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Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line

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Well-Defined Lagrangian Flows for Absolutely Continuous Curves of Probabilities on the Line Graduate Theses, Dissertations, and Problem Reports 2016 Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Amsaad, Mohamed H., "Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line" (2016). Graduate Theses, Dissertations, and Problem Reports. 5098. https://researchrepository.wvu.edu/etd/5098 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H. Amsaad Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Adrian Tudorascu, Ph.D., Chair Harumi Hattori, Ph.D. Harry Gingold, Ph.D. Tudor Stanescu, Ph.D. Charis Tsikkou, Ph.D. Department Of Mathematics Morgantown, West Virginia 2016 Keywords: Continuity Equation, Lagrangian Flow, Optimal Transport, Wasserstein metric, Wasserstein space. Copyright c 2016 Mohamed H. Amsaad ABSTRACT Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the line Mohamed H Amsaad Doctor Of Philosophy in Mathematics It is known from Fluid Mechanics [25], [39], [57], [62], [66], [75], [83], [84], [90], [91], [93], [105], [108], that the time-evolution of a probability measure describing some physical quantity (such as the density of a fluid) is related to the velocity of the fluid by the continuity equation. This is known as the Eulerian description of fluid flow. Dually, the Lagrangian description uses the flow of the velocity field to look at the individual trajectories of particles. In the case of flows on the real line, only recently has it been discovered [70], [92] that ``some sort'' of dual Lagrangian flow consisting of monotone maps is always available to match an Eulerian flow. The uniqueness of this ``monotone flow'' among all possible ``flows'' (quotation marks used precisely because traditionally it cannot be called a ``flow'' unless it is unique) of the fluid velocity which is the centerpiece of this dissertation. The Lagrangian description of absolutely continuous curves of probability measures on the real line is analyzed in this thesis. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated to the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated to a given velocity; that is, they provide uniqueness for the solution of the continuity equation within a certain class of curves. Our proposed approach is based on the connection between the flow equation and one-dimensional Optimal Transport. This is based on joint work of the author with A. Tudorascu [21], in which some results on well- posedness (in the one-dimensional case) have been achieved. The results are presented in major conferences and published in a highly ranked mathematical journal. iv Dedication I'd like to dedicate this work. To my optimal parents Hassan Amsaad and Mabruka Omran for their immeasurable support over the years during all my life. To my beloved wife Fatma Naser for taking care of our kids while I was finishing this dissertation. To my children Hassan, Raghad and Jihad, whose love made my desire to finish my PhD every day and night I stayed away from them. To my deer siblings Somia, Fathi, Khaled, Naima, Nader, Jamal, Ahlam and Aisha. To all of my teachers, colleagues and friends for all of their support and encouragement during studies in school and college. I want further to dedicate this thesis to my grandfather Mohamed Amsaad. v Acknowledgments First and foremost, I would like to express my gratitude to Allah (God) Almighty for giving me the strength, knowledge, ability and time to undertake this research study and to persevere and complete it satisfactorily. Without his blessings, this achievement would not have been possible. I would then like to thank Dr. Adrian D. Tudorascu, for giving me the opportunity to work with him. He constantly supervised my work and gave a lot of suggestions regarding my research. I would also like to thank Dr. Harumi Hattori, Dr. Harry Gingold, Dr. Tudor Stanescu and Dr. Charis Tsikkou for being members on my committee. I have been very fortunate to have had the opportunity to take courses with some of them, and their teaching has been essential to my understanding of the subject. I would like to thank the Department of Mathematics of WVU for its support regarding the results obtained in this dissertation. I would like to extend my sincere appreciation to Dr. Harvey Diamond and Dr. Casian Pantea for their great help over my study duration. I wish to thank everybody with whom I had stimulating discussions that led to the inception of this thesis, and their suggestions for indicating ways that led to clarify the presentation, concepts and results. Last but not least, I would like to express my gratitude to my family for their constant guidance and support. First to my beloved wife and my children who care about my successes for pushing me forward through all the stresses and deadlines, while also dealing with their own. I would be remiss if I did not also express my especially grateful to my parents, brothers, and sisters for giving me the motivation to pursue higher studies and gave me a lot of valuable suggestions which are helpful during my stay here in US. I want to say thanks to all of my professors, colleagues and friends who support me all the time and made my stay at WVU so memorable. Their support seemingly has no limit and has been much appreciated throughout my life. Finally, I gratefully acknowledge support from CBIE grant No. 1515 as well as funds from the State of WV and Libyan Ministry of Higher Education. vi Contents Approval Page 2 Acknowledgments v List of Figures ix Notation x Preface 1 1 An overview on Lagrangian and Eulerian descriptions of fluid flows 16 1.1LagrangianvsEulerianpointsofview...................... 17 1.2TheclassicaltheoryofLagrangianflows.................... 20 1.3TheContinuityEquationinthesmoothsetting................ 27 1.3.1 WeakandMeasure–ValuedSolutions.................. 28 1.3.2 Existence and uniqueness of smooth solutions ............. 31 1.3.3 ConnectiontotheTransportEquation................. 36 1.4TheContinuityEquationoutofthesmoothsetting.............. 43 1.4.1 Existenceofweaksolutionsofthecontinuityequation......... 44 1.4.2 VectorfieldswithSobolevspatialregularity.............. 45 1.4.3 Vector fields with BV spatialregularity................. 53 2 Theory of regular Lagrangian flows 55 2.1 The probabilistic point of view .......................... 56 2.1.1 Probabilistic solutions of the continuity equation ............ 57 2.1.2 Probabilistic representation of the measure–valued solutions ..... 60 2.2 ODE uniqueness vs PDE uniqueness . ..................... 61 2.3Ambrosio’snotionofLagrangianflows..................... 63 2.3.1 ThereqularLagrangianflow....................... 63 2.3.2 Lv–Lagrangianflows........................... 64 2.3.3 Well–posednessofLagrangianflows................... 67 2.4DiPerna–Lions’notionofLagrangianflows................... 71 3 Theory of optimal transportations 73 3.1Transportproblemintheclassicalframework................. 74 3.2KantorovichrelaxationofMongeproblem................... 77 CONTENTS vii 3.2.1 Statment of the primal optimization problem .............. 78 3.2.2 Existence of optimal transport plans and couplings .......... 83 3.2.3 Dual optimization problem ........................ 88 3.3 Regularity of Brenier’s map and its applications ................ 95 3.3.1 Existence and uniqueness of Brenier’s maps .............. 96 3.3.2 Monge–Amp´ereequationandregularity................. 102 3.3.3 Brenier’spolarfactorizationandrearrangements............ 104 4 Unidimensional optimal transports 109 4.1 Monotonicity of optimal transports on the real line .............. 110 4.1.1 Inverse distribution functions ...................... 110 4.1.2 Theoptimalityoftransportplansforthequadraticcost........ 123 4.1.3 Monotone transport maps and plans on R ............... 126 4.1.4 Theoptimalityofmonotonemapsforaconvexcost.......... 131 4.2 Monge–Kantorovich metrics on the real line .................. 137 4.2.1 The pth–Wasserstein space and Lp–Wasserstein distance ........ 138 4.2.2 Topology of the pth–Wasserstein space . ................ 142 4.2.3 Convergence in sense
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