Table Of ContentTohru Ozawa
Editor
Collected Papers
in Honor of Yoshihiro
Shibata
Tohru Ozawa
Editor
Collected Papers in Honor
of Yoshihiro Shibata
Previously published in Journal of Mathematical Fluid Mechanics
“Special Issue: Collected Papers in Honor of Yoshihiro Shibata”
Volume 24, issue 1, February 2022, Volume 24, issue 2, May 2022
and Volume 24, issue 3, August 2022
Editor
Tohru Ozawa
Department of Applied Physics
Waseda University
Tokyo, Japan
Spinoff from journal: “Journal of Mathematical Fluid Mechanics” Volume 24, issue 1, February 2022, Volume
24, issue 2, May 2022 and Volume 24, issue 3, August 2022
ISBN 978-3-031-19251-7
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Contents
Preface .............................................................................................................................................. vii
Global Wellposedness of the Primitive Equations with Nonlinear Equation
of State in Critical Spaces ................................................................................................................. 1
Tim Binz and Matthias Hieber: Journal of Mathematical Fluid Mechanics 2022,
2022:36 (19, March 2022) DOI: 10.1007/s00021-022-00672-9
On the Global Existence for the Compressible Euler–Riesz System ........................................... 19
R. Danchin and B. Ducomet: Journal of Mathematical Fluid Mechanics 2022,
2022:48 (30, March 2022) DOI: 10.1007/s00021-022-00664-9
Rotation Problem for a Two-Phase Drop ...................................................................................... 45
I. V. Denisova and V. A. Solonnikov: Journal of Mathematical Fluid Mechanics
2022, 2022:40 (21, March 2022) DOI: 10.1007/s00021-022-00662-x
On the Stokes-Type Resolvent Problem Associated with Time-Periodic
Flow Around a Rotating Obstacle .................................................................................................. 71
Thomas Eiter: Journal of Mathematical Fluid Mechanics 2022, 2022:52
(15, April 2022) DOI: 10.1007/s00021-021-00654-3
Euler System with a Polytropic Equation of State as a Vanishing Viscosity Limit ................... 89
Eduard Feireisl, Christian Klingenberg, and Simon Markfelder: Journal of Mathematical
Fluid Mechanics 2022, 2022:67 (02, June 2022) DOI: 10.1007/s00021-022-00690-7
On the Hydrostatic Approximation of Compressible Anisotropic
Navier–Stokes Equations–Rigorous Justification ....................................................................... 111
Hongjun Gao, Šárka Nečasová, and Tong Tang: Journal of Mathematical
Fluid Mechanics 2022, 2022:86 (20 July 2022) DOI: 10.1007/s00021-022-00717-z
A Route to Chaos in Rayleigh–Bénard Heat Convection ........................................................... 129
Chun-Hsiung Hsia and Takaaki Nishida: Journal of Mathematical Fluid Mechanics
2022, 2022:38 (19, March 2022) DOI: 10.1007/s00021-022-00659-6
Existence of Weak Solution to the Nonstationary Navier-Stokes Equations
Approximated by Pressure Stabilization Method ....................................................................... 147
Hiroki Kikuchi, Takayuki Kubo, and Ranmaru Matsui: Journal of Mathematical
Fluid Mechanics 2022, 2022:80 (09, July 2022) DOI: 10.1007/s00021-022-00711-5
Resolvent Estimates for a Compressible Fluid Model of Korteweg Type
and Their Application ................................................................................................................... 165
Takayuki Kobayashi, Miho Murata and Hirokazu Saito: Journal of Mathematical
Fluid Mechanics 2021, 2022:12 (18, December 2021) DOI: 10.1007/s00021-021-00646-3
v
vi Contents
Rate of the Enhanced Dissipation for the Two-jet Kolmogorov
Type Flow on the Unit Sphere ...................................................................................................... 207
Yasunori Maekawa and Tatsu-Hiko Miura: Journal of Mathematical Fluid
Mechanics 2022, 2022:92 (30, July 2022) DOI: 10.1007/s00021-022-00718-y
Reacting Multi-component Fluids: Regular Solutions in Lorentz Spaces ................................ 259
Piotr Bogusław Mucha and Tomasz Piasecki: Journal of Mathematical
Fluid Mechanics 2022, 2022:37 (19, March 2022) DOI: 10.1007/s00021-022-00670-x
Global Well Posedness for a Q-tensor Model of Nematic Liquid Crystals ............................... 281
Miho Murata and Yoshihiro Shibata: Journal of Mathematical Fluid Mechanics
2022, 2022:34 (16, March 2022) DOI: 10.1007/s00021-022-00677-4
Maximal Regularity for Compressible Two-Fluid System......................................................... 311
Tomasz Piasecki and Ewelina Zatorska: Journal of Mathematical Fluid Mechanics
2022, 2022:39 (19, March 2022) DOI: 10.1007/s00021-022-00671-w
Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet
Boundary Condition for the Temperature .................................................................................. 335
Milan Pokorný: Journal of Mathematical Fluid Mechanics 2022, 2022:17
(13, January 2022) DOI: 10.1007/s00021-021-00655-2
A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions
to the Navier–Stokes Equations .................................................................................................... 351
G. Seregin: Journal of Mathematical Fluid Mechanics 2022, 2022:18
(24, January 2022) DOI: 10.1007/s00021-021-00656-1
Spatial Pointwise Behavior of Time-Periodic Navier–Stokes Flow Induced
by Oscillation of a Moving Obstacle ............................................................................................ 369
Toshiaki Hishida: Journal of Mathematical Fluid Mechanics 2022, 2022:102
(08, September 2022) DOI: 10.1007/s00021-022-00729-9
Preface
This volume is dedicated to Professor Yoshihiro Shibata on the occasion of his
seventieth birthday.
YoshihiroShibatawasbornonthe28thofAugust,1952inTokyo. Heiscurrently
full professor at the Department of Mathematics of Waseda University, a position
thatheholdssince1997. HeisalsoAdjunctProfessorattheMechanicalEngineering
andMaterialsScienceDepartmentoftheUniversityofPittsburgh,honoraryposition
that he holds since 2016. He authored more than 150 papers, with his first one
published in 1979. A number of his contributions represent a landmark in different
areas of nonlinear PDE’s.
Yoshihiro Shibata launched his brilliant career with studies on initial-boundary
value problems for linear hyperbolic systems and, shortly afterwards, nonlinear hy-
perbolic systems such as those associated to classical nonlinear thermoelastic and
viscoelastic models. A fundamental tool that he introduced in this study is en-
ergy decay estimates for the linearized systems by means of the resolvent analysis
[1]. Actually, the celebrated paper of Hirokazu Iwashita on the Lp approach to the
Navier-Stokes initial-boundary value problem in exterior domains was, in fact, com-
pleted under Shibata’s supervision, where the local energy decay and the cut-off
techniques of Shibata played a fundamental role.
The significant impact of Professor Shibata’s work on the Navier-Stokes equa-
tions already occurred with his first contributions to this difficult area of research.
Precisely, he proved algebraic decay properties of solutions to the Oseen equations
in a three-dimensional exterior domain [2] with its application to the global well-
posednessfortheNavier-Stokesprobleminathree-dimensionalexteriordomainwith
non-zero data at infinity [3]. Using the results of [2] and [3], Galdi, Heywood and
Shibata [4] solved the so called “starting problem” proposed by Finn, which was left
unresolved for more than 30 years. His method for the algebraic decay properties
of the Stokes and Oseen equations is based on the cut-off technique with estimates
of the linearized equations and local energy decay, which has been developed in the
study of hyperbolic exterior domain problems [1]. By using a similar technique, in
a joint paper with Toshiaki Hishida, he also proved algebraic decay properties and
the global well-posedness for the rotating problem of the Navier-Stokes equations in
three dimensions [5], a result, the latter that was entirely far from being obvious.
vii
viii Preface
This cut-off technique turned out to be crucial also in the study of free boundary
problems and compressible viscous fluids [6], [7], [8].
Another major contribution of Shibata is the R-bounded solution operators’
method. The method is effectively applied to obtain the L − L maximal regu-
p q
larity for the initial value problem with non-trivial boundary data of a system of
parabolic or even hyperbolic-parabolic mixed equations, including the Stokes equa-
tions in both incompressible and compressible cases. Especially, its application to
the free boundary problems has generated a great number of interesting results
[6], [9]. Moreover, he showed that combination of R-bounded solution operators’
method with the operator valued de Leeuw transference theorem is quite useful for
the maximal regularity of periodic problems [10].
Very recently, he has introduced a new method to prove the L −Bs maximal
1 q;1
regularitytheoremfortheStokesequationswithnon-homogeneousboundarydatain
both compressible and incompressible cases, which will be published shortly. This
proves that Professor Shibata continues to be very active also at the time of his
retirement.
Yoshihiro Shibata has been a member of the editorial board of the Journal of
Mathematical Fluid Mechanics since 2008. He was a spokesperson international
externship, JSPS-DFG, Waseda-TU Darmstadt, “Mathematical Fluid Dynamics”
from 2009 till 2014, together with Professor Mathias Hieber, TU of Darmstadt,
Germany. Moreover, he was the leader of a part of the CREST (Core Research for
Evolutional Science and Technology) – SBM (Search for Breakthrough by Math-
ematics) from 2009 till 2014 (some research results were found in [11]), and the
leader of Mathematics and Physics Unit of Waseda University of the Top Global
University Project organized by Japan Society for the Promotion of Science from
2014 till 2020. He has been the chief organizer of the International Workshop of
Mathematical Fluid Mechanics in Waseda University every year since 2009. This
is a part of his significant contribution to the community of Mathematical Fluid
Mechanics.
ThisspecialissueoftheJournalofMathematicalFluidMechanicscollectspapers
byworldwiderenownexpertsandleadingresearchersintheareamathematicalfluid
mechanics. We would like to thank all the authors for their excellent contribution.
We also wish to thank the reviewers for their critical and enlightening comments on
the original papers.
We are extremely pleased to dedicate this volume to Professor Yoshihiro Shibata
with friendship and admiration.
Giovanni P. Galdi
Tohru Ozawa
Preface ix
References
[1] Y. Shibata, On the global existence of classical solutions of second order fully
nonlinear hyperbolic equations with first order dissipation in the exterior do-
main, Tsukuba J. Math., 7 (1) (1983), 1-68.
[2] T. Kobayashi and Y. Shibata, On the Oseen equation in exterior domains,
Math. Ann., 310 (1) (1998), 1-45.
[3] Y. Shibata, On an exterior initial-boundary value problem for Navier-Stokes
equations, Quart. Appl. Math., 57 (1) (1999), 117-155.
[4] G. P. Galdi, J.G. Heywood, and Y. Shibata, On the global existence and
convergence to steady state of Navier-Stokes flow past an obstacle that is
started from rest, Arch. Rational. Mech. Anal., 138 (4) (1997), 307-318.
[5] T.HishidaandY.Shibata, L −L estimateoftheStokesoperatorandNavier-
p q
Stokesflowsintheexteriorofarotatingobstacle,Arch. RationalMech. Anal.,
193 (2) (2009), 339-421.
[6] Y. Shibata, R-boundedness, Maximal regularity and Free boundary problems
for the Navier-Stokes equations, Mathematical Analysis of the Navier-Stokes
Equations, G.P.GaldiandY.Shibata(Eds.),193-462, LectureNotesinMath-
ematics, 2254, Springer, 2020.
[7] T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations
of motion of compressible viscous and heat-conductive gases in an exterior
domain in R3, Commun. Math. Phys., 200 (3) (1999), 621-659.
[8] Y. Shibata and Xin Zhang, The L −L decay estimate for the multidimen-
p q
sionalcompressibleflowwithfreesurfaceintheexteriordomain,J.Differential
Equations 325 (2022).
[9] Y. Shibata, New thought on Matsumura-Nishida theory in the Lp−Lq maxi-
mal regularity framework, J. Math. Fluid Mech. 24 (3) (2022), Paper No.66,
23pp.
[10] T. Either, M. Kyed and Y. Shibata, On periodic solutions for one-phase and
two phase problems of the Navier-Stokes equations, J. Evol. Equ. 21 (2021),
2955-3014.
[11] Y. Shibata and Y. Suzuki (Eds.), Mathematical fluid dynamics, present and
future, Springer Proceedings in Mathematics and Statistics, Vol 183, Springer,
2016, xix+613pp.
J.Math.FluidMech.(2022)24:36
(cid:2)c 2022TheAuthor(s) Journal of Mathematical
1422-6928/22/020001-18
Fluid Mechanics
https://doi.org/10.1007/s00021-022-00672-9
Global Wellposedness of the Primitive Equations with Nonlinear Equation of State
in Critical Spaces
TimBinzandMatthiasHieber
CommunicatedbyT.Ozawa
DedicatedtoYoshihiroShibataontheoccasionofhis70th-Birthday
Abstract.Thisarticleinvestigatestheprimitiveequationswithnonlinearequationsofstate.Aglobal,strongwell-posedness
resultforthissetofequationsisestablishedforinitialdatalyingincriticalspacesprovidedthatthedensity,dependingon
temperature,salinityandpressure,satisfiescertainregularityassumptions.Theseassumptionsareinparticularsatisfiedfor
theTEOS-80formulationoftheequationofstate.
MathematicsSubjectClassification.Primiary:35Q35,Secondary:76D03,47D06,86A05.
Keywords.Primitiveequations,Nonlinearequationofstate,Criticalspaces.
1. Introduction
Theprimitiveequationsfortheoceanandtheatmospherewithlinearequationofstatearederivedfrom
the Navier–Stokes equations by assuming a hydrostatic balance for the pressure term in the vertical
direction. The analysis of these equations was pioneered by Lions, Temam and Wang in their articles
[12–14],wheretheyprovedtheexistenceofaglobal,weaksolutiontotheseequations.Theiruniqueness
remains an open problem until today. A breakthrough result concerning global strong well-posedness
of the primitive equations for initial data in H1 was shown by Cao and Titi [2] using energy methods.
DifferentapproachesbasedonthetheoryofevolutionequationsandmaximalregularityareduetoHieber
and Kashiwabara [8], and Giga, Gries, Hieber, Hussein and Kashiwabara [5]. The vertical momentum
equationofthefluidisrepresentedduetothehydrostaticbalanceas
∂zπ+gρ=0,
whereπ denotesthepressureandρthedensityofthefluid,respectively,andg>0istheaccelerationof
gravity.Thespecialcaseofthelinearequationofstate
ρ=ρ −α(τ−τ)+β(σ−σ)
0
was investigated in [2,7,11]. Here τ and σ denote temperature and salinity, and ρ , τ and σ given
0
constants. In this article we consider the primitive equations with a general nonlinear equation of state
that relates temperature τ, salinity σ and pressure π to the density ρ. Following Section 1.8 of [18], the
physicalpropertiesofseawaterinthermodynamicalequilibriumaredescribedbyanequationofstatefor
ρoftheform
ρ=R(π,τ,σ). (1.1)
Thisarticleispartofthetopicalcollection“YoshihiroShibata”editedbyTohruOzawa.
A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License
at https://link.springer.com/10.1007/s00021-022-00672-9.
Chapter 1 was originally published as Binz, T. & Hieber, M. Journal of Mathematical Fluid Mechanics (2022) 24: 36. https://
doi.org/10.1007/s00021-022-00672-9.
Reprinted from the journal 1
