Hybrid Function Spaces, Heat and Navier-stokes Equations

195 Pages·2015·1.953 MB·English

by  Hans Triebel

EMS Tracts in Mathematics 24
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Hans Triebel
Hybrid Function Spaces,
Heat and Navier-Stokes
Equations
Author:
Hans Triebel
Friedrich-Schiller-Universität Jena
Fakultät für Mathematik und Informatik
Mathematisches Institut
07737 Jena
Germany
E-mail: [email protected]
2010 Mathematical Subject Classification: 46-02, 46E35, 42B35, 42C40, 35K05, 35Q30, 76D03, 76D05
Key words: Function spaces, Morrey spaces, heat equations, Navier-Stokes equations
ISBN 978-3-03719-150-7
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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© European Mathematical Society 2015
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9 8 7 6 5 4 3 2 1
Preface
This book is the continuation of [T13]. Our aim is twofold. First we develop the
theory of hybrid spaces LrAs .Rn/ which are between the nowadays well-known
p;q
globalspacesAs .Rn/ with A 2 fB;Fgand theirlocalization (orMorreyfication)
p;q
LrAs .Rn/ as considered in detail in [T13]. Spaces As .Rn/ cover (fractional)
p;q p;q
Sobolevspaces,(classical)BesovspacesandHo¨lder-Zygmundspaces,whereaslocal
Morrey spaces Lr.Rn/ are special cases of the local spaces LrAs .Rn/. In [T13]
p p;q
weappliedthetheoryofspacesLrAs .Rn/tononlinearheatequationsandNavier-
p;q
Stokes equations. But this caused some problems which will be discussed in the
Introduction(Chapter 1) below. It came out quite recently that it is more natural in
this contextto switch from local spaces LrAs .Rn/ to hybrid spaces LrAs .Rn/.
p;q p;q
ThisagainwillbeilluminatedintheIntroductionbelow.
It is the second aim of this book to apply the theory of global spaces As .Rn/
p;q
andhybridspacesLrAs .Rn/totheNavier-Stokesequations
p;q
@ uC.u;r/u(cid:2)(cid:2)uCrP D0 inRn(cid:3).0;T/, (0.1)
t
divuD0 inRn(cid:3).0;T/; (0.2)
u.(cid:4);0/Du inRn; (0.3)
0
intheversionof
@ u(cid:2)(cid:2)uCPdiv.u˝u/D 0 inRn(cid:3).0;T/, (0.4)
t
u.(cid:4);0/D u inRn; (0.5)
0
reducedtothescalarnonlinearheatequations
@ v(cid:2)Dv2(cid:2)(cid:2)v D0 inRn(cid:3).0;T/; (0.6)
t
v.(cid:4);0/Dv inRn; (0.7)
0
(cid:2) (cid:3)
where0 < T (cid:5) 1. Hereu.x;t/ D u1.x;t/;:::;un.x;t/ in(0.1)–(0.5)istheun-
knownvelocityandP.x;t/theunknown(scalar)pressure,whereasv.x;t/in (0.6),
(0.7)isascalarfunction,2(cid:5)n2N. Recall@ D@=@t,@ D@=@x ifj D 1;:::;n;
t j j
(cid:4) (cid:5) Xn
.u;r/u k D uj@ uk; k D1;:::;n; (0.8)
j
jD1
Xn
divuD @ uj; rP D.@ P;:::;@ P/; (0.9)
j 1 n
jD1
andby(0.2)
Xn
.u;r/uDdiv.u˝u/; div.u˝u/k D @ .ujuk/: (0.10)
j
jD1
vi Preface
Furthermore,PistheLerayprojector
Xn
.Pf/k Dfk CR R fj; k D1;:::;n; (0.11)
k j
jD1
basedonthe(scalar)Riesztransforms
(cid:6) (cid:7) Z
(cid:3) _ y
R g.x/Di kbg .x/Dc lim k g.x(cid:2)y/dy; x 2Rn: (0.12)
k j(cid:3)j n"#0 jyj(cid:2)" jyjnC1
In (0.4), (0.5) there is no need to care about (0.2) any longer. But if, in addition,
divu D 0thendivu D 0inourcontext(mildsolutionsbasedonfixedpointasser-
0
tions). Inthescalarequation(0.6)weusedtheabbreviation
Xn
Df D @ f: (0.13)
j
jD1
As mentioned above we dealt in [T13] with the above equations in the context
ofthe globalspacesAs .Rn/. Rigorousreductionof(0.1)–(0.3)to (0.4), (0.5)and
p;q
finally to (0.6), (0.7) requires a detailed study of the nonlinearity u 7! u2 and of
boundedness of Riesz transforms in the underlying spaces. In [T13] we tried to
extend this theory to some local spaces LrAs .Rn/. But one needs some modifi-
p;q
cations, especially a replacement of the Riesz transforms by some truncated Riesz
transforms. In the Introductionbelowwe repeatthe aboveconsiderationsin greater
details and discuss in particular this somewhat disturbing (but unavoidable) point.
ThehybridspacesLrAs .Rn/preservemanydesirablepropertiesofthelocalspaces
p;q
LrAs .Rn/ but avoid the above-indicated shortcomings. They are between global
p;q
spaces and local spaces, which may justify calling them hybrid spaces. They coin-
cidewiththewell-studiedspacesAs;(cid:2) .Rn/,(cid:4) D 1 C r,includingtheglobalspaces
p;q p n
As;0.Rn/DAs .Rn/asspecialcases.
p;q p;q
Chapter1istheannouncedIntroductionwherewereturntotheabovedescription
in greater details and with some references. Chapter 2 deals with local and global
Morrey spaces Lr.Rn/, Lr.Rn/, their duals and preduals and, in particular, with
p p
the question whether the Riesz transforms R in (0.12) are bounded maps in these
k
spaces and what they look like. This chapter is self-contained and we hope that it
is of interestfor researchersin this field. In Chapter3 we developthe theoryof the
hybridspacesLrAs .Rn/ as neededfor ourabove-outlinedpurposes. It comes out
p;q
thatmanybasicpropertiesforthelocalspacesLrAs .Rn/canbetransferredeasily
p;q
from [T13]to the hybridspacesLrAs .Rn/. We concentrateon some new aspects
p;q
which will be crucial in the context described above. Similarly we carry over and
complementinChapter4thetheoryofheatequationsintheglobalspacesAs .Rn/
p;q
asdevelopedin[T13]tothehybridspacesLrAs .Rn/. Chapter5dealswithNavier-
p;q
Stokes equations especially in the version (0.4), (0.5) in hybrid spaces. Then one
is in a rather comfortable position, clipping together related assertions of the two
Preface vii
precedingchapters. Butagainweaddafewnewaspects. Inparticular,iftheadmitted
initialdataareinfrared-dampedthentherelatedlocalsolutionsoftheNavier-Stokes
equationscan be extendedglobally in time. These considerationswill be continued
inChapter6nowspecifiedtotheglobalspacesAs .Rn/andextendedtothespaces
p;q
Sr A.Rn/withdominatingmixedsmoothness. Wediscussconditionsfortheinitial
p;q
data in terms of Haar wavelets, Faber bases, and sampling in connection with the
hyperboliccross,ensuringsolutionsoftheNavier-Stokesequationswhichareglobal
intime. FurthermoreweaddsomecommentsabouttheinfluenceoflargeReynolds
numbers. Thischapterislargelyindependentoftheprecedingconsiderations.
We assume that the reader has a working knowledge about basic assertions for
the spacesAs .Rn/. But to make this bookindependentlyreadablewe providere-
p;q
latednotation,facts,anddetailedreferences.Formulaearenumberedwithinchapters.
Furthermoreineachchapteralldefinitions,theorems,propositions,corollariesandre-
marksarejointlyandconsecutivelynumbered.Referencesareorderedbynames,not
bylabels,whichroughlycoincide,butmayoccasionallycauseminordeviations. The
bracketednumbers following the items in the Bibliographymark the page(s)where
thecorrespondingentryisquoted.Allunimportantpositiveconstantswillbedenoted
byc(withadditionalmarksifthereareseveralc’sinthesameformula). Toavoidany
misunderstandingwefixouruseof(cid:6)(equivalence)asfollows. LetI beanarbitrary
indexset. Then
a (cid:6)b for i 2I (equivalence) (0.14)
i i
fortwosets ofpositivenumbersfa W i 2 Igandfb W i 2 Igmeansthatthereare
i i
twopositivenumbersc andc suchthat
1 2
c a (cid:5)b (cid:5)c a foralli 2I:
1 i i 2 i
Contents
1 Introduction 1
2 Morreyspaces 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Definitionsandpreliminaries . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Morreyspaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 DualMorreyspaces . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Besovspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Mainassertions . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Furtherembeddings . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.4 Non-separabilityanddensity . . . . . . . . . . . . . . . . . . 23
2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.1 Mainassertions . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4.2 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Caldero´n-Zygmundoperators . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 Mainassertions . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.3 Distinguishedrepresentations . . . . . . . . . . . . . . . . . . 36
2.6 Haarbases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.1 Preliminariesanddefinitions . . . . . . . . . . . . . . . . . . 39
2.6.2 Mainassertions . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6.3 Littlewood-Paleytheorem . . . . . . . . . . . . . . . . . . . . 43
3 Hybridspaces 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Globalspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.2 Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.3 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.4 Multiplicationalgebras . . . . . . . . . . . . . . . . . . . . . 54
3.2.5 Spacesondomains . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Definitionsandbasicproperties . . . . . . . . . . . . . . . . . . . . . 56
3.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.2 Basicproperties . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.1 Waveletcharacterizations . . . . . . . . . . . . . . . . . . . . 63
3.4.2 Atomiccharacterizations . . . . . . . . . . . . . . . . . . . . 66
3.4.3 Fourier-analyticalcharacterizations . . . . . . . . . . . . . . . 68
Contents ix
3.4.4 Haarwavelets . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.5 Morreyspaces,revisited . . . . . . . . . . . . . . . . . . . . . 78
3.4.6 Meyerwavelets . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.5 EquivalentnormsandFouriermultipliers . . . . . . . . . . . . . . . . 82
3.5.1 Equivalentnorms . . . . . . . . . . . . . . . . . . . . . . . . 82
3.5.2 FouriermultipliersandRiesztransforms . . . . . . . . . . . . 88
3.6 Someproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6.1 Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6.2 Multiplicationalgebras . . . . . . . . . . . . . . . . . . . . . 95
3.6.3 Morreycharacterizations . . . . . . . . . . . . . . . . . . . . 96
3.6.4 Pointwisemultipliersanddiffeomorphisms . . . . . . . . . . . 99
3.6.5 Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.6.6 Thermiccharacterizations:commentsandproposals . . . . . . 102
4 Heatequations 112
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Homogeneousheatequations . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Inhomogeneousheatequations . . . . . . . . . . . . . . . . . . . . . 114
4.4 Nonlinearheatequations . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.1 Specialcases . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.2 Generalcases . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.3 Strongsolutions . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.4.4 Commentsandexamples . . . . . . . . . . . . . . . . . . . . 122
5 Navier-Stokesequationsinhybridspaces 125
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Specialcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Generalcases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Commentsandexamples . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.1 Supercriticalspaces . . . . . . . . . . . . . . . . . . . . . . . 132
5.5.2 Infrared-dampedinitialdata . . . . . . . . . . . . . . . . . . . 133
5.5.3 Globalsolutions . . . . . . . . . . . . . . . . . . . . . . . . . 137
6 Navier-Stokesequationsinglobalspaces 143
6.1 HaarwaveletsandReynoldsnumbers. . . . . . . . . . . . . . . . . . 143
6.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1.2 Someembeddings . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1.3 Mainassertions . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.1.4 Reynoldsnumbers . . . . . . . . . . . . . . . . . . . . . . . . 154
6.1.5 Oscillation,persistency,latticestructureofinitialdata . . . . . 157
6.2 Initialdatainspaceswithdominatingmixedsmoothness . . . . . . . 159
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2.2 Definitionsandbasicproperties . . . . . . . . . . . . . . . . . 162
6.2.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
x Contents
6.2.4 Sobolevspaces. . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2.5 Besovspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.2.6 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Bibliography 175
Symbols 183
Index 185