Advances in Mathematical Fluid Mechanics

Rolf Rannacher · Adelia SequeiraEditors

Advances in MathematicalFluid Mechanics

Dedicated to Giovanni Paolo Galdion the Occasion of his 60th Birthday

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EditorsRolf RannacherInstitut fur Angewandte MathematikUniversitat HeidelbergIm Neuenheimer Feld 293/29469120 HeidelbergGermanyrannacher@iwr.uni-heidelberg.de

Adelia SequeiraDepartment of MathematicsCentre for Mathematics and its ApplicationsInstituto Superior Tecnico/UTLAv. Rovisco Pais, 11049-001 LisboaPortugaladelia.sequeira@math.ist.utl.pt

ISBN 978-3-642-04067-2 e-ISBN 978-3-642-04068-9DOI 10.1007/978-3-642-04068-9Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010920233

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Dedicated to Giovanni Paolo Galdi, on theoccasion of his sixtieth birthday

Giovanni Paolo Galdi

Foreword

The present volume celebrates the 60th birthday of Professor Giovanni Paolo Galdiand honors his remarkable contributions to research in the field of MathematicalFluid Mechanics. The book contains a collection of 35 peer reviewed papers, withauthors from 20 countries, reflecting the worldwide impact and great inspirationby his work over the years. These papers were selected from invited lectures andcontributed talks presented at the International Conference on Mathematical FluidMechanics held in Estoril, Portugal, May 21–25, 2007 and organized on the occa-sion of Professor Galdi’s 60th birthday. We express our gratitude to all the authorsand reviewers for their important contributions.

Professor Galdi devotes his career to research on the mathematical analysis of theNavier-Stokes equations and non-Newtonian flow problems, with special emphasison hydrodynamic stability and fluid-particle interactions, impressing the worldwidemathematical communities with his results. His numerous contributions have laiddown significant milestones in these fields, with a great influence on interdisci-plinary research communities. He has advanced the careers of numerous youngresearchers through his generosity and encouragement, some directly through intel-lectual guidance and others indirectly by pairing them with well chosen senior col-laborators. A brief review of Professor Galdi’s activities and some impressions bycolleagues and friends are included here.

This project could not have been successfully concluded without the generoussupport of some collaborators and several Portuguese institutions. Special thanksare due to Joao Janela for the careful preparation of the final version of this book,and also to Thomas Wick for his precious help. The financial and technical sup-port of Fundacao para a Ciencia e a Tecnologia (FCT), Fundacao Calouste Gul-benkian, Fundacao Luso-Americana para o Desenvolvimento (FLAD) and of Centrode Matematica e Aplicacoes (CEMAT), Instituto Superior Tecnico, are gratefullyacknowledged. Finally, a special thanks to Springer-Verlag for accepting to publishthis work.

On behalf of all collaborators and friends of Professor G.P. Galdi we wish himmany more years of continued high energy, great enthusiasm and further impressivemathematical achievements.

Heidelberg, Germany Rolf RannacherLisboa, Portugal Adelia Sequeira

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Short Biography

Giovanni P. Galdi was born in Naples, Italy, on January 3, 1947, where he receivedhis University degree (Laurea) in Physics from the University of Naples in 1971.

He is, currently, the William Kepler Whiteford Professor of Engineering andProfessor of Mathematics at the University of Pittsburgh. He is also Adjunct Facultyat the Tata Institute of Fundamental Research in Mumbai, India. Before joiningthe faculty at the University of Pittsburgh in the Fall semester 1999, in the years1980–1985 he was Professor at the Department of Mathematics of the Universityof Naples (Italy) and, from 1985 until 1998, he was Professor at the Institute ofEngineering of the University of Ferrara (Italy).

Professor Galdi founded and organized the School of Engineering of the Uni-versity of Ferrara in 1989, where he was the Dean from 1989 until 1995. He hasbeen Visiting Professor in several academic institutions, including the University ofGlasgow (Scotland), University of Minnesota (USA), University of Paderborn (Ger-many), University of Pretoria (South Africa), TIFR, Bangalore (India), Fudan Uni-versity, Shanghai (China), University of Waseda, Tokyo (Japan), Czech Academyof Science (Czech Republic), Steklov Institute of Mathematics, the St PetersburgBranch (Russia), University of Paris-Sud XI, Orsay (France), Instituto SuperiorTecnico, Lisbon (Portugal), and University of Pisa (Italy).

In the years 2003 and 2009 he was awarded the Mercator-Gastprofessuren fromDeutschen Forschungsgemeinschaft (DFG).

He is a member of the Editorial Board of several scientific Journals, includingEuropean Journal of Mechanics B/Fluids. He is also co-founder and Editor in Chiefof the Journal of Mathematical Fluid Mechanics, and of the Series Advances inMathematical Fluid Mechanics, published by Birkhauser-Verlag, Basel, Boston.

Professor Galdi has (co) authored over 130 original research papers and 6books, and (co) edited 13 books, dedicated mostly to hydrodynamic and mag-netohydrodynamic stability, mathematical theory of the Navier-Stokes equations,non-Newtonian fluid mechanics and fluid-solid interactions. In particular, his two-volume book “An Introduction to the Mathematical Theory of the Navier-StokesEquation”, first published with Springer-Verlag in 1994, is a classical milestone inthe steady-state theory of the Navier-Stokes equations.

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Paolo Galdi – The Man and the Mathematician

During the 1960s I worked hard on the Navier-Stokes equations, writing a numberof papers. Then I sensed that the problems weren’t getting easier, on top of whichcompetition was moving in, so I looked for other things to do. My instincts soonfound expression in reality when G.P. Galdi appeared on the scene; he would havebeen a formidable competitor and I’m happy I didn’t have to face that challenge. Hehad a voracious appetite for knowledge, and I recall repeated requests for reprints ofmy papers. At that time, e-mail was not yet suitable for exchanging manuscripts, andreprints were generally sent as hard copies through postal services. He requested anearly paper, which I dutifully sent him. Postal service between USA and Italy wasrather slow, and apparently he lost patience, as shortly later he requested the samepaper again. I assumed then that he wanted a later paper, and sent him that one. Thatseems to have arrived prior to the one he wanted, as I then got still another requestfrom him. Again there was confusion, and I sent him still another paper. Ever since,he has been repeatedly accusing me (publicly) of sending him things he didn’t want.

Never mind! It is clear that he read all the papers (also those written by others),absorbed their content, and then carried the theory further in new ways that haveleft a permanent imprint with his signature. A case close to my heart is his 1997paper with Heywood and Shibata, in which the dissertation problem that I gaveto John Heywood about 1965, on which Heywood at the time made deep initialprogress, was finally solved. Beyond that is his impressive recasting of my ownresults on exterior problems into function space settings, and his extensions of theresults to rotating and more general periodic motions. Galdi became – and continuesto be – a central figure in describing and clarifying some of the most profoundproblems in modern hydromechanics. He has established himself as one of the fewtop contributors to a theory that has attracted worldwide interest and activity, andas a person who has stimulated and encouraged the creative achievements of manyothers. He is in fact in large part responsible for the present worldwide interest,by calling attention to the beauty, depth and underlying unity of the many openproblems.

Stanford, California Robert Finn

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xii Paolo Galdi – The Man and the Mathematician

As many people have commented, “Paolo Galdi is a very special person”. Hisenthusiasm, generosity, kindness and discernment have made him a most valu-able colleague to the whole community of researchers working on topics in fluiddynamics.

Paolo’s own contributions to fluid dynamics have been very significant and wideranging. As an Editor of the Handbook of Mathematical Fluid Dynamics, I was verypleased to include an article by Paolo “On the motion of a rigid body in a viscousfluid” which covered an interesting sub-class of important fluid applications, andillustrated Paolo’s versatility.

Through hard work and dedication, Paolo, John Heywood and Rolf Rannacherhave created a major journal in the field, The Journal of Mathematical Fluid Dynam-ics, which has encouraged the resurgence of a classical topic in mathematics that isagain taking center stage in partial differential equations.

I was honored and delighted to be part of the wonderful celebration of Paolo’s60th birthday that Adelia Sequeira and her colleagues organized in Estoril in 2007.

Los Angeles, California Susan Friedlander

I am not sure when I first met Paolo. I think it was in 1976 when I published my bookon the stability of fluid motions. The mathematicians in Naples, under SalvatoreRionero, had taken an interest in the energy theory of stability. Paolo was a studentof Rionero. I had published papers on that subject which led to my 1976 book on thestability of fluid motions. I was greatly stimulated to go in this direction by papersof James Serrin. I think that Mariolina Padula, Paolo’s wife then, was also Rionero’sstudent. She was, in any case, very active in math and she and Paolo would studytogether. Paolo and Mariolina came to the US I think in 1976. I do not know howtheir trip was financed. In any case, my wife then (Ellen) and I had a little party inour house which was on a lake in Minnesota. James Serrin was also a guest. I thinkit was in 1976 because Paolo reminded me that my book had arrived from Springerthat very day. I think that Paolo and Mariolina were very impressed to meet withpersons they though so great in an environment so different that via Mezzacannone.This is the time that Paolo and I became fast friends. I like to say that people fromMinneapolis and Naples are natural friends since Minneapolitans can be thought tobe small persons from Naples.

The next stage of our friendship developed in successive trips to Italy. These tripswere arranged by Professor Rionero. In the first trip, Paolo and Mariolina invitedme to their home. There, I learned about Paolo’s special talents in the arts as apainter specializing in portraits of Donald Duck and as a fine pianist specializingin Chopin. I loved Naples. For years in the early 1980s I taught in the summerschool in Ravello. These were very pleasant summers. In 1982, I practiced thereto run the original marathon in Greece. I would run up to Valico di Chiunzi andback, chased always by angry Italian dogs. The further south you go in Europe, thegreater are the number of female math students and there were many nice ragazzi

Paolo Galdi – The Man and the Mathematician xiii

in Ravello. I remember one handsome Italian mathematician from Bologna tellingme that his marriage license was not valid in the South. Later, it was not valid in thenorth.

I had many intimate conversations with Paolo in the cafes in Ravello. At thistime he was drifting to a more rigorous approach to mathematics. I urged himto continue his studies on the applied side, warning him that if he followed hisdesire he would have to compete with fine mathematicians much better preparedthan he. At that time, he was working on energy theory on unbounded domains.To complete his results, he needed some powerful Navier Stokes theorems. I thinkthe work of Leray was involved in his theorems. He was intense about mathemat-ics and had gone beyond the energy theory of stability which frankly is a theorywhich demands that you know how to use the divergence theorem and integrateby parts. Gradually, our trajectories in science grew apart. Paolo was encouragedto prepare his now well known Springer monograph on the Navier Stokes equa-tions by Clifford Truesdell, who at an earlier time was also my mentor but later mytormentor.

In 1991, Paolo and Salvatore visited me again in Minneapolis. I had developeda theory for miscible liquids and showed that mixtures of incompressible mixingliquids are compressible and obtained a new theory of diffusion. The velocity is notsolenoidal. Paolo found that a certain combination of the velocity and expansionvelocity was divergence free. It turns out that this combination is equivalent toa volume averaged velocity. It is a great result, which we used in all subsequentpapers.

Paolo is a very special, outgoing and supportive person. He has that magic per-sonality which radiates interest and concern about others wherever he goes. Heengages all persons and elevates their level of well being. Maybe this is why hehas so many friends all over the world and at the great birthday meeting celebratedby this volume.

I know three mathematicians from the applied side who developed a burningdesire to be the master of rigorous mathematics. This goal is at the top of a mountain.One of these mathematicians in Klaus Kirchgassner, another is Edward Fraenkeland the third is Paolo. It was a difficult journey, but Paolo has reached the summitwithout forgetting the applied side. It is my pleasure to wish happy 60th birthday tothis good friend, great man and fine mathematician.

Minneapolis, Minnesota Daniel D. Joseph

Naples, via Mezzocannone, 8, early Seventies. It is here, in this stern and monu-mental environment where Tommaso d’Aquino gave his theological lectures, thatI, a young student of Mathematics, take my first steps in the field of mathemati-cal research. Professor Salvatore Rionero introduces and guides me into this worldwhere at first I feel awe. Further than him, I owe his young co-worker GiovanniPaolo Galdi my quick and complete adaption, so that I could feel quite at ease

xiv Paolo Galdi – The Man and the Mathematician

among those great staircases and those lecture halls where one could smell theodor of history. There had lived and worked mathematicians of the level of ErnestoCesaro, Roberto Marcolongo, Mauro Picone and Renato Caccioppoli and there stillworked in that period great masters like Alfredo Franchetta, Carlo Miranda andCarlo Tolotti.

Paolo, as I called him at once since at once we became friends, was for me amost precious guide towards that world which seemed to me very far and difficultto reach. Paolo addressed me to the journals he deemed more suitable for my kindof research, gave me his advice, but above all stimulated me with his observationsand his sparkling conversation. Our friendship and our work relations consolidatedin time. I always remember the wonderful evenings passed together, when Paolo, asa true showman, was the life and soul of the company with his pleasantness and hisskill as a piano player; or the football games played on Saturday in a small pitchon one of the finest hills of Naples. I always remember the days Marina and I spentat the seaside together with Paolo, Mariolina, Adriana and Giovanni, swimming inthe crystal clear waters of Calabria and playing on the beach. Memories the timewill never wipe out, even though life has assigned to each of us a different road tofollow.

Caserta, Italy Remigio Russo

Christian Simader met Paolo Galdi for the first time in 1988, exactly 20 years ago.He describes this meeting as follows: “In the spring 1987 I was a visitor at theUniversity of Catania in Sicily, where I gave several lectures. One was devoted tothe Helmholtz decomposition of vector fields. Shortly before this visit, HermannSohr and I found an elementary proof of this theorem. Professor Giuseppe Mulonesuggested I contact Professor Galdi from Ferrara who, at the time, was writing abook on the Navier-Stokes equations. In April 1988, my wife and I intended toparticipate in an intensive Italian language course at Venice. Shortly before we leftfor Venice, I realized that Ferrara is close to Venice, so I immediately contacted Pro-fessor Galdi who invited me to come to Ferrara. At that time I was already familiarwith many of his papers, but I had never met him in person. These papers impressedme deeply because of their clearness, accuracy and profoundness. Therefore, I hadthe impression that the author had to be a mature mathematician, much older than Iwas. Our first appointment was in a hotel in Ferrara. Precisely at the time of ourappointment, a young couple entered the lounge and was obviously looking forsomeone. After some seconds, I asked the man if he was Professor Galdi. He saidyes and I introduced my wife and myself. He was very surprised – and told me thathe thought vice versa that I had to be much older. From his mathematical studieswith Professor Carlo Miranda at Naples, he knew my old Springer Lecture Notesfrom 1972 which in fact was an English translation (at least I regarded it so) of mythesis from 1968. So we laughed a lot and spent a very nice evening together, enjoy-ing a wonderful dinner. My wife and I had the impression of a very sympathetic and

Paolo Galdi – The Man and the Mathematician xv

cultured couple, interested in many topics – clearly very much in mathematics. Thisimpression was deepened in the following days, especially when we met their threechildren. Briefly expressed, it was reciprocal sympathy at the first view. At that time,the joint work with Hermann Sohr on the L-Dirichlet problem for the Laplacian inexterior domains was in progress. The problem turned out to be much more difficultthan we expected. Though clearly we had Stokes’ system in mind, we regarded thisproblem as a first step and as a type of model to find the appropriate function spaces.At that time, we worked with function spaces which in certain cases (1 < q < n)turned out to be too “small” and we had to impose a certain compatibility conditionon the data such that we could prove existence and uniqueness of weak L-solutionsin exterior domains. I presented a lecture on the results Hermann and I had achieved.Afterwards Paolo (meanwhile, we had begun to use first names) told me, that he wasstudying the corresponding problem for Stokes’ system. Some weeks after returningto Bayreuth, Paolo called me and asked again for our compatibility condition whichturned out to be different from the one he found. So I returned to Ferrara and westarted to jointly work on the exterior Stokes’ system. It turned out that we have asimilar mathematical taste and a similar way in regarding problems. These similari-ties, simplified our collaboration very much. But there were also differences: Paoloknew a magnitude more about Stokes’ problem than I did and he was much fasterin thinking and calculating. But it was a very happy collaboration and within somedays we solved the problem completely and the joint paper appeared 1992 in theArchive. But to tell the truth, 90% of the paper is due to Paolo. In the sequel, I oftenvisited Ferrara which also gave me the opportunity to meet many truly sympathetic,excellent mathematicians from all over the world. In 1991 Deutsche Forschungsge-meinschaft (DFG) installed a research group “Equations of Hydrodynamics” at theUniversity of Bayreuth (members: Professors F. Busse, C. G. Simader, M. Wiegner,W. von Wahl (all Bayreuth) and H. Sohr (Paderborn)) which worked until the spring1998. This group provided us the financial basis to invite Paolo to Bayreuth andPaderborn, where a fruitful collaboration with Hermann Sohr started too. This wasvery important since at that time Hermann did (not yet) like to travel.

Now, we can continue to jointly describe our impressions of Paolo. In May1992, mainly Professors I. Straskraba and R. Salvi organized a meeting on Navier-Stokes equations in the wonderful Villa Monastero in Varenna. This meeting wasfollowed by many other meetings in the castle of Thurnau near Bayreuth (1992),Cento/Italy (1993), Funchal/Madeira (1994), Toulon-Hyeres/France (1995), Preto-ria/South Africa (1996) and again in Varenna (1997). It was surely Paolo’s influencethat both very young mathematicians as well as very experienced mathematicianswere included for these scientific gatherings. In addition to an enormous numberof very important papers, to Paolo’s persistent merit are the two volumes of “AnIntroduction to the Mathematical Theory of the Navier-Stokes Equations” whichwas finished in 1992 and appeared in 1994. At that time, these books representedthe state of the art in this field. His book is both very precise and readable. Besides,including an excellent preface, there is an introduction to each chapter where thereader is clearly introduced to the aims of the chapter and the underlying ideas.Due to our knowledge, many young people became fascinated by this book for the

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Navier-Stokes equations. We are very much indebted to Paolo as a friend and as anoutstanding scientist. We wish him (and us too) the chance to celebrate many furtherbirthdays together, such as the 70th, 80th, 90th and. . . .

Bayreuth, Germany Christian G. SimaderPaderborn, Germany Hermann Sohr

Contents

Isotropically and Anisotropically Weighted Sobolev Spaces for theOseen Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Cherif Amrouche and Ulrich Razafison

A New Model of Diphasic Fluids in Thin Films . . . . . . . . . . . . . . . . . . . . . . . . 25Guy Bayada, Laurent Chupin, and Berenice Grec

On the Global Integrability for Any Finite Power of the Full Gradientfor a Class of Generalized Power Law Models p < 2 . . . . . . . . . . . . . . . . . . . 37Hugo Beirao da Veiga

Steady Flow Around a Floating Body: The Rotationally Symmetric Case . 43Josef Bemelmans and Mads Kyed

On a Stochastic Approach to Eddy Viscosity Models for Turbulent Flows . 55Luigi C. Berselli and Franco Flandoli

Numerical Study of the Significance of the Non-Newtonian Nature ofBlood in Steady Flow Through a Stenosed Vessel . . . . . . . . . . . . . . . . . . . . . . . 83Tomas Bodnar and Adelia Sequeira

A Priori Convergence Estimates for a Rough Poisson-Dirichlet Problemwith Natural Vertical Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Eric Bonnetier, Didier Bresch, and Vuk Milisic

Vortex Induced Oscillations of Cylinders at Low and IntermediateReynolds Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Roberto Camassa, Bong Jae Chung, Philip Howard, Richard McLaughlin,and Ashwin Vaidya

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One-dimensional Modelling of Venous Pathologies: Finite Volume andWENO Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147Nicola Cavallini and Vincenzo Coscia

On the Energy Equality for Weak Solutions of the 3D Navier-StokesEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Alexey Cheskidov, Susan Friedlander, and Roman Shvydkoy

The ( p − q) Coupled Fluid-Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Luisa Consiglieri

A Potential-Theoretic Approach to the Time-Dependent Oseen System . . . 191Paul Deuring

Regularity of Weak Solutions for the Navier-Stokes Equations ViaEnergy Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Reinhard Farwig, Hideo Kozono, and Hermann Sohr

Looking for the Lost Memory in Diffusion-Reaction Equations . . . . . . . . . . 229Jose Augusto Ferreira and Paula de Oliveira

Maximum Principle and Gradient Estimates for Stationary Solutions ofthe Navier-Stokes Equations: A Partly Numerical Investigation . . . . . . . . . . 253Robert Finn, Abderrahim Ouazzi, and Stefan Turek

A Study of Shark Skin and Its Drag Reducing Mechanism . . . . . . . . . . . . . . 271Elfriede Friedmann, Julia Portl, and Thomas Richter

Stability of Poiseuille Flow in a Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . 287Antony A. Hill and Brian Straughan

Towards a Geometrical Multiscale Approach to Non-Newtonian BloodFlow Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295Joao Janela, Alexandra Moura, and Adelia Sequeira

The Role of Potential Flow in the Theory of the Navier-Stokes Equations . 311Daniel D. Joseph

Small Perturbations of Initial Conditions of Solutionsof the Navier-Stokes Equations in the L3-Norm and Applications . . . . . . . . 319Petr Kucera

Streaming Flow Effects in the Nearly Inviscid Faraday Instability . . . . . . . . 329Elena Martın and Jose M. Vega

Contents xix

The Dirichlet Problems for Steady Navier-Stokes Equations in Domainswith Thin Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339Yuliya V. Namlyeyeva, Sarka Necasova, and Igor Igorievich Skrypnik

Existence of Weak Solutions to the Equations of Natural Convectionwith Dissipative Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367Joachim Naumann and Jorg Wolf

A Weak Solvability of the Navier-Stokes Equation with Navier’sBoundary Condition Around a Ball Striking the Wall . . . . . . . . . . . . . . . . . . 385Jirı Neustupa and Patrick Penel

On the Influence of an Absorption Term in Incompressible Fluid Flows . . . 409Hermenegildo B. de Oliveira

Adaptive FE Eigenvalue Computation with Applications toHydrodynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425Rolf Rannacher

Numerical Simulation of Laminar Incompressible Fluid-StructureInteraction for Elastic Material with Point Constraints . . . . . . . . . . . . . . . . . 451Mudassar Razzaq, Jaroslav Hron, and Stefan Turek

On Stokes’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473Remigio Russo

On a C0 Semigroup Associated with a Modified Oseen Equation withRotating Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513Yoshihiro Shibata

A New Approach to the Regularity of Weak Lq-Solutions of Stokes andSimilar Equations via the Cosserat Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 553Christian G. Simader

Large Time Behavior of Energy in Some Slowly Decreasing Solutions ofthe Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573Zdenek Skalak

A Selected Survey of the Mathematical Theory of 1D Flows . . . . . . . . . . . . . 581Ivan Straskraba

A Numerical Method for Nonstationary Stokes Flow . . . . . . . . . . . . . . . . . . . 589Werner Varnhorn

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A New Criterion for Partial Regularity of Suitable Weak Solutions tothe Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613Jorg Wolf

An In Vitro Device for Evaluation of Cellular Response to Flows Foundat the Apex of Arterial Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631Zijing Zeng, Bong Jae Chung, Michael Durka, and Anne M. Robertson

Contributors

Cherif Amrouche Laboratoire de Mathematiques Appliquees, UMR CNRS5142, Universite de Pau et des Pays de l’Adour, IPRA, 64000 Pau, France,cherif.amrouche@univ-pau.fr

Guy Bayada INSA de Lyon, Institut Camille Jordan CNRS UMR 5208 &Lamcos CNRS UMR 5259, Bat. L. de Vinci, F-69621 Villeurbanne, France,guy.bayada@insa-lyon.fr

Josef Bemelmans Institut fur Mathematik RWTH-Aachen, Aachen, Germany,bemelmans@instmath.rwth-aachen.de

Luigi C. Berselli Dipartimento di Matematica Applicata “U. Dini,” Universita diPisa, I-56127 Pisa, Italy, berselli@dma.unipi.it

Tomas Bodnar Department of Technical Mathematics, Faculty of MechanicalEngineering, Czech Technical University, 121 35 Prague 2, Czech Republic,bodnar@marian.fsik.cvut.cz

Eric Bonnetier LJK-IMAG, UMR 5523 CNRS, 38041 Grenoble cedex 9, France,Eric.Bonnetier@imag.fr

Didier Bresch LAMA, UMR 5127 CNRS, Universite de Savoie, 73217 LeBourget du Lac cedex, France, Didier.Bresch@univ-savoie.fr

Roberto Camassa Department of Mathematics, University of North Carolina,Chapel Hill, NC 27599, USA, camassa@email.unc.edu

N. Cavallini Research Center “Mathematics for Technology”, Universityof Ferrara, Scientific-Technological Campus, 44100 Ferrara, Italy,nicola.cavallini@unife.it

Alexey Cheskidov Department of Mathematics, University of Chicago, Chicago,IL 60637, USA, acheskid@math.uchicago.edu

Bong Jae Chung Department of Marine Sciences, University of North Carolina,Chapel Hill, NC 27599, USA, bjchung@email.unc.edu

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xxii Contributors

Bong Jae Chung Department of Marine Sciences, University of NorthCarolina-Chapel Hill, Chapel Hill, NC 27599, USA, bjchung@email.unc.edu

Laurent Chupin INSA de Lyon, Institut Camille Jordan CNRS UMR 5208,F-69621 Villeurbanne, France, laurent.chupin@insa-lyon.fr

Luisa Consiglieri Department of Mathematics and CMAF, Sciences Faculty,University of Lisbon, 1749-016 Lisboa, Portugal, lcconsiglieri@fc.ul.pt

Vincenzo Coscia Department of Mathematics, University of Ferrara, 44100Ferrara, Italy, cos@unife.it

Hugo Beirao da Veiga Dipartimento di Matematica Applicata “U. Dini”Universita di Pisa, 56127-Pisa, Italy, bveiga@dma.unipi.it

Paula de Oliveira CMUC-Department of Mathematics, University of Coimbra,Coimbra, Portugal, poliveir@mat.uc.pt

Hermenegildo B. de Oliveira FCT – Universidade do Algarve, Campus deGambelas 8005-139 Faro, Portugal, holivei@ualg.pt

Paul Deuring Universite Lille Nord de France, F-59000 Lille, France; ULCO,LMPA, F-62228 Calais, France, Paul.Deuring@lmpa.univ-littoral.fr

Michael Durka Department of Mechanical Engineering and Materials Science,University of Pittsburgh, Pittsburgh, PA 15261, USA, mjd4+@pitt.edu

Reinhard Farwig Department of Mathematics, Darmstadt University ofTechnology, 64283 Darmstadt, Germany, farwig@mathematik.tu-darmstadt.de

Jose Augusto Ferreira CMUC-Department of Mathematics, University ofCoimbra, Coimbra, Portugal, ferreira@mat.uc.pt

Robert Finn Department of Mathematics, Stanford University, Stanford, CA,USA, finn@math.stanford.edu

Franco Flandoli Dipartimento di Matematica Applicata “U. Dini,” Universita diPisa, I-56127 Pisa, Italy, flandoli@dma.unipi.it

Susan Friedlander Department of Mathematics, Statistics and Computer Science,University of Illinois, Chicago, IL 60607, USA, susan@math.northwestern.edu

Elfriede Friedmann Department of Applied Mathematics, University Heidelberg,69120 Heidelberg, Germany, friedmann@iwr.uni-heidelberg.de

Berenice Grec Ecole Centrale de Lyon, Institut Camille Jordan CNRS UMR5208, F-69621 Villeurbanne, France, berenice.grec@insa-lyon.fr

Antony A. Hill School of Mathematical Sciences, University of Nottingham,Nottingham, NG7 2RD, UK, antony.hill@nottingham.ac.uk

Philip Howard Department of Statistics, University of North Carolina, ChapelHill, NC 27599, USA, pdhoward@email.unc.edu

Contributors xxiii

Jaroslav Hron Institute of Mathematics, Charles University, Prague, CzechRepublic, hron@karlin.mff.cuni.cz

Joao Janela Department of Mathematics/ISEG and CEMAT/IST, 1200 Lisboa,Portugal, jjanela@iseg.utl.pt

Daniel D. Joseph University of Minnesota, Minneapolis, MN; Universityof California, Irvine, CA, USA, joseph@aem.umn.edu

Hideo Kozono Mathematical Institute, Tohoku University, Sendai, 980-8578Japan, kozono@math.tohoku.ac.jp

Petr Kucera Department of Mathematics, Faculty of Civil Engineering, CzechTechnical University, 16629 Prague 6, Czech Republic, kucera@mat.fsv.cvut.cz

Mads Kyed Institut fur Mathematik, RWTH-Aachen, Aachen, Germany,kyed@instmath.rwth-aachen.de

Elena Martın E. T. S. Ingenieros Industriales, Universidad de Vigo, Vigo, Spain,emortega@uvigo.es

Richard McLaughlin Department of Mathematics, University of North Carolina,Chapel Hill, NC 27599, USA, rmm@email.unc.edu

Vuk Milisic LJK-IMAG, UMR 5523 CNRS, 38041 Grenoble cedex 9, France,Vuk.Milisic@imag.fr

Alexandra Moura CEMAT/IST, 1049-001 Lisboa, Portugal, almoura@math.ist.utl.pt

Yuliya V. Namlyeyeva Institute of Applied Mathematics and Mechanics of NASof Ukraine, 83114 Donetsk, Ukraine, namleeva@iamm.ac.donetsk.ua

Joachim Naumann Mathematical Institute, Humboldt University Berlin, 10099Berlin, Germany, jnaumann@math.hu-berlin.de

Sarka Necasova Mathematical Institute of Academy of Sciences, 11567 Prague1, Czech Republic, matus@math.cas.cz

Jirı Neustupa Mathematical Institute of the Czech Academy of Sciences, 115 67Praha 1, Czech Republic, neustupa@math.cas.cz

Abderrahim Ouazzi Institute of Applied Mathematics, TU Dortmund, D-44227Dortmund, Germany, abderrahim.ouazzi@math.uni-dortmund.de

Patrick Penel Departement de Mathematique & Laboratoire “Systemes NavalsComplexes”, Universite du Sud Toulon–Var, 83957 La Garde cedex, France,penel@univ-tln.fr

Julia Portl Interdisciplinary Center for Scientific Computing (IWR), UniversityHeidelberg, 69120 Heidelberg, Germany, julia.portl@iwr.uni-heidelberg.de

Rolf Rannacher Institute of Applied Mathematics, University of Heidelberg,D-69120 Heidelberg, Germany, rannacher@iwr.uni-heidelberg.de

xxiv Contributors

Ulrich Razafison MAPMO, UMR CNRS 6628, Federation Denis Poisson, bat.Mathematiques, B. P. 6759, Orleans cedex 2, France, ulrich.razafison@math.cnrs.fr

Mudassar Razzaq Institute of Applied Mathematics, TU Dortmund, Germany,mrazzaq@math.uni-dortmund.de

Thomas Richter Institute of Applied Mathematics, University Heidelberg, 69120Heidelberg, Germany, thomas.richter@iwr.uni-heidelberg.de

Anne M. Robertson Department of Mechanical Engineering and MaterialsScience, McGowan Institute for Regenerative Medicine, Center for VascularRemodeling and Regeneration (CVRR); University of Pittsburgh, Pittsburgh, PA15261, USA, rbertson@pitt.edu

Remigio Russo Dipartimento di Matematica, Seconda Universita di Napoli, 81100Caserta, Italy, remigio.russo@unina2.it

Adelia Sequeira Department of Mathematics and CEMAT, Instituto SuperiorTecnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal,adelia.sequeira@math.ist.utl.pt

Yoshihiro Shibata Department of Mathematics, Research Institute of Science andEngineering, Waseda University, Tokyo 169-8555, Japan, yshibata@waseda.jp

Roman Shvydkoy Department of Mathematics, Statistics and Computer Science,University of Illinois, Chicago, IL 60607, USA, shvydkoy@math.uic.edu

Christian G. Simader Universitat Bayreuth, D-95447 Bayreuth, Germany,Christian.Simader@uni-bayreuth.de

Zdenek Skalak Czech Technical University, 16612 Prague 6, Czech Republic,skalak@mat.fsv.cvut.cz

Igor Igorievich Skrypnik Institute of Applied Mathematics and Mechanics ofNAS of Ukraine, 83114 Donetsk, Ukraine, iskrypnik@iamm.donbass.com

Hermann Sohr Faculty of Computer Science, Electrical Engineering andMathematics, University of Paderborn, 33098 Paderborn, Germany,hsohr@math.uni-paderborn.de

Ivan Straskraba Mathematical Institute, Academy of Sciences of the CzechRepublic, 11567 Prague 1, Czech Republic, strask@math.cas.cz

Brian Straughan Department of Mathematical Sciences, Durham University,Durham, DH1 3LE, UK, brian.straughan@durham.ac.uk

Stefan Turek Institute of Applied Mathematics, TU Dortmund, Germany,ture@featflow.de

Ashwin Vaidya Department of Mathematics, University of North Carolina,Chapel Hill, NC 27599, USA, avaidya@email.unc.edu

Contributors xxv

Werner Varnhorn Fachbereich Mathematik, Universitat Kassel, 34109 Kassel,Germany, varnhorn@mathematik.uni-kassel.de

Jose M. Vega E. T. S. Ingenieros Aeronauticos, Universidad Politecnica deMadrid, Madrid, Spain, vega@fmetsia.upm.es

Jorg Wolf Mathematical Institute, Humboldt University Berlin, 10099 Berlin,Germany, jwolf@math.hu-berlin.de

Zijing Zeng Department of Mechanical Engineering and Materials Science,University of Pittsburgh, Pittsburgh, PA 15261, USA, ziz3+@pitt.edu