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Functional Approach to Nonlinear Models of Water Flow in Soils MATHEMATICAL MODELLING: Theory and Applications VOLUME 21 This series is aimed at publishing work dealing with the definition, development and application of fundamental theory and methodology, computational and algorithmic implementations and comprehensive empirical studies in mathematical modelling. Work on new mathematics inspired by the construction of mathematical models, combining theory and experiment and furthering the understanding of the systems being modelled are particularly welcomed. Manuscripts to be considered for publication lie within the following, non-exhaustive list of areas: mathematical modelling in engineering, industrial mathematics, control theory, operations research, decision theory, economic modelling, mathematical programmering, mathematical system theory, geophysical sciences, climate modelling, environmental processes, mathematical modelling in psychology, political science, sociology and behavioural sciences, mathematical biology, mathematical ecology, image processing, computer vision, artificial intelligence, fuzzy systems, and approximate reasoning, genetic algorithms, neural networks, expert systems, pattern recognition, clustering, chaos and fractals. Original monographs, comprehensive surveys as well as edited collections will be considered for publication. Editor: R. Lowen (Antwerp, Belgium) Editorial Board: J.-P. Aubin (Université de Paris IX, France) E. Jouini (Université Paris IX - Dauphine, France) G.J. Klir (New York, U.S.A.) P.G. Mezey (Saskatchewan, Canada) F. Pfeiffer (München, Germany) A. Stevens (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany) H.-J. Zimmerman (Aachen, Germany) The titles published in this series are listed at the end of this volume. Functional Approach to Nonlinear Models of Water Flow in Soils by Gabriela Marinoschi Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest, Romania AC.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-10 1-4020-4879-3 (HB) ISBN-13 978-1-4020-4879-1 (HB) ISBN-10 1-4020-4880-7 (e-book) ISBN-13 978-1-4020-4880-7 (e-book) Published by Springer, P.O. 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Contents Foreword ...................................................... ix Introduction-motivation ....................................... xi Part I Modelling water infiltration in soils 1 Brief overview of unsaturated flow concepts ............... 3 1.1 Some basic definitions in the unsaturated flow............... 3 1.2 Richards’ equation....................................... 6 1.3 Presentation of the empirical hydraulic models .............. 11 1.4 Comments.............................................. 15 2 Settlement of the mathematical models of nonhysteretic infiltration ................................................ 17 2.1 Physical context and mathematical hypotheses .............. 18 2.2 Strongly nonlinear saturated-unsaturated diffusive model ..... 22 2.3 Weakly nonlinear saturated-unsaturated diffusive model ...... 28 2.4 Quasi-unsaturated model ................................. 29 2.5 Degenerate models....................................... 33 2.6 Extensions of the functions below the field capacity .......... 34 2.7 Dimensionless form of the diffusive models.................. 35 2.8 Comments.............................................. 36 Part II Analysis of infiltration models 3 Basic existence theorems for evolution equations with monotone operators in Hilbertsp.a.c.e.s.................. 43 3.1 The semigroup approach ................................. 43 3.2 Nonlinear m-accretive operators in Hilbert spaces............ 46 3.3 The Cauchy problem within the semigroup approach......... 48 vi Contents 3.4 The Cauchy problem within the variational approach ........ 60 3.5 Comments.............................................. 65 4 Functional approach to the quasi-unsaturated infiltration model ..................................................... 67 4.1 Basic hypotheses for the quasi-unsaturated model ........... 68 4.2 Preliminary results ...................................... 70 4.3 Weakly nonlinear conductivity. Homogeneous Dirichlet boundary conditions ..................................... 76 4.4 Strongly nonlinear conductivity. Homogeneous Dirichlet boundary conditions..................................... 101 4.5 Weakly nonlinear conductivity. Nonhomogeneous Dirichlet boundary conditions..................................... 105 4.6 Comments..............................................128 5 Functional approach to the saturated-unsaturated infiltration model ..........................................133 5.1 Basic hypotheses for the saturated-unsaturated model........134 5.2 The approximating problem ..............................141 5.3 The original problem.....................................166 5.4 The weak solution in the pressure form.....................177 5.5 Existence of the free boundary ............................185 5.6 Uniqueness of the weak solution ...........................198 5.7 Comments..............................................200 6 Specific problems in infiltration............................205 6.1 Analysis of the diffusivity-degenerate model.................205 6.2 Analysis of the porosity-degenerate model ..................209 6.3 Analysis of an infiltration hysteretic model..................224 6.4 Comments..............................................238 Part III Inverse problems in infiltration 7 Identification of the boundary conditions from recorded observations ..............................................243 7.1 Basic concepts in the theory of optimal control..............243 7.2 The identification problem settlement ......................245 7.3 Identification using time average observations ...............248 7.4 Case of a plane soil surface ...............................271 7.5 Identification problem using final time observations ..........273 7.6 Comments..............................................276 Contents vii Part IV Appendix A Background tools ..........................................281 A.1 Some definitions and results in Banach spaces...............281 A.2 Lp spaces and Sobolev spaces .............................284 A.3 Vectorial distributions and Wk,p spaces ....................291 A.4 Operators in Banach spaces...............................295 A.5 Convex functions and subdifferential mappings ..............299 A.6 Various formulas ........................................303 References.....................................................305 Index..........................................................313 Foreword ... a pure mathematician does what he can do as well as he should, whilst an applied mathematician does what he should do as well as he can... (Gr. C. Moisil Romanian mathematician, 1906-1973) Flows in porous media were initially the starting point for the study which has evolved into this book, because the acquirement and improving of know- ledgeabouttheanalysisandcontrolofwaterinfiltrationandsolutespreading arechallenginganddemandingpresentissuesinmanydomains,likesoilscien- ces, hydrology, water management, water quality management, ecology. The mathematical modelling required by these processes revealed from the begin- ning interesting and difficult mathematical problems, so that the attention was redirected to the theoretical mathematical aspects involved. Then, the qualitative results found were used for the explanation of certain behaviours of the physical processes which had made the object of the initial study and for giving answers to the real problems that arise in the soil science practice. In this way the work evidences a perfect topic for an applied mathematical research. Thisbookwaswrittenintheframeworkofmyresearchactivitywithinthe Institute of Mathematical Statistics and Applied Mathematics of the Roma- nianAcademy.SomeresultswereobtainedwithintheprojectCNCSIS33045/ 2004-2006, financed by the Romanian Ministry of Research and Education. In a preliminary form, part of the results included here were lecture notes for master and Ph.D. students during the scientific stages (November- December 2003 and May-June 2004) of the author at the Center for Optimal Control and Discrete Mathematics belonging to the Central China Normal University in Wuhan. ix x Foreword The book addresses mathematicians, applied mathematicians and all re- searchersinterestedinmathematicalproblemssusceptibletobesolvedwithin the semigroup and variational approaches, in particular applied to ground- water flows, and can be used as a basis for a graduate course in Applied Mathematics. This work is a result of the suggestion made by Professor Viorel Barbu, to whom I also owe my initiation in this elegant domain of mathematics and theunderstandingoftheperspectivewhichthefunctionalapproachconfersto appliedproblems.Itakethisopportunitytoexpressmygratitudeforthefruit- fuldiscussionsandobservations,aswellasforthepermanentencouragements provided during the elaboration of this work. I am indebted to Professor Mimmo Iannelli for the helpful mathematical discussions I have had with him in the last years. Also, I would like to thank Springer and in particular Prof. R. Lowen for the publication of my book, and Marlies Vlot, Marieke Mol and Werner Hermens for their kind assistance. Gabriela Marinoschi January 2006, Bucharest Introduction-motivation This book is a work of applied mathematics focusing on the functional study ofthenonlinearboundaryvalueproblemsrelatingtothewaterflowinporous media and it was written with the belief that the abstract theory may be sometimes easier and richer in consequences for applications than standard classicalapproachesare.Thevolumedealswithdiffusiontypemodelsandem- phasizes the mathematical treatment of their nonlinear aspects. An unifying approachtodifferentboundaryvalueproblemsmodellingthewatermovement in porous media is presented, and the high degree of generality and abstrac- tion,kepthoweverwithinreasonablelimits,isrewardedbytherichnessofthe results obtained in this way. Water infiltration, and transport and diffusion of solutes in porous media, are two underground flow processes whose study is of great importance due tothestrongimpacttheyhaveonthehumanlife.Watersuppliedbyrainfalls, irrigationorleakagesfromotherwaterbodiescrossesthesoilcarryingwithit various soluble substances provided by surface or subsurface sources and can reach thephreatic aquifer from where thedrinkablewater isextracted. These processes evolve in time and in principle the problem is to detect the state of the system at any given time when knowing its initial state and the laws that governthesystemchangeswithtime.Themathematicalmodelforsuchasys- tem is an evolution equation, in the most cases a partial differential equation (PDE). Water infiltration and solute dispersion are not the only processes that develop in the underground, but we restricted the study especially to the first one because it is basic and the models describing it are fundamental in the theory of parabolic PDEs, being valid, with slight modifications, for solute dispersion, too. The practical demands lie generally on the quantitative and numerical study of the system evolution, namely on its solution, but the mathematical point of view occurs before this and directs the interest to the proof of facts that allow the approach to make sense. These are the existence, uniqueness and the regularity properties of the solutions. The final intention is to apply theseresultstorealphysicalsystems,soitmustbetakenintoaccountthatthe xi