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Lecture Notes in Mathematics
C.I.M.E. Foundation Subseries
Volume 2272
Editors-in-Chief
Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
SeriesEditors
KarinBaur,UniversityofLeeds,Leeds,UK
MichelBrion,UGA,Grenoble,France
CamilloDeLellis,IAS,Princeton,NJ,USA
AlessioFigalli,ETHZurich,Zurich,Switzerland
AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany
DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA
IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK
AngelaKunoth,UniversityofCologne,Cologne,Germany
ArianeMézard,IMJ-PRG,Paris,France
MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg
SylviaSerfaty,NYUCourant,NewYork,NY,USA
GabrieleVezzosi,UniFI,Florence,Italy
AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany
Moreinformationaboutthissubseriesathttp://www.springer.com/series/3114
ˇ
Tristan Buckmaster (cid:129) Suncˇica Canic´ (cid:129)
Peter Constantin (cid:129) Alexander A. Kiselev
Progress in Mathematical
Fluid Dynamics
Cetraro, Italy 2019
Luigi C. Berselli (cid:129) Michael Ru˚žicˇka
Editors
Authors
TristanBuckmaster SuncˇicaCˇanic´
DepartmentofMathematics DepartmentofMathematics
PrincetonUniversity UniversityofCalifornia
Princeton,NJ,USA Berkeley,CA,USA
PeterConstantin AlexanderA.Kiselev
DepartmentofMathematics DepartmentofMathematics
PrincetonUniversity DukeUniversity
Princeton,NJ,USA Durham,NC,USA
Editors
LuigiC.Berselli MichaelRu˚žicˇka
DepartmentofMathematics Abteilungfu¨rAngewandteMathematik
UniversityofPisa Albert-Ludwigs-Universita¨tFreiburg
Pisa,Italy Freiburg,Germany
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
C.I.M.E.FoundationSubseries
ISBN978-3-030-54898-8 ISBN978-3-030-54899-5 (eBook)
https://doi.org/10.1007/978-3-030-54899-5
MathematicsSubjectClassification:76-06,35Q30,76A02,76F02,74F10
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicencetoSpringerNatureSwitzerland
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Preface
Fluid mechanics is an extremely active area of science, with interdisciplinary
interplay of pure and applied mathematics, physics, geophysics, engineering,
medicine, and biology. The lectures delivered at the CIME School on “Progress
in Mathematical Fluid Dynamics” in Cetraro, June 2019 were devoted to some
foundational issues regarding the Euler and Navier–Stokes equations and related
reducedmodels(asexistence,uniqueness,formationofsingularities,andvanishing
viscositylimits),butalsototheproblemofafluidinteractingwithmovingdomain
with elastic boundaries. More precisely, the questions addressed span from the
basic problems of existence of weak and more regular solutions, the analysis of
potential singularities, qualitative and quantitative results about the behavior in
specialsingularscenarios,tothemodelingandnumericalanalysisofthesolutions
of FSI problems. Each of the four speakers gave a set of 6h of classes, with
introductory material and more specialized content in the final lectures. As an
additionalactivity,asessionofshortcommunicationshasbeenorganizedduringthe
school,wheresomeselectedseniorstudentspresentedtheirownongoingresearch
results.
Hereabriefsummaryofthecontributionsinthisvolume:
ThevolumecontainsthelecturesbyTristanBuckmaster(PrincetonUniversity),
Suncˇica Cˇanic´ (University of California, Berkeley), Peter Constantin (Princeton
University),andAlexanderA.Kiselev(DukeUniversity).
Prof. T. Buckmaster in the contribution: “Heuristic Approach to Convex Inte-
gration for the Euler Equations” presents several recent groundbreaking results
concerning the use of convex integration to construct wild weak solutions of the
incompressible Euler and Navier–Stokes equations. This chapter contains a quite
informalintroductionfocusingon the main ideas, more than on the technicalities,
andwillbeparticularlyusefulforstudentsstartingtheirworkinthisfield.
Prof. S. Cˇanic´ in the contribution: “Fluid–Structure Interaction with Incom-
pressible Fluids” presents some mathematical techniques developed to study the
existence of weak solutions for a class of fluid–structure interaction problems. In
particular, the interaction is between viscous, incompressible fluids and elastic,
viscoelastic, or composite structures. The proofs are constructive, being based on
v
vi Preface
semidiscretizing the coupled problem in time, with the Rothe method, and an
operator splitting method. Then convergence of the scheme is proved by using a
recent generalization of the Aubin–Lions–Simon compactness lemma, which can
beusefulalsoinothercontexts.
Prof. P. Constantin in the contribution: “Regularity and Inviscid Limits in
HydrodynamicModels”discussestheproblemofvanishingviscositylimitandthe
related low regularity bounds, uniform in viscosity. This is done for the vorticity
in the Yudovichclass in two-space dimensions. He also considers the problem of
construction of multi-scale steady solutions of the Navier–Stokes equations with
a given power-law energy spectrum (in a remarkable way this includes also the
KolmogorovK41case).Resultsarenotlimitedtotheproblemwithoutboundaries,
buttheycanbeextendedtoanydomaininthree-spacedimensions.
Prof. A. Kiselev in the contribution: “Small Scale Creation in Active Scalars”
focusesonsmall-scaleformationinsolutionsoftheincompressiblefluiddynamics
and associated models. First, there is a review of the two-dimensional case and
thenthethree-dimensionalscenariobyHouandLuoisdiscussedwithgreatdetail.
Examplesonthesurfacequasi-geostrophic(SQG)equation,withemphasisonpatch
solutionsarealsostudied.
Itisourpleasuretothankthelecturersfortheirenthusiasticparticipationinthe
school,fordeliveringtheirlectures,andalsoforwritingupthenotes.Wealsothank
all participants for their keen scientific and social contribution to the success of
theschool.WethanktheFondazioneCIMEanditsscientificcommitteeforgiving
usthe opportunityto organizethis eventandespecially Elvira Mascoloand Paolo
Salani,DirectorandScientificSecretaryoftheCIMEFoundation,fortheirhelpin
thepreparationoftheCIMESchoolandofthisvolume.Wegratefullyacknowledge
the Universities of Freiburg and Pisa and the CIME Foundation, for the financial
support.
Pisa,Italy LuigiC.Berselli
Freiburg,Germany MichaelRu˚žicˇka
February2020
Contents
1 AHeuristicApproachtoConvexIntegration
fortheEulerEquations ..................................................... 1
TristanBuckmasterandVladVicol
2 Fluid-StructureInteractionwithIncompressibleFluids ................ 15
SuncˇicaCˇanic´
3 RegularityandInviscidLimitsinHydrodynamicModels .............. 89
PeterConstantin
4 SmallScaleCreationinActiveScalars.................................... 125
AlexanderA.Kiselev
vii
Chapter 1
A Heuristic Approach to Convex
Integration for the Euler Equations
TristanBuckmasterandVladVicol
Abstract The purpose of these lecture notes is to employ a heuristic approach
in designing a convex integration scheme that produces non-conservative weak
solutionstotheEulerequations.
1.1 ConvexIntegration as aMathematical TooltoResolve
Onsager’s Conjecture
In these lecture notes, we aim to outline how a convexintegrationcan be used to
constructnon-conservativeweaksolutionstotheEulerequations:
∂ v+div(v⊗v)+∇p =0,
t
(1.1)
divv =0.
We will restrictourselvesto consideringthe Euler equationson the periodictorus
T3fortimest ∈(−1,1).Itiseasytocheck,afterasimpleintegrationbyparts,that
forsmoothsolutionstotheEulerequation,thekineticenergy,definedby
(cid:2)
1
E(t):= |v(t)|2 dx,
2 T3
is conserved. This calculation however does not hold for weak solutions. Indeed,
the theory of turbulence naturally leads one to study the existence of dissipative
T.Buckmaster((cid:2))
DepartmentofMathematics,PrincetonUniversity,Princeton,NJ,USA
e-mail:[email protected]
V.Vicol
CourantInstituteforMathematicalSciences,NewYorkUniversity,NewYork,NY,USA
e-mail:[email protected]
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusive 1
licencetoSpringerNatureSwitzerlandAG2020
L.C.Berselli,M.Ru˚žicˇka(eds.),ProgressinMathematicalFluidDynamics,
LectureNotesinMathematics2272,https://doi.org/10.1007/978-3-030-54899-5_1
2 T.BuckmasterandV.Vicol
weaksolutionstotheEulerequations.IfoneviewstheEulerequationasaninviscid
limit of the Navier–Stokesequations,then formally,if onetakes the inviscid limit
of turbulent solutions, then one expects to obtain solutions to the Euler equations
that dissipate kinetic energy,and are thereforenecessarily weak solutions(see for
exampleSection2of[5]andthereferenceswithinforamoredetaileddiscussion).
The postulate of dissipation of kinetic energy at the inviscid limit is sometimes
referredtointheliteratureasthezerothlawofturbulence.
In[20],Onsagerfamouslyconjecturedthefollowingdichotomy:
Conjecture1.1(Onsager’sConjecture)
(a) Any weak solution v belongingto the Hölder space Cθ for θ > 1 conserves
3
kineticenergy.
(b) Foranyθ < 1 thereexistweaksolutionsv ∈Cθ whichdissipatekineticenergy.
3
Part(a)wasresolvedbyConstantin,EandTitiin[9],followingapartialresolution
ofEyinkin[16](seealso[7,15]formorerefinedresults).Thefirstresulttowards
proving Part (b) was the construction of non-conservative L2 weak solution to
the Euler equationsby Scheffer [21]. While the solutionsconstructed by Scheffer
were non-conservative, they could not be classed a dissipative since they did not
satisfythepropertyofnon-increasingenergy.Thefirstexampleofdissipativeweak
solutions to the Euler equations was due to Shnirelman in [22] (cf. [11, 12]).
Motivated in part by the convex integration scheme of Nash, employed in order
toconstructexoticcounter-examplestotheC1 isometricembeddingproblem[19],
DeLellisandSzékelyhidiJr.in[13,14],madesignificantprogresstowardsPart(b)
1−
by constructing dissipative Hölder C10 continuous weak solutions to the Euler
equations. Then after a series on advancements [1–3, 10, 17], Isett resolved the
conjecture in [18]. However, like the original paper of Scheffer [21], the weak
solutions constructed by Isett [18] were not strictly dissipative. This technical
issue was resolved in a paper by the authors in collaboration with De Lellis and
SzékelyhidiJr.[4],inwhichtheprecisestatementofPart(b)wasproven.
Instead of considering the more difficult problem of proving Part (b), let us
considerthesimplerproblemofconstructingnon-trivial,non-conservative,Hölder
continuousweaksolutions:
Theorem1.1 ForsomeHölderexponentβ,thereanon-trivialweaksolutiontothe
Eulerequationsv ∈C((−1,1);Cβ(T3))withcompactsupportintime.
Thepurposeofthesenotesistoprovideanoutlineofhowtogoaboutconstructing
a convex integration scheme in order prove Theorem 1.1. The outline will track
closely with the approachtaken in Section 5 of the review paper [5], which itself
isbasedontheworks[2,11,12,17].Inthispresentation,weeschewmathematical
rigorinfavorroughheuristics.Thiswillallowustobetterillustratethemainideas
thatgointodesigninga convexintegrationofthetypepioneeredbyDe Lellisand
Székelyhidi Jr. in [13], without getting caught up in the nitty gritty technicalities
thatarigorousapproachentails.
