medium equation, the total variation flow and a convection-diffusion equation. This book is based mainly on results from the papers [14], [15], [16], [17], [68], [78], [79], [80], [120], [121]and[140]. First, let us briefly introduce the prototype of nonlocal problems that will be considered in this monograph. Let J : RN → R be a nonnegative, radial, continuous function with J(z) dz =1. RN Nonlocal evolution equations of the form (0.1) ut(x, t)=(J ∗ u − u)(x, t)= J(x − y)u(y, t) dy − u(x, t), RN and variations of it, have been recently widely used to model diffusion processes. More precisely, as stated in [106], if u(x, t) is thought of as a density at a point x at time t and J(x − y) is thought of as the probability distribution of jumping − ∗ from location y to location x,then RN J(y x)u(y, t) dy =(J u)(x, t)istherate at which individuals are arriving at position x from all other places and −u(x, t)= − − RN J(y x)u(x, t) dy is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation (0.1). Equation (0.1) is called nonlocal diffusion equation since the diffusion of the density u at a point x and time t depends not only on u(x, t) and its derivatives, but also on all the values of u in a neighborhood of x through the convolution term J ∗u. This equation shares many properties with the classical heat equation, ut =Δu, such as: bounded stationary solutions are constant, a maximum principle holds for both of them and, even if J is compactly supported, perturbations propagate with infinite speed, [106].
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