New Directions in Mathematical Fluid Mechanics: The Alexander V. Kazhikhov Memorial Volume

434 Pages·2010·6.34 MB·English

by  G. V. Alekseev

Advances in Mathematical Fluid Mechanics
Series Editors
Giovanni P. Galdi John G. Heywood Rolf Rannacher
Department of Mechanical Department of Mathematics Institut für Angewandte Mathematik
Engineering and Materials University of British Columbia Universität Heidelberg
Science Vancouver BC Im Neuenheimer Feld 293/294
University of Pittsburgh Canada V6T 1Y4 69120 Heidelberg
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Advances in Mathematical Fluid Mechanics is a forum for the publication of high quality monographs, or collections
of works, on the mathematical theory of fluid mechanics, with special regards to the Navier-Stokes equations. Its
mathematical aims and scope are similar to those of the Journal of Mathematical Fluid Mechanics. In particular,
mathematical aspects of computational methods and of applications to science and engineering are welcome as
an important part of the theory. So also are works in related areas of mathematics that have a direct bearing on
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New Directions in
Mathematical Fluid
Mechanics
The Alexander V. Kazhikhov Memorial Volume
Andrei V. Fursikov
Giovanni P. Galdi
Vladislav V. Pukhnachev
Editors
Birkhäuser
Basel · Boston · Berlin
Editors:
Andrei V. Fursikov Giovanni P. Galdi
Department of Mechanics Department of Mechanical Engineering
and Mathematics and Materials Science
Moscow State University University of Pittsburgh
Vorob’evy Gory 630 Benedum Engineering Hall
119991 Moscow Pittsburgh, PA 15261
Russia USA
e-mail: [email protected] e-mail: [email protected]
Vladislav V. Pukhnachev
Lavrentyev Institute of Hydrodynamics
Lavrentyev prospect 15
630090 Novosibirsk
Russia
e-mail: [email protected]
2000 Mathematics Subject Classification: 76 (76N, 76D, 76E, 76B), 35, 93, 49
Library of Congress Control Number: 2009935413
Bibliographic information published by Die Deutsche Bibliothek:
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the internet at <http://dnb.ddb.de>
ISBN 978-3-0346-0151-1 Birkhäuser Verlag, Basel – Boston – Berlin
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Contents
Preface ................................................................... vii
Scientific Portraitof
Alexander Vasilievich Kazhikhov .......................................... ix
G.V. Alekseev and D.A. Tereshko
Boundary Control Problems for Stationary Equations
of Heat Convection .................................................. 1
Y. Amirat and V. Shelukhin
Homogenization of the Poisson–BoltzmannEquation ................. 23
S.N. Antontsev and N.V. Chemetov
Superconducting Vortices: Chapman Full Model ..................... 41
D. Bresch, E.D. Fern´andez-Nieto, I.R. Ionescu and P. Vigneaux
Augmented LagrangianMethod and Compressible Visco-plastic
Flows: Applications to Shallow Dense Avalanches .................... 57
D. Bresch, B. Desjardins and E. Grenier
Oscillatory Limits with Changing Eigenvalues: A Formal Study ...... 91
A.Yu. Chebotarev
Finite-dimensional Control for the Navier–Stokes Equations .......... 105
H. Beira˜o da Veiga
On the Sharp Vanishing Viscosity Limit
of Viscous Incompressible Fluid Flows ............................... 113
E. Feireisl and A. Novotn´y
Small P´eclet Number Approximation as a Singular Limit
of the Full Navier-Stokes-FourierSystem with Radiation ............. 123
E. Feireisl and A. Vasseur
New Perspectives in Fluid Dynamics: Mathematical Analysis
of a Model Proposed by Howard Brenner ............................ 153
vi Contents
J. Frehse and M. R˚uˇziˇcka
Existence of a Regular Periodic Solution to the
Rothe Approximation of the Navier–Stokes Equation
in Arbitrary Dimension .............................................. 181
A.V. Fursikov and R. Rannacher
Optimal Neumann Control for the Two-dimensional
Steady-state Navier-Stokes equations ................................ 193
S. Itoh, N. Tanaka and A. Tani
On Some Boundary Value Problem for the Stokes Equations
with a Parameter in an Infinite Sector ............................... 223
A. Khludnev
Unilateral Contact Problems Between an Elastic Plate
and a Beam ......................................................... 237
W. Layton and A. Novotn´y
On Lighthill’s Acoustic Analogy for Low Mach Number Flows ....... 247
A.E. Mamontov
On the Uniqueness of Solutions to Boundary Value Problems
for Non-stationary Euler Equations .................................. 281
M. Padula
On Nonlinear Stability of MHD Equilibrium Figures ................. 301
V.V. Pukhnachev
Viscous Flows in Domains with a Multiply Connected Boundary ..... 333
E.V. Radkevich
Problems with Insufficient Information about
Initial-boundary Data ............................................... 349
V.A. Solonnikov
On the Stability of Non-symmetric Equilibrium Figures
of Rotating Self-gravitating Liquid not Subjected
to Capillary Forces .................................................. 379
V.N. Starovoitov and B.N. Starovoitova
Dynamics of a Non-fixed Elastic Body ............................... 415
Preface
On November 3, 2005, Alexander Vasil’evich Kazhikhov left this world, untimely
and unexpectedly.
Hewasoneofthemostinfluentialmathematiciansinthemechanicsoffluids,
andwillberememberedforhisoutstandingresultsthathad,andstillhave,acon-
siderablysignificantinfluenceinthefield.Amonghismanyachievements,werecall
that he was the founder of the modern mathematical theory of the Navier-Stokes
equations describing one- and two-dimensionalmotions of a viscous, compressible
and heat-conducting gas. A brief account of Professor Kazhikhov’s contributions
to science is provided in the following article “Scientific portrait of Alexander
Vasil’evich Kazhikhov”.
This volume is meant to be an expression of high regard to his memory,
from most of his friends and his colleagues. In particular, it collects a selection of
papers that representthe latestprogressin a number ofnew important directions
of Mathematical Physics,mainly of Mathematical Fluid Mechanics.These papers
are written by world renownedspecialists. Most of them were friends, students or
colleagues of Professor Kazhikhov, who either worked with him directly, or met
him many times in official scientific meetings, where they had the opportunity of
discussing problems of common interest.
We shallnotgivethe detaileddescriptionofthe resultspresentedinthis vol-
ume,but,rather,weshallonlygiveashortlistofthemainareaswheretheseresults
havebeenobtained.Theseareasrangefromboundaryvalueproblemsfordifferent
types of fluid dynamic equations, to certain models describing the properties of
compressibleflows,tolimitsofdifferentkindwithvanishingparameters.Theyalso
include control problems for fluid flow, stability problems for equilibrium figures
of a liquid, problems connected with elastic bodies, Poisson-Boltzmannequation,
and problems with insufficient information about initial and/or boundary data.
In many articles the reader will find an account of the state-of-the-artof the
correspondingdisciplinesthatmayserveasastimulatingstartingpointforfurther
research.
Alsoforthisreason,webelievethatthisvolumecouldbehelpfultospecialists
aswellastoresearcherswhowouldliketobecomeacquaintedwithcertainaspects
of Mathematical Fluid Mechanics.
Moscow, Pittsburgh, and Novosibirsk,September 2009
Andrei V. Fursikov Giovanni P. Galdi Vladislav V. Pukhnachev
Aleksander V. Kazhikov: August 28, 1946–November 3, 2005
Scientific Portrait of
Alexander Vasilievich Kazhikhov
Alexander Vasilievich Kazhikhov was born on the 28th of August in 1946 in the
village of Proskokovo,KemerovoRegion. In 1947 his family moved to the town of
Kolyvan,NovosibirskRegion,wherehegraduatedfromasecondaryschool.In1964
AlexanderKazhikhoventeredthe MechanicsandMathematicsDepartmentatthe
NovosibirskStateUniversity.Upongraduationin1969,hecontinuedhiseducation
asa postgraduatestudent. In1971Alexander Kazhikhovdefended his Candidate
of Science thesis entitled “Global solvability of some boundary value problems in
hydrodynamics”, and occupied a position at the Theoretical Department of the
Institute for Hydrodynamicsofthe SiberianBranchofthe SovietUnionAcademy
of Sciences, where he had been working as a full-time scientific researcher for the
wholeofhisacademiccareer.ThedegreeofDoctorofSciencewasawardedtohim
in 1982on the basis of a successful defence of the thesis entitled “Boundary value
problems for the viscous gas equations and equations of nonhomogeneous fluids”.
Alexander Kazhikhov published about 80 scientific works, the monograph
“Boundary value problems in mechanics of nonhomogeneous fluids” written in
1983 in collaboration with S.N. Antontsev and V.N. Monakhov being the most
known among them. All of Alexander Kazhikhov’s works belong to the field of
mathematical hydrodynamics. This field descends from the works of L. Euler,
who in 1750 derived his famous equations of an ideal liquid. Later C.L. Navier
(1822) and G.G. Stokes (1845) generalized Euler’s equations taking into account
viscosity effects. Since then, great progresshas been achieved in understanding of
the classical models of fluids. However, to date, there still remains unanswered a
set of important mathematical questions regarding solvability of these equations,
aswellasuniquenessandstabilityoftheirsolutions.Differentialequationsoffluid
mechanics are of the greatest interest in applied mathematics, due to numerous
applications in meteorology, aerodynamics, thermodynamics, physics of plasma,
andmanyotherfields.ThecontributionofAlexanderKazhikhovisverysignificant
and has been widely internationally recognized.
Three major directions can be distinguished in Kazhikhov’s studies.
He constructedthe theory ofboundary value problems for the Navier-Stokes
equationsofone-dimensionalmotionofviscousheat-conductinggas.Hispioneering
results in this field were established in the 1970s, by means of a priori estimates
techniques. Alexander Kazhikhov became a master of those techniques.
x Scientific Portrait of A.V. Kazhikhov
Further, one of the first ever results on global solvability for equations of
multi-dimensionalmotionsofviscousgasisduetohim;itwaspublishedin1995in
ajointarticlewithV.A.Weigant.Also,Kazhikhovestablishedthefundamentalsof
the contemporarytheory of viscous nonhomogeneous incompressible fluids, which
was solidly demonstrated in the world-famous monograph of P.-L. Lions entitled
“Mathematical topics in fluid mechanics” (1996).
One more set of remarkable results of Alexander Kazhikhov relates to the
classical Euler equations of an ideal incompressible liquid. It must be emphasized
that the question about well-posedness of boundary value problems for Euler’s
equations is nontrivial, even within the local setting. The theory built by N.M.
Gunter, L. Lichtenstein, W. Wolibner, and N.E. Kochin left open an important
questiononwell-posednessoftheboundaryvalueproblemonflowofliquidthrough
a given domain. In particular, it was unclear whether it was legitimate to impose
boundary conditions on the velocity vector at the entrance of the flow region.
Kazhikhov justified the well-posedness of this boundary value problem and of
some closely related formulations proposed for modeling of liquid flows through
given domains.
As a matter of fact, the words “for the first time” characterize many of
Alexander Kazhikhov’s results. The notion of renormalized solutions to the dif-
ferential mass conservation law was introduced in 1989 by P.-L. Lions and R.J.
DiPernaandremainsrootedinoneoftheearlierworksofKazhikhovonequations
of viscous nonhomogeneous fluids (1974). This notion aims at improving conver-
gence of approximate solutions. In his work, Kazhikhov found that the weakly
convergent sequence of approximate densities converges, in fact, strongly. This
was verified by an analysis of the equation whose solution is the weak limit of the
sequence of squares of approximate densities. Further generalizationof this result
became a basis of contemporary theory. The generalization consists in replacing
the quadratic expression with a proper convex function.
The scientific achievements of Alexander Kazhikhov were honored, in 1978
and1984,bythePrizeoftheSiberianBranchoftheSovietUnionAcademyofSci-
ences,and,in1989,bythe SilverMedalofthe AllSovietIndustrialExposition.In
2003 he became Laureate of the Lavrentiev Prize of the Russian Academy of Sci-
ences.Within the community ofspecialists onthe Navier-Stokesequations he was
outstanding for his persistent tackling of the most complicated and fundamental
problems.
Alexander Kazhikhov frequently lectured abroad. He participated in all sig-
nificant international conferences on mathematical hydrodynamics. One of the
conferenceswasorganizedbyJapanesemathematiciansinFukuokainhonorofhis
50th anniversary.
AlexanderKazhikhovdevotedmuchtime andefforttoscientific,administra-
tive,andpedagogicalactivities.Hewasamemberoftheeditorialboardsof“Jour-
nal of Mathematical Fluid Mechanics”, “Siberian Mathematical Journal”, and
“Vestnik –QuarterlyJournalofNovosibirskState University”.Formany yearshe
actively participated in functions of various dissertation expertise councils and of