Honours Thesis

116 Pages·2014·1.03 MB·English

by  Angus Griffith

A tent space approach to mild solutions
of Navier–Stokes equations with rough
initial data
Angus Griffith
October 2014
A thesis submitted in partial fulfilment for the degree of
Bachelor of Philosophy (Honours).

Acknowledgements
First and foremost I would like to thank my supervisor Pierre Portal. His enthusiasm for
mathematics has been inspirational and I thank him for spending many hours sharing his
knowledge with me. Alan McIntosh and John Urbas I thank for being excellent teachers and
always challenging me to aim higher.
Theinputfrommyproofreadingteam; AlexAmenta,KowshikBettadapura,andSvenPistre
has been invaluable. I thank Dorothey Frey for answering my questions about the proof of
Theorem 5.11. From how to change a rocker cover gasket to harmonic analysis on manifolds, I
thank Lashi for teaching me many things and being a great friend.
I thank my friends in the basement. In particular, Chaturi, Hannah, Nam and Sam, for
putting up with me and always being supportive; in addition, Benedict, Josh, Simon, Hannah
and Zaiga, for the fun times and great conversations.
I would also like to thank my family for always supporting me.

Contents
1 Introduction 1
2 Singular integral operators 3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Spatial assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Maximal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Calderón–Zygmund decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Weak Lp spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.1 Marcinkiewicz interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.2 Riesz–Thorin interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Boundedness of singular integral operators . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Calderón–Zygmund operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.1 The Riesz transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8.2 Lp boundedness of CZO operators . . . . . . . . . . . . . . . . . . . . . 26
α
2.8.3 Littlewood–Paley theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Functions of bounded mean oscillation 33
3.1 Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Maximal characterisation of Hp . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 Atomic decomposition of H1 . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.3 Singular integrals and H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 The space BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Classical BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Duality of H1 and BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Tent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Tent spaces in parabolic scaling . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Tent spaces and H1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Tent spaces and BMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Cauchy problems 55
4.1 Abstract Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Semigroup theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Solutions to the Cauchy problems. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Notions of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2 Homogeneous Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.3 Inhomogeneous Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Maximal regularity and PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.5 Maximal regularity operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5.1 L2 boundedness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.2 Tent space boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Nonlinear Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5 Navier–Stokes equations 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 Solutions to the Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 The Leray projection operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1 Riesz transform interpretation . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2 The Oseen kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.3 Well definedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Picard fixed point method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 The Koch and Tataru result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Appendix A Schur’s lemma 97
A.1 Schur’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Schur’s lemma for positive operators . . . . . . . . . . . . . . . . . . . . . . . . . 97
Appendix B Distribution theory 101
B.1 Fréchet spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2 Schwartz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.3 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.4 Tempered distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Bibliography 104
Notation
x.y There exists c>0 such that x cy
≤
A B A is a subset of B (not necessarily proper subset)
⊂
1 characteristic function of the set E
E
A Lebesgue measure of a set A
| |
B(x,r) ball of radius r centred at x
dx Lebesgue measure
supp(u) support of the function u
p p/(p 1) whenever p (1, ), 1 = , and =1.
′ ′ ′
− ∈ ∞ ∞ ∞
∂
∂ = the ith partial derivative
i
∂x
i
∆=∂2+ +∂2 Laplacian on Rn
1 ··· n
∂
= j Riesz transform
j
R √∆
(A,B) bounded operators from A into B, (A)= (A,A)
L L L
ρ(T) resolvent set of T
σ(T) spectrum of T
S the positive sector of angle µ, z C 0 : Arg(z) <µ
µ+
{ ∈ \{ } | | }
D the diagonal (x,x):x Rn R2n
{ ∈ }⊂
R nonnegative real numbers x R:x 0
+
{ ∈ ≥ }
K either R or C
x x2+ +x2 where x=(x ,...x ) Rn
| | 1 ··· n 1 n ∈
Ck(Ω) space of Ck functions with compact support in Ω
c p
1/p
ℓp space of C valued sequences with norm x := x p
k kℓp j 1| j|
≥
Lp(X,µ,F) Lebesgue space of functions with values in F (cid:16) (cid:17)
P
Lp space of functions that lie in Lp(K) for every compact K Rn
loc
⊂
Wk,p Sobolev spaces
Hp real Hardy spaces
BMO functions of bounded mean oscillation
Schwartz functions
S
tempered distributions
′
S
distributions
′
D
M centred Hardy–Littlewood maximal function
M˜ uncentered Hardy–Littlewood maximal function
smooth maximal function
ϕ
M
nontangential smooth maximal function
M∗ϕ
maximal regularity operator of A
A
M

Chapter 1
Introduction
In this thesis we examine the Navier–Stokes equations in n spatial dimensions in a rather
restricted setting: we consider a viscous, homogeneous, incompressible fluid that fills the entire
space and feels no external forces. The equations that describe the evolution of the velocity
~u(x,t) of such a fluid at time t and position x are
∂ ~u=∆~u P (~u ~u),
t
− ∇ ⊗ (1.1)
 ~u=0,
∇·
where the Leray projector P is defined by Pf~:=f~ 1( f~).
−∇∆ ∇·
The focus of this thesis is the existence of global mild solutions for the Navier–Stokes equa-
tions. We are interested in the Cauchy problem for the Navier–Stokes equations; given some
initial velocity field ~u (x) we seek a solution ~u(x,t) that satisfies (1.1) in an appropriate mild
0
sense with initial value ~u(x,0) = ~u (x). That is, we look for a solution to the corresponding
0
integral equation
t
~u=et∆~u e(t s)∆P (~u ~u)ds,
0 −
− ∇· ⊗ (1.2)
 Z0
 ~u0 =0.
∇·
We solve (1.2) by lookingfor a fixed point of the transform
t
~v et∆~u e(t s)∆P (~v ~v)ds. (1.3)
0 −
7→ − ∇· ⊗
Z0
The Picard contraction principle then gives a solution of (1.3). Given a Banach space of initial
data and a space of solutions E such that (x,t) et∆u (x) E for all u the existence
0 0
E 7→ ∈ ∈E
of solutions in E reduces to asking whether
t
B(~u,~v):= e(t s)∆P (~u ~v)ds (1.4)
−
∇· ⊗
Z0
defines a bounded bilinear operator from En En to En.
×
In2001KochandTataru[KT01]provedwell-posednessofNavier–StokesequationsinBMO−1
(the space of derivatives of functions with bounded mean oscillation) by showing boundedness
1
2 CHAPTER 1. INTRODUCTION
of the bilinear transform (1.4) in an appropriate Banach space. The goal of this thesis is to
establish their global existence result for data in BMO−1. We show that for every sufficiently
small ~u0 (BMO−1)n there exists a mild solution ~u to the Navier–Stokes equations that lies
∈
in a certain subspace of the tent space T ,2.
∞
Theorem 1.1. There exits ǫ > 0 such that for all ~u0 ∈ (BMO−1)n with k~u0kBMO−1 < ǫ and
~u =0 there exists a mild solution ~u of the Navier–Stokes equations on Rn+1 :=Rn R :
∇· 0 + × +
t
~u=et∆~u e(t s)∆P (~u ~u)ds, (1.5)
0 −
− ∇ ⊗
Z0
so that
(i) sup sup √t~u(x,t) < ,
t>0 x∈Rn| | ∞
1
(ii) sup sup t n/2 t es∆~u(x,s)2dsdx 2 < ,
t>0 x∈Rn − 0 B(x,√t)| | ∞
(cid:16) (cid:17)
(iii) lim ~u(x,t)=~u (x) RforRa.e. x Rn.
t 0 0
→ ∈
Chapter 2 begins by establishing the necessary background theory. The focus is on finding
sufficientconditionstoestablishtheboundedness,inseveralfunctionspaces,ofsingularintegral
operators of the form
Tf(x):= k(x,y)f(y)dµ(y) f , x / supp(f).
ZRn ∀ ∈S ∈
Here,k :Rn Rn D CiscalledthekerneloftheoperatorT andissingularonthediagonal
× \ →
D = (x,x):x Rn .
{ ∈ }
The space of functions of bounded mean oscillation (BMO) is introduced in chapter 3. We
begin by introducing the Hardy spaces and eventually characterise BMO as the dual of the
Hardy space H1; along the way we present several equivalent characterisations of both H1
and BMO. The chapter ends by introducing tent spaces in parabolic scaling and pointing out
connections to H1 and BMO.
Inchapter 4wediscusstheCauchyproblem,anabstractformulationofinitialvalueproblems.
Theaimofthischapteristointroducetheconceptofmaximalregularityandestablishmaximal
regularity of the heat equation in the tent space T ,2. This will allow us to treat the Navier–
∞
Stokes equations as a nonlinear perturbation of the heat equation on that space. This program
isfinallycarriedoutinchapter 5. TheproofofTheorem 1.1wepresentisbasedonrecentwork
by Auscher and Frey [AF13]. It differs from the original proof by Koch and Tataru [KT01]
in that it relies more heavily on tent space theory and does not use the self-adjointness of the
Laplacian. Thehopeisthatabetterunderstandingofthisresultwillallowtreatmentofrelated
problems such as stochastic Navier–Stokes equations.