Continuity of the solution map of the Euler equations in H\"older spaces and weak norm inflation in Besov spaces

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by  Gerard Misiołek

CONTINUITY OF THE SOLUTION MAP OF THE EULER
EQUATIONS IN HO¨LDER SPACES AND WEAK NORM
INFLATION IN BESOV SPACES
6
1
0 GERARDMISIOL EKANDTSUYOSHIYONEDA
2

n
Abstract. We construct an example showing that the solution map of the
a EulerequationsisnotcontinuousintheH¨olderspaceC1,α forany0<α<1.
J
Onthe other handweshow that itiscontinuous when restrictedtothelittle

6 H¨oldersubspacec1,α. Weapplythelattertoprovenorminflationforsolutions
ofthevorticityequations inBesovspaces nearthecriticalspaceB21,1.
]
P
A
.
h
t 1. Introduction
a
m We study the Cauchy problem for the Euler equations of an incompressible and
[ inviscid fluid

1 u +u u= p, t 0, x Rn
t
·∇ −∇ ≥ ∈
v
(1.1) divu=0
4
2 u(0)=u
0
0
1 whereu=u(t,x)andp=p(t,x)denotethevelocityfieldandthepressurefunction
0 of the fluid respectively. The first rigorous results for (1.1) were proved in the
.
1 framework of Ho¨lder spaces by Gyunter [17], Lichtenstein [22] and Wolibner [30].
0 More refined results using a similar functional setting were obtained subsequently
6 by Kato [18], Swann [28], Bardos and Frisch [1], Ebin [14], Chemin [8], Constantin
1
[10]andMajdaandBertozzi[23]amongothers. Themainfocusinthesepaperswas
:
v on existence and uniqueness of C1,α solutions and the question of continuity with
Xi respect to initial conditions was not explicitly addressed. Recall that, according to
the definition of Hadamard, a Cauchy problem is said to be locally well-posed in
r
a a Banach space X if for any initial data in X there exists a unique solution which
persists for some T >0 in the space C([0,T),X) and which depends continuously
on the data. Otherwise the problem is said to be ill-posed.
Systematic studies of ill-posedness of the Cauchy problem (1.1) are of a more
recentdate andconcerna wide range ofphenomena including graduallossof regu-
larityofthesolutionmap,energydissipationandnon-uniquenessofweaksolutions,
see e.g. Yudovich [31], Koch [21], Morgulis, Shnirelman and Yudovich [26], Eyink
[16], Constantin, E and Titi [11] or Shnirelman [27]. Recently, Bardos and Titi [2]
found examples of solutions in Ho¨lder spaces Cα and the Zygmund space B1
∞,∞
which exhibit an instantaneous loss of smoothness in the spatial variable for any
Date:January7,2016.
2000 Mathematics Subject Classification. Primary35Q35;Secondary35B30.
Key words and phrases. Eulerequations,H¨olderspaces,solutionmap.
1
2 GERARDMISIOL EKANDTSUYOSHIYONEDA
0 < α < 1. Similar examples in logarithmic Lipschitz spaces logLipα were con-
structed by the authors in [24]. Solutions that fail to be continuous with respect
tothe time variableweredescribedbyCheskidovandShvydkoy[9]. Morerecently,
in a series of papers Bourgain and Li [5, 6] constructed smooth solutions which
exhibit norm inflation in borderline spaces such as Wn/p+1,p for any 1 p <
≤ ∞
and Bn/p+1 for any 1 p < and 1 < q as well as in the standard spaces
p,q
≤ ∞ ≤ ∞
Ck and Ck−1,1 for any integer k 1; see also Elgindi and Masmoudi [15] and [25].
≥
As observed in [6] the cases Ck and Ck−1,1 are particularly intriguing in view of
the classical existence and uniqueness results mentioned above.
Ourmaingoalinthispaperistoclarifythepictureoflocalwell-posednessinthe
sense of Hadamard for the Euler equations in Ho¨lder spaces. We present a simple
examplebasedonaDiPerna-Majdatypeshearflowwhichshowsthatingeneralthe
data-to-solution map of (1.1) is not continuous in C1,α for any 0<α<1. On the
other hand, we show that continuity of this map is restored if the Cauchy problem
isrestrictedtothesocalledlittleHo¨lderspacec1,α. Thefailureofcontinuityinour
example does not seem to be related to the norm inflation mechanism described
in [6] which essentially relies on unboundedness of the double Riesz transform in
L∞. Rather, it can be explained by the fact that smooth functions are not dense
in the standard C1,α spaces. We point out that continuity of the solution map for
the Euler equations in Sobolev spaces Ws,p for p 2 and s >2/p+2 is of course
≥
well known(see e.g., Ebin and Marsden[13], Kato and Lai [19] or Kato and Ponce
[20], see alsoRemark 4.3). However,we couldnotfind the correspondingresultfor
c1,α in the literature althoughit should be familiar to the experts in the field. Our
main results can be stated as follows.
Theorem 1.1. The solution map oftheincompressible Euler equations (1.1)is not
continuous as a map from C1,α(R3) to C([0,T),C1,α(R3)) for any 0<α<1.
Theorem 1.2. The incompressible Euler equations (1.1) are locally well-posed in
the sense of Hadamard in the little Ho¨lder space c1,α(Rn) for any 0 < α < 1 and
n=2 or 3.
AsanapplicationofTheorem1.2weproveanorminflationresultforthevorticity
equations that involves a family of Besov spaces. Although this result is weaker
than the norm inflation described by Bourgain and Li our methods can be applied
intheborderlineend-pointspacessuchasB2 (R2)whichliejustoutsidethe range
2,1
ofthespacesconsideredin[5]. Recallthatexistenceanduniquenessresultsfor(1.1)
in B2 (R2) are already known, cf. e.g. Vishik [29] or Chae [7]. The proof uses
2,1
continuity of the data-to-solution map in c1,α as well as several technical lemmas
provedinour earlierpaper [25]. In this respectthe presentpaper canbe viewedas
a continuation of [25].
Theorem 1.3. Let M be an increasing sequence of positive numbers. There
j
ր∞
existsasequenceofsmoothrapidlydecayinginitialdata u˜ ∞ andtwosequences
{ 0,j}j=1
of indices r ∞ and q ∞ with r 2 and q 1 such that
{ j}j=1 { j}j=1 j → j →
u˜ .1 and u˜ (t) >M for some 0<t<M−3.
k 0,jkBr2j,qj k j kBr2j,qj j j
Inthenextsectionwerecallthebasicsetupandnotation. InSection3weprove
Theorem1.1 by constructinga shearflow counterexampleinthe C1,α space. Local
Hadamard well-posedness in c1,α is shown in Section 4. The proof of Theorem 1.3
is given in Section 5.
THE SOLUTION MAP OF THE EULER EQUATIONS 3
2. The basic setup: function spaces and diffeomorphisms
Let ψ S(Rn) be any function of Schwartz class satisfying 0 ψ 1 and
0 0
suppψ ∈ξ Rn :1/2 ξ 2 and such that ≤ ≤
0
⊂{ ∈ ≤| |≤ }
ψ(ξ)=1, for anyξ =0
l
6
l∈Z
X
where ψ(ξ) = ψ (2−lξ). For any s > 0 and 1 p,q let Bs (Rn) denote the
l 0 ≤ ≤ ∞ p,q
inhomogeneous Besov space equipped with the norm
(2.1) f = f + f
k kBps,q k kLp k kB˙ps,q
where the homogeneous semi-norm is given by
1/q
2slq ψ f q if 1 q <
k l∗ kLp ≤ ∞
(2.2) kfkB˙ps,q = (cid:18)Xl∈sZup2sl bψl f Lp(cid:19) if q =
l∈Z k ∗ k ∞
for any f ∈S′(Rn). In particular, if sb=k+α is not an integer then B∞s ,∞ is the
Ho¨lder space Ck,α(Rn) with the standard norm
ϕ = ϕ +[Dkϕ]
k kk,α k kCk α
where
Dβϕ(x) Dβϕ(y)
[Dkϕ] = sup| − |, 0<α<1, k N.
α x y α ∈
x6=y
|βX|=k | − |
Let ck,α(Rn) denote the closed subspace of Ck,α(Rn) consisting of those functions
whose derivatives satisfy the vanishing condition
Dβϕ(x) Dβϕ(y)
(2.3) lim sup | − | =0
h→00<|x−y|<h |x−y|α
foranymulti-index β =k. Itiswellknownthatck,α(Rn)isaninterpolationspace
| |
containing the smooth functions as a dense subspace, cf. e.g. [3].
In what follows we will use an alternative formulation of the fluid equations in
terms of particle trajectories and vorticity. Any sufficiently smooth velocity field
u solving (1.1) has a flow which traces out a curve t η(t,x) of diffeomorphisms
→
starting at the identity configuration e(x) = x with initial velocity u . Using the
0
incompressibilityconstraintdetDη(t,x)=1andthe Biot-Savartlawtheequations
satisfied by the flow can be written in the form
dη
(2.4) (t,x)= K η(t,x) η(t,y) ω(t,η(t,y))dy, t 0, x Rn
n
dt ZRn − ≥ ∈
η(0,x)=x (cid:0) (cid:1)
where ω =curlu is the vorticity1 and the kernel K is given by
n
1 x x
(2.5) K (x)= 2 , 1 , x R2
2 2π − x2 x2 ∈
(cid:18) | | | | (cid:19)
1Ifn=2wecanidentifythevorticityofuwiththefunctionω=∇⊥·uandifn=3withthe
vectorfieldω=∇×u.
4 GERARDMISIOL EKANDTSUYOSHIYONEDA
and
1 x y
(2.6) K (x)y = × , x,y R3.
3 4π x3 ∈
| |
For our purposes it will be sufficientto take asa configurationspaceof the fluid
the set of those diffeomorphisms of Rn which differ fromthe identity by a function
of class c1,α. Let
U = η :Rn Rn :η =e+ϕ , ϕ c1,α(Rn) and ϕ <δ
δ η η η 1,α
→ ∈ k k
n o
where δ >0 is chosen small enough so that
1
(2.7) inf detDη(x)> .
x∈Rn 2
Clearly, U can be identified with an open ball centered at the origin in c1,α(Rn).
δ
The next two lemmas collect some elementary properties of compositions and in-
versions of diffeomorphisms in U that will be used in Section 4.
δ
Lemma 2.1. Let 0 < α < 1. Suppose that η and ξ are in U and ψ c1,α(Rn).
δ
∈
Then ξ η and η−1 are also of class c1,α and we have
◦
(2.8) ψ η .C ψ and ψ η−1 .C ψ
1,α 1,α 1,α 1,α
k ◦ k k k k ◦ k k k
where C >0 depends only on δ and α.
Proof. First, observe that ψ η = ψ and D(ψ η) = Dψ Dη
∞ ∞ ∞ ∞ ∞
k ◦ k k k k ◦ k k k k k
and therefore the first of the inequalities in (2.8) follows at once from
(2.9) [D(ψ η)] [Dψ η] Dη + Dψ η [Dη]
α α ∞ ∞ α
◦ ≤ ◦ k k k ◦ k
[Dψ] Dη 1+α+ Dψ [Dη]
≤ αk k∞ k k∞ α
and [Dη] = [Dϕ ] where η = e + ϕ with ϕ < δ. Similarly, we have
α η α η η 1,α
k k
ψ η−1 = ψ and from (2.7) and Dη−1 =(Dη)−1 η−1 we get
∞ ∞
k ◦ k k k ◦
D(ψ η−1) . Dψ Dη
∞ ∞ ∞
k ◦ k k k k k
and
(2.10) [(Dη)−1] = (detDη)−1adj(Dη) .(1+ Dη 2 )[Dη]
α α k k∞ α
which, in turn, with the h(cid:2)elp of (2.9) yields (cid:3)
(2.11) [D(ψ η−1)] [Dψ] Dη 1+α+ Dψ (1+ Dη 2 )[Dη] .
◦ α ≤ αk k∞ k k∞ k k∞ α
From these bounds we obtain the second of the inequalities in (2.8).
Finally, observe that if ξ U with ξ = e+ϕ then ξ η = e+ϕ where
δ ξ ξ◦η
∈ ◦
ϕ =ϕ +ϕ η. Therefore, using (2.8) we get
ξ◦η η ξ
◦
ϕ . ϕ + ϕ
ξ◦η 1,α η 1,α ξ 1,α
k k k k k k
and combining (2.8) with (2.9) and the vanishing condition (2.3) we conclude that
ϕ c1,α(Rn). Similarly, we also have η−1 =e+ϕ where ϕ = ϕ η−1.
ξ◦η ∈ η−1 η−1 − η◦
Applyingthesecondoftheestimatesin(2.8)togetherwith(2.11)and(2.3)wefind
again that ϕ c1,α(Rn). (cid:3)
η−1
∈
THE SOLUTION MAP OF THE EULER EQUATIONS 5
Lemma 2.2. Let 0<α<1. Suppose that η,ξ and ζ are in U . Then
δ
(2.12) ξ η ζ η .C ϕ ϕ
1,α ξ ζ 1,α
k ◦ − ◦ k k − k
and for any ψ c2,α(Rn) we have
∈
(2.13) ψ η ψ ξ .C ψ ϕ ϕ
1,α 2,α η ξ 1,α
k ◦ − ◦ k k k k − k
where C >0 depends only on δ and α. Furthermore, the functions ξ,η ξ η and
→ ◦
η η−1 are continuous in the Ho¨lder norm topology.
→
Proof. From the estimates of Lemma 2.1 we obtain as before
ξ η ζ η . ϕ ϕ + Dη D(ϕ ϕ )
1,α ξ ζ ∞ ∞ ξ ζ ∞
k ◦ − ◦ k k − k k k k − k
+[Dη] D(ϕ ϕ ) + Dη 1+α[D(ϕ ϕ )]
αk ξ− ζ k∞ k k∞ ξ− ζ α
. 1+ Dη + Dη 1+α+[Dη] ϕ ϕ
k k∞ k k∞ α k ξ− ζk1,α
(cid:16) (cid:17)
whichimpliestheestimatein(2.12). Ontheotherhand,using(2.8)andthealgebra
property of Ho¨lder functions we have
1
ψ η ψ ξ Dψ rη+(1 r)ξ (η ξ) dr . Dψ η ξ
1,α 1,α 1,α 1,α
k ◦ − ◦ k ≤ k − − k k k k − k
Z0
(cid:0) (cid:1)
which gives (2.13) since η ξ =ϕ ϕ . From (2.12) and (2.13) we conclude that
η ξ
composition of diffeomorp−hisms in U− is continuous with respect to ξ and η.
δ
Finally, using the second of the inequalities in (2.8) we have
ξ−1 η−1 . ξ−1 η e .
1,α 1,α
k − k k ◦ − k
By density, givenany ε>0 pick a smooth ζ :Rn Rn suchthat ζ ξ−1 <ε
1,α
→ k − k
and estimate the above expression further by
ξ−1 η ζ η + ζ η ζ ξ + ζ ξ ξ−1 ξ .
1,α 1,α 1,α
k ◦ − ◦ k k ◦ − ◦ k k ◦ − ◦ k
The first and the third of these terms can be bounded using the first inequality in
(2.8)byCε. Forthemiddletermweuse(2.13)tobounditby ζ η ξ . (cid:3)
2,α 1,α
k k k − k
3. A 3D shear flow in C1,α
InthissectionweproveTheorem1.1byconstructingaC1,α shearflowforwhich
the data-to-solution map of (1.1) fails to be continuous. Shear flow solutions were
introduced in [12]. They were used recently in [2] to exhibit instantaneous loss of
smoothness of the Euler equations in Cα for any 0<α<1.
Proof of Theorem 1.1. Let t 0 and consider
≥
u(t,x)= f(x ),0,h(x tf(x )) and v(t,x)= g(x ),0,h(x tg(x ))
2 1 2 2 1 2
− −
where f, g a(cid:0)nd h are bounded real-(cid:1)valued functions o(cid:0)f one variable of class C(cid:1)1,α
with any 0<α<1. It is not difficult to verify that both u and v satisfy the Euler
equations with initial conditions
u (x)= f(x ),0,h(x ) and v (x)= g(x ),0,h(x ) .
0 2 1 0 2 1
Given any ε>0 w(cid:0)e can arrange s(cid:1)o that f and g sa(cid:0)tisfy (cid:1)
u v = f g <ε
0 0 1,α 1,α
k − k k − k
and then choose h such that
h′(x )= x α for all 2a x 2a
1 1 1
| | − ≤ ≤
6 GERARDMISIOL EKANDTSUYOSHIYONEDA
where a=max f , g .
∞ ∞
k k k k
Next, weestimate the normofthe differenceofthe correspondingsolutions. For
(cid:8) (cid:9)
any 0<t 1 we have
≤
u(t) v(t) = f g + h( tf()) h( tg())
k − k1,α k − k1,α ·− · − ·− · 1,α
h( tf((cid:13))) h( tg()) (cid:13)
≥ ∇ ·− ·(cid:13) − ·− · 0,α (cid:13)
=(cid:13)h′(cid:0)( tf()) h′( tg()) (cid:1)(cid:13).
(cid:13) ·− · − ·− · 0,α(cid:13)
Letb=min f , g . I(cid:13)t isclear thatthe normon(cid:13)the righthandside canbe
k k∞ k k∞ (cid:13) (cid:13)
bounded below by
(cid:8) (cid:9)
x tf(x )α x tg(x )α y tf(y )α y tg(y )α
1 2 1 2 1 2 1 2
| − | −| − | − | − | −| − |
sup
x,y∈x[6=−yb,b]2 (cid:12)(cid:12)(cid:12)(cid:0) |x(cid:1)−y(cid:0)|α (cid:1)(cid:12)(cid:12)(cid:12)
Evaluatingthisexpressionatx =y =cwith b c bwegetafurtherestimate
2 2
− ≤ ≤
from below by
(x tf(c)α x tg(c)α) (y tf(c)α y tg(c)α)
1 1 1 1
sup | | − | −| − | − | − | −| − | |
x y α
x,yx∈1[6=−yb1,b]2 | 1− 1|
andevaluatingonce againatthe points x =tg(c)andy =tf(c)we obtaina final
1 1
lower bound
tσ g(c) f(c)α+tα f(c) g(c)α
| − | | − | =2
≥ tα f(c) g(c)α
| − |
which proves Theorem 1.1. (cid:3)
4. Local well-posedness in c1,α
We turn to the question of well-posedness of (1.1) in the sense of Hadamard.
As mentioned in the Introduction, local existence and uniqueness results in Ho¨lder
spacesarewellknownandourcontributionhereconcernsonlythecontinuityprop-
ertyofthesolutionmap. Tothisendwewillmakesomeadjustmentsintheapproach
based on the particle-trajectory method of [23]. We first state Theorem 1.2 more
precisely as follows.
Theorem 4.1. Let 0 < α < 1. For any divergence free vector field u c1,α(Rn)
0
∈
with compactly support vorticity there exist T >0 and a unique solution u of (1.1)
such that the map u u is continuous from c1,α(Rn) to C([0,T),c1,α(Rn)).
0
→
Proof of Theorem 4.1. We will concentrate on the two-dimensional case since the
arguments in the three-dimensional case are very similar (the necessary modifica-
tions will be described below). We begin by constructing the Lagrangianflow as a
unique solution of an ordinary differential equation in U .
δ
Since in two dimensions the vorticity is conserved by the flow2 we can rewrite
equations (2.4) in the form
dη
(4.1) (t,x)= K η(t,x) η(t,y) ω (y)dy =:F (η )(x),
dt R2 2 − 0 u0 t
Z
η(0,x)=x (cid:0) (cid:1)
2Thatis,ω(t,η(t,x))=ω0(x).
THE SOLUTION MAP OF THE EULER EQUATIONS 7
where ω = ⊥ u andK is givenby (2.5). In orderto apply Picard’smethod of
0 0 2
∇ ·
successive approximations it is sufficient to show that the right hand side of (4.1)
is locally Lipschitz continuous in U . The first task is thus to establish that F
δ u0
maps into c1,α.
Changing variables in the integral we have
(4.2) F (η)(x)=(K ω˜ ) η(x)
u0 2∗ 0 ◦
where ω˜ = ω η−1detDη−1 and η U . Using (2.5) and (2.7) and estimating
0 0 δ
◦ ∈
directly we obtain
(4.3) F (η) = K ω˜ C ω
k u0 k∞ k 2∗ 0k∞ ≤ k 0k∞
whereC dependsonthe sizeofthesupportofω . Next,differentiatingF in(4.2)
0 u0
with respect to the x variable gives
(4.4) DF (η)(x)=D(K ω˜ ) η(x)Dη(x)
u0 2∗ 0 ◦
1
= Tω˜ ω˜ J η(x)Dη(x)
0 0
− 2 ◦
(cid:16) (cid:17)
0 1
where J = is the standard 2 2 symplectic matrix and T is a singular
−1 0
×
integraloperatorof the Calderon-Zygmundtype with the matrixkernel DK (x)=
(cid:0) (cid:1) 2
Ω(x)/x2 where Ω is homogeneous of degree zero
| |
1 Ω(x y) 2x x x2 x2
Tf(x)= p.v. − f(y)dy, Ω(x)= x−2 1 2 2− 1 .
2π ZR2 |x−y|2 | | (cid:18)x22−x21 −2x1x2(cid:19)
Standard estimates in Ho¨lder spaces for such operators3give
(4.5) DF (η) C Tω˜ η + ω˜ η Dη C ω˜ +[ω˜ ]
k u0 k∞ ≤ k 0◦ k∞ k 0◦ k∞ k k∞ ≤ k 0k∞ 0 α
and (cid:0) (cid:1) (cid:0) (cid:1)
[DF (η)] Tω˜ 1/2ω˜ J η [Dη] + Tω˜ 1/2ω˜ J η Dη
u0 α ≤k 0− 0 ◦ k∞ α 0− 0 ◦ αk k∞
(4.6) ≤C(cid:0)kω˜0k∞+[ω˜0]α(cid:1) [Dϕη]α+CkD(cid:2)η(cid:0)k1∞+α Tω˜0−1/(cid:1)2ω˜0J(cid:3) α
C(cid:0) ω˜0 ∞+[ω˜0]α(cid:1)[Dϕη]α+C(1+ Dϕ(cid:2)η ∞)1+α[ω˜0]α.(cid:3)
≤ k k k k
Furthermore, since(cid:0)from a direct c(cid:1)omputation using (2.7) we have
(4.7) [ω˜ ] =[ω η−1detDη−1] .[Dϕ ] ω +[ω ]
0 α 0 α η α 0 ∞ 0 α
◦ k k
combining these estimates with (4.3) and the fact that η U we get
δ
∈
(4.8) F (η) .C ω +[ω ] .
k u0 k1,α k 0k∞ 0 α
To show that F maps U into c1,α((cid:0)R2) it suffices n(cid:1)ow to observe that (4.6)
u0 δ
together with (4.7) yield
DF (η)(x) DF (η)(y)
lim sup | u0 − u0 | =0
h→00<|x−y|<h |x−y|α
since both ϕ and ω are in c1,α(R2) by assumption.
η 0
3Seee.g.,[23],Chap. 4.
8 GERARDMISIOL EKANDTSUYOSHIYONEDA
Finally,differentiatingF in(4.1)withrespecttoηinthedirectionw c1,α(R2)
u0 ∈
we obtain
d
(4.9) δ F (η)(x)= F (η+rw)(x)
w u0 dr u0 r=0
(cid:12)
(cid:12)
= DK2 η(x) η(cid:12)(y) (w(x) w(y))ω0(y)dy
ZR2 − −
(cid:0) (cid:1)
which again can be bounded directly using standard Ho¨lder estimates by
(4.10) δ F (η) .C ω +[ω ] w .
k w u0 k1,α k 0k∞ 0 α k k1,α
Inparticular,itfollowsthatF hasab(cid:0)oundedGateau(cid:1)xderivativeinU andhence
u0 δ
is locally Lipschitz by the mean value theorem.
We nowturntothe questionofdependence ofthe solutionsof (1.1)onu . Note
0
that since ω = ⊥ u the initial velocity u appears as a parameter on the right
0 0 0
∇ ·
handsideof(4.1). Moreover,sincethedependenceislinearitfollowsthatcontinuity
(and, in fact, differentiability) of the map u F is an immediate consequence
0 → u0
ofthe estimate in(4.8). Applying the fundamentaltheoremofordinarydifferential
equations (with parameters) for Banach spaces we find that there exist T >0 and
a unique Lagrangian flow η C([0,T),U ) which depends continuously (in fact,
δ
∈
differentiably) on u . Using the equations in (4.1) we find that the same is true
0
of the time derivative η˙ C([0,T),c1,α(R2)). It follows therefore that the vector
field u = η˙ η−1 belongs∈to C([0,T),c1,α(R2)) C1([0,T),cα(R2)) and a routine
◦ ∩
calculation shows that it is divergence free.
Next, suppose that u and v are two divergence free vector fields in c1,α(R2)
0 0
and let η(t) and ξ(t) be the corresponding Lagrangian flows solving the Cauchy
problem (4.1) in U with initial vorticities ⊥ u and ⊥ v respectively. Given
δ 0 0
∇ · ∇ ·
anyε>0 andusingthe factthatsmoothfunctions aredense inc1,α we canchoose
φ in C∞([0,T) R2) such that
ε
×
sup φ (t) η˙(t) <ε.
ε 1,α
k − k
0≤t≤T
Applying this together with (2.8) and (2.13) we estimate
u v = η˙ η−1 ξ˙ ξ−1
1,α 1,α
k − k k ◦ − ◦ k
η˙ η−1 φ η−1 + φ η−1 φ ξ−1 + φ ξ−1 ξ˙ ξ−1
ε 1,α ε ε 1,α ε 1,α
≤k ◦ − ◦ k k ◦ − ◦ k k ◦ − ◦ k
. η˙ φ + φ η−1 ξ−1 + φ ξ˙ .
ε 1,α ε 2,α 1,α ε 1,α
k − k k k k − k k − k
Thefirsttermofthelastlineisclearlyboundedbyε. Themiddletermconvergesto
zeroby Lemma 2.2 (continuity of the inversionmap) andthe fact that η converges
toξ inc1,α wheneveru convergestov sincethe Lagrangianflowsη andξ depend
0 0
continuously on the initial velocities. To dispose of the last term we use (4.1) and
the triangle inequality
φ ξ˙ φ η˙ + η˙ ξ˙
ε 1,α ε 1,α 1,α
k − k ≤k − k k − k
ε+ F (η) F (η) + F (η) F (ξ) .
≤ k u0 − v0 k1,α k v0 − v0 k1,α
THE SOLUTION MAP OF THE EULER EQUATIONS 9
Applying (4.8) we obtain
F (η) F (η) = K η(t, ) η(t,y) ⊥ u (y) ⊥ v (y) dy
k u0 − v0 k1,α R2 2 · − ∇ · 0 −∇ · 0 1,α
Z
.(cid:13)(cid:13)k∇⊥·(u(cid:0)0−v0)k∞+ ∇(cid:1)⊥(cid:0)·(u0−v0) α (cid:1) (cid:13)(cid:13)
. u v
0 0 1,α (cid:2) (cid:3)
k − k
and using (4.10) we get
1 d
F (η) F (ξ) = F rη+(1 r)ξ dr
k v0 − v0 k1,α dr v0 − 1,α
Z0
(cid:13) 1 (cid:0) (cid:1) (cid:13)
(cid:13) (cid:13)
δ F rη+(1 r)ξ dr
≤ k η−ξ v0 − k1,α
Z0
. v η ξ(cid:0) , (cid:1)
0 1,α 1,α
k k k − k
wherethelatterconvergestozerobycontinuousdependence oftheflowsonu and
0
v as before. This completes the proof of Theorem 4.1 when n=2.
0
In the three-dimensional case the flow equations (2.4) take only a slightly more
complicated form
dη
(4.11) (t,x)= K η(t,x) η(t,y) Dη(t,y)ω (y)dy =:G (η )(x)
dt R3 3 − 0 u0 t
Z
η(0,x)=x (cid:0) (cid:1)
where ω = u , K is given by (2.6) and consequently the derivative δ G in
the direc0tion∇w× c01,α(3R3) has an extra term w u0
∈
(4.12) δ G (η)(x)= DK η(x) η(y) (w(x) w(y))Dη(y)ω (y)dy
w u0 R3 3 − − 0
Z
(cid:0) (cid:1)
+ K η(x) η(y) Dw(y)ω(y)dy.
3
R3 −
Z
(cid:0) (cid:1)
As before, applying standardHo¨lderian estimates we obtain the analogues of (4.8)
and (4.10) and the proof proceeds as in the two dimensional case. (cid:3)
Remark 4.2. Theorem 4.1 remains valid if the initial vorticity has noncompact
support and satisfies some suitable decay conditions at infinity. In Section 5 we
will apply it under the assumption ω L1(R2). In this case we only have to
0
∈
replace the bound in (4.3) with
(4.3’) F (η) . ω˜ + ω˜
k u0 k∞ k 0k∞ k 0kL1
and those in (4.5), (4.6) with
(4.5’) DF (η) . ω˜ + ω˜
k u0 k∞ k 0k0,α k 0kL1
1+α
(4.6’) [DF (η)] . ω˜ + ω˜ [Dϕ ] + 1+ Dϕ [ω˜ ]
u0 α k 0k0,α k 0kL1 η α k ηk∞ 0 α
andadjusttheestimates((cid:0)4.8)and(4.10)acc(cid:1)ordingly. T(cid:0)herestofthe(cid:1)proofremains
unchanged.
Remark 4.3. We also mention in passing that an alternative proof of Theorem 4.1
could be given based on the following commutator inequalities
Ds(fg) fDsg . f g + f g
k − k∞ k kB˙∞1 ,∞k kB˙∞s−,1∞ k kB˙∞s ,∞k kB˙∞0 ,∞
10 GERARDMISIOL EKANDTSUYOSHIYONEDA
and
Ds(v g) (v )Dsg . v g .
∞ C1,s C1,s
k ·∇ − ·∇ k k k k k
where Ds = ( ∆)s/2 is the fractional Laplacian with 0 < s < 1. Inequalities of
−
this type were recently obtained in [4]. However, using this approach we could
only establish local well-posedness of (1.1) in the space c2,α for 0 < α < 1, under
suitabledecayconditions. BesidescontinuityofthedoubleRiesztransformsandthe
density property our proof required that the vorticity of u be at least of class C1,α
and hence that u belong to c2,α (in order to apply the second of the commutator
estimates above). A similar proof also works in Ws,p for p 2 and s>2/p+2.
≥
5. Proof of Theorem 1.3: norm inflation near B2
p,1
Our aim in this section is to exhibit a norm inflation type mechanism which
involves a family of Besov spaces. On the one hand the result stated in Theorem
1.3 is weaker than the norm inflation results obtained by Bourgain and Li. On
the other hand, our method is applicable in the borderline function spaces that
wereleftout ofthe analysisin[5,6]. The proofinvolvesconstructinga Lagrangian
flow with a large deformation gradient and a high-frequency perturbation of the
correspondinginitialvorticity. Italsoreliesonthecontinuityresultforthesolution
map in the little Ho¨lder space of Section 4.
Itwillbeconvenienttoworkwiththevorticityequationswhichintwodimensions
have the form
(5.1) ω +u ω =0, t 0, x R2
t
·∇ ≥ ∈
ω(0)=ω
0
where u = ⊥∆−1ω and ω = ∂ u ∂ u . We first proceed to choose the initial
1 2 2 1
∇ −
vorticityω fortheCauchyproblem(5.1). Givenanysmoothradialbumpfunction
0
0 φ 1 with support in the ball B(0,1/4) let
≤ ≤
(5.2) φ (x ,x )= ε ε φ(x ε ,x ε ).
0 1 2 1 2 1 1 2 2
− −
ε1,Xε2=±1
Clearly,the function φ is odd with respect to both x and x . Given any M 1,
0 1 2
≫
r>0 and q >0 define
(5.3) ω0(x)=ω0M,N,r,q(x)=M−2N−q1 φk(x)
0≤k≤N
X
whereN =1,2... andφk(x)=2(−1+2r)kφ0(2kx). Notethatthe supportsofφk are
disjoint and compact with
(5.4) suppφ B (ε 2−k,ε 2−k),2−(k+2) .
k 1 2
⊂
ε1,ε[2=±1 (cid:0) (cid:1)
Next, we have
Lemma 5.1. If 1<q < and 2<r < with q r. For any integer N >0 we
∞ ∞ ≤
have
(5.5) ω + ω .M−2.
0 W1,r 0 B1
k k k k r,q