Escher J., Guidotti P., Hieber M. et al. (editors) Parabolic Problems: The Herbert Amann Festschrift
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Progress in Nonlinear Differential Equations
and Their Applications
Editor
Haim Brezis
Universit
´
e Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Internationale Superiore di Studi Avanzati, Trieste
Felix Browder, Rutgers University, New Brunswick
Luis Caffarelli, The University of Texas, Austin
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Mariano Giaquinta, University of Pisa
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
Sergiu Klainerman, Princeton University
Robert Kohn, New York University
P. L. Lions, University of Paris IX
Jean Mawhin, Universit
´
e Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
A. Bahri, Rutgers University, New Brunswick
Volume 80
For further volumes:
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The Herbert Amann Festschrift
Parabolic Problems
Yoshihiro Shibata
Christoph Walker
Jan rüss
Piotr Mucha
Matthias Hieber
Joachim Escher
Gieri Simonett
Patrick Guidotti
Editors
B.
Wojciech Zajączkowski
W. P
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ISBN 978-3-0348-0074-7 e-ISBN 978-3-0348-0075-4
DOI 10.1007/978-3-0348-0075-4
Cover design: deblik, Berlin
2010 Mathematics Subject Classification: 35, 47, 76
Joachim Escher
Institut für Angewandte Mathematik
Gottfried Wilhelm Leibniz Universität Hannover
Welfengarten 1
30167 Hannover, Germany
escher@ifam.uni-hannover.de
Matthias Hieber
Fachbereich Mathematik
Technische Universität Darmstadt
Schlossgartenstr. 7
64289 Darmstadt, Germany
hieber@mathematik.tu-darmstadt.de
Jan W. Prüss
Institut für Mathematik
Martin-Luther-Universität Halle-Wittenberg
06099 Halle (Saale), Germany
jan.pruess@mathematik.uni-halle.de
Gieri Simonett
Department of Mathematics
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240, USA
gieri.simonett@vanderbilt.edu
Institute of Mathematics
Polish Academy of Sciences
Sniadeckich 8
00-956 Warsaw, Poland
W.Zajaczkowski@impan.pl
Patrick Guidotti
Department of Mathematics
University of California, Irvine
340 Rowland Hall
Irvine, CA 92697-3875, USA
gpatrick@math.uci.edu
Piotr Mucha
Institute of Applied Mathematics and Mechanics
Warsaw University
ul. Banacha 2
02-097 Warszawa Poland
mucha@hydra.mimuw.edu.pl.
Yoshihiro Shibata
Department of Mathematics
Waseda University
3-4-1 Okubo, Shinjuku-ku
Tokyo 169-8555, Japan
yshibata@waseda.jp
Christoph Walker
Institut für Angewandte Mathematik
Gottfried Wilhelm Leibniz Universität Hannover
Welfengarten 1
30167 Hannover, Germany
walker@ifam.uni-hannover.de
Editors
Wojciech Zajączkowski
B.
Contents
Preface .................................................................. ix
H. Abels
Double Obstacle Limit for a Navier-Stokes/Cahn-Hilliard System . . . . 1
M. Pires and A. Sequeira
Flows of Generalized Oldroyd-B Fluids in Curved Pipes ............. 21
P. Auscher and A. Axelsson
Remarks on Maximal Regularity .................................... 45
G. Bizhanova
On the Classical Solvability of Boundary Value Problems for Parabolic
Equations with Incompatible Initial and Boundary Data . . . . . . . . . . . . 57
D. Bothe
On the Maxwell-Stefan Approach to Multicomponent Diffusion . . . . . . 81
T. Cie´slak and Ph. Lauren¸cot
Global Existence vs. Blowup in a One-dimensional
Smoluchowski-Poisson System ....................................... 95
R. Cross, A. Favini and Y. Yakubov
Perturbation Results for Multivalued Linear Operators .............. 111
W. Desch and S.-O. Londen
Semilinear Stochastic Integral Equations in L
p
...................... 131
B. Ducomet and
ˇ
S. Neˇcasov´a
On the Motion of Several Rigid Bodies in an Incompressible
Viscous Fluid under the Influence of Selfgravitating Forces . . . . . . . . . . 167
J. Escher, Martin Kohlmann and B. Kolev
Geometric Aspects of the Periodic μ-Degasperis-Procesi Equation . . . 193
R. Farwig, H. Kozono and H. Sohr
Global Leray-Hopf Weak Solutions of the Navier-Stokes Equations with
Nonzero Time-dependent Boundary Values . . . . . . . . . . . . . . . . . . . . . . . . . . 211
H.O. Fattorini
Time and Norm Optimality of Weakly Singular Controls . . . . . . . . . . . . 233
vi Contents
G.P. Galdi and M. Kyed
Asymptotic Behavior of a Leray Solution around
a Rotating Obstacle ................................................ 251
M. Geissert and H. Heck
A Remark on Maximal Regularity of the Stokes Equations . . . . . . . . . . 267
D. Guidetti
On Linear Elliptic and Parabolic Problems in Nikol’skij Spaces . . . . . . 275
P. Gwiazda and A.
´
Swierczewska-Gwiazda
Parabolic Equations in Anisotropic Orlicz Spaces with
General N-functions ................................................ 301
R. Haller-Dintelmann and J. Rehberg
Maximal Parabolic Regularity for Divergence Operators
on Distribution Spaces .............................................. 313
T. Hishida
On the Relation Between the Large Time Behavior of the Stokes
Semigroup and the Decay of Steady Stokes Flow at Infinity . . . . . . . . . 343
B. Kaltenbacher, I. Lasiecka and S. Veljovi´c
Well-posedness and Exponential Decay for the Westervelt
Equation with Inhomogeneous Dirichlet Boundary Data . . . . . . . . . . . . . 357
N.V. Krylov
On Divergence Form Second-order PDEs with Growing
Coefficients in W
1
p
Spaces without Weights .......................... 389
K. Laidoune, G. Metafune, D. Pallara and A. Rhandi
Global Properties of Transition Kernels Associated with
Second-order Elliptic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
M. Lewicka
Metric-induced Morphogenesis and Non-Euclidean Elasticity:
Scaling Laws and Thin Film Models ................................ 433
A. Lunardi
Compactness and Asymptotic Behavior in Nonautonomous
Linear Parabolic Equations with Unbounded Coefficients in R
d
...... 447
J. Maas and J. van Neerven
Gradient Estimates and Domain Identification for Analytic
Ornstein-Uhlenbeck Operators ...................................... 463
T. Nau and J. Saal
R-sectoriality of Cylindrical Boundary Value Problems . . . . . . . . . . . . . . 479
J. Pr¨uss and G. Simonett
Analytic Solutions for the Two-phase Navier-Stokes Equations
with Surface Tension and Gravity ................................... 507
Contents vii
J. Pr¨uss and M. Wilke
On Conserved Penrose-Fife Type Models ............................ 541
F. Ragnedda, S. Vernier Piro and V. Vespri
The Asymptotic Profile of Solutions of a Class of Singular
Parabolic Equations ................................................ 577
W.M. Ruess
Linearized Stability for Nonlinear Partial Differential
Delay Equations .................................................... 591
R. Schnaubelt and M. Veraar
Stochastic Equations with Boundary Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 609
G. Seregin
A Note on Necessary Conditions for Blow-up of Energy
Solutions to the Navier-Stokes Equations ............................ 631
S. Shimizu
Local Solvability of Free Boundary Problems for the
Two-phase Navier-Stokes Equations with Surface Tension
in the Whole Space ................................................. 647
V.A. Solonnikov
Inversion of the Lagrange Theorem in the Problem of Stability
of Rotating Viscous Incompressible Liquid .......................... 687
G. Str¨ohmer
Questions of Stability for a Parabolic-hyperbolic System . . . . . . . . . . . . 705
Prof. Herbert Amann
Preface
Herbert Amann studied at the universities of Freiburg, Basel, and M¨unchen in
the early 1960s. In 1965 he received his doctoral degree under the supervision of
Joachim Nitsche from the University of Freiburg. At that time, Herbert Amann’s
research revolved around the use of Monte Carlo simulations in connection with
the resolution of elliptic problems [1]. His research interests then shifted toward the
area of nonlinear integral equations, with a particular focus on the Hammerstein
equation [2, 3]. In 1970 Herbert Amann moved from Freiburg to Bloomington,
Indiana, and, the following year, to Lexington, Kentucky, where he held visiting
professor positions. During the years spent in the US, his interests evolved toward
nonlinear elliptic problems and the use of topological methods for their analysis.
He was appointed full professor at the Ruhr-Universit¨at Bochum in 1972 where he
continued these investigations. Of this time are some of his most frequently cited
and influential research papers about the topological degree [4, 5], the sub- and
supersolution method [6, 7, 8], and multiplicity of solutions for nonlinear elliptic
problems [9, 10]. Of outstanding importance is his consistently highly cited review
article [11] on fixed point theory in ordered Banach spaces.
Herbert Amann moved to the Christian-Albrechts-Universit¨at zu Kiel in
1978, and then to the Universit¨at Z¨urich in 1979. During his tenure in Z¨urich, he
continued his studies on qualitative features of nonlinear elliptic boundary value
problems [12, 13], and then immersed himself in the study of nonlinear para-
bolic problems. A deep and careful understanding of the fundamental properties
of general evolution systems together with the development of the interpolation-
extrapolation framework were an important breakthrough in the study of nonlin-
ear parabolic problems [14, 15, 16]. The full strength of this abstract approach is
apparent in the dynamic theory for general quasilinear systems of parabolic type
[17, 18, 19, 20]. A successful implementation in applications, like, e.g., coagulation-
fragmentation processes [21], requires a thorough insight into the theory of function
spaces and multiplier results, particularly also in the Banach space valued setting.
Among the most important contributions in this context are [20, 22, 23, 24, 25, 35].
In recent years, Herbert Amann also contributed to the development of the theory
of maximal regularity. His comprehensive view on complex structures allowed him
to derive far-reaching results on Navier-Stokes equations, non-Newtonian fluids,
image processing, and evolution equations with memory [26, 27, 28, 29]. Besides
more than 100 research papers, Herbert Amann also has written important mono-
graphs [30, 31] and successful text books [32, 33, 34].