Remarks on a fractional diffusion transport equation with applications to the dissipative quasi-geostrophic equation

0.39 MB·English

by  Diego Chamorro

Remarks on a fractional diffusion transport equation
with applications to the
dissipative quasi-geostrophic equation
Diego Chamorro
December 6, 2011
Abstract
1
1 Inthisarticle we studyH¨olderregularity Cγ forsolutions of atransport equation based inthedissipative quasi-geostrophic
0
equation. Following an idea of A. Kiselev and F. Nazarov presented in [10], we will use the molecular characterization of local
2
Hardy spaces hσ in order to obtain information on H¨older regularity of such solutions. This will be done by following the

c evolution of molecules in a backward equation. We will also study global existence, Besov regularity for weak solutions and a
e
maximum principle (or positivity principle) and we will apply these results to thedissipative quasi-geostrophic equation.
D
Keywords: quasi-geostrophic equation, H¨older regularity, Hardy spaces.
5

]
P
1 Introduction
A
.
h
Thedissipativequasi-geostrophicequationhasbeenstudiedbymanyauthors,notonlybecauseofitsownmathematical
t
a importance, but also as a 2D model in geophysical fluid dynamics and because of its close relationship with other
m
equations arising in fluid dynamics. See [5], [14] and the references given there for more details. This equation has
[ the following form:

5 ∂tθ(x,t)= ∇·(uθ)(x,t)−Λ2αθ(x,t)
v (QG)
9 α 
1  θ(x,0)= θ0(x)
9
for0<α<1andt∈[0,T]. Hereθ isareal-valuedfunctionandthevelocityuisdefinedbymeansofRiesztransforms
3
. in the following way:
7
0 u=(−R2θ,R1θ).
0
1 RecallthatRieszTransformsR aregivenbyR θ(ξ)=−iξjθ(ξ) forj =1,2andthatΛ2α =(−∆)α isthe Laplacian’s
: j j ξ
v fractional power defined by the formula | |
Xi dΛ[2αθ(ξ)=|bξ|2αθ(ξ)
r
a
where θ denotes the Fourier transform of θ. b
It isbclassical to consider three cases in the analysis of dissipative quasi-geostrophic equation following the values
of diffusion parameter α. Case 1/2 < α is called sub-critical since diffusion factor is stronger than nonlinearity. In
this case, weaksolutions wereconstructedby S. Resnick in[16] andP.Constantin& J.Wu showedin[4]that smooth
initial data gives a smooth global solution.
The critical case is given when α=1/2. Here, P. Constantin, D. C´ordoba & J. Wu studied in [6] global existence
in Sobolev spaces, while global well-posedness in Besov spaces has been treated by H. Abidi & T. Hmidi in [1]. Also
in this case and more recently A. Kiselev, F. Nazarov & A. Volberg showed in [11] that any regular periodic data
generates a unique C solution.
∞
Finally, the case when 0<α<1/2 is calledsuper-critical partiallybecause it is harder to work with than the two
othercases,but mostly because the diffusiontermis weakerthanthe nonlinearterm. In this lastcase,weak solutions
for initial data in Lp or in H˙ 1/2 were studied by F. Marchand in [15].
−
In this article, following L. Caffarelli & A. Vasseur in [2], we will study a special version of the dissipative quasi-
geostrophicequation(QG) . TheideaistoreplacetheRieszTransform-basedvelocityubyanewvelocityv toobtain
α
1
the following n-dimensional fractional diffusion transport equation for 0<α≤1/2:
∂ θ(x,t)= ∇·(vθ)(x,t)−Λ2αθ(x,t)
t

(T) θ(x,0)= θ (x) (1)
α  0
div(v)= 0.

In (1) functions θ and v are such that θ : Rn × [0,T] −→ R and v : Rn × [0,T] −→ Rn. Remark that the
velocity v is now a given data for the problem and we will always assume that v is divergence free and belongs to
L ([0,T];bmo(Rn)).
∞
We fix once and for all the parameter α∈]0,1/2] in order to study the critical and the super-critical case.
Themaintheorempresentedinthisarticlestudiestheregularityofthesolutionsofthefractionaldiffusiontransport
equation (T) with α∈]0,1/2]:
α
Theorem 1 (Ho¨lder regularity) Let θ be a function such that θ ∈ L (Rn). If θ(x,t) is a solution for the
0 0 ∞
equation (T) with α ∈]0,1/2], then for all time 0 < t < T, we have that θ(·,t) belongs to the Ho¨lder space Cγ(Rn)
α
with 0<γ <2α.
Since the velocity v in equation (T) belongs to L ([0,T];bmo(Rn)), it would be quite simple to adapt this result to
α ∞
the quasi-geostrophic equation (QG) . See section 7 for details.
α
The conclusion of this theorem is surprising when applied to the quasi-geostrophic equation: it asserts, that even
when the diffusion term is weaker than the transport term we still obtain a small smoothing effect. Furthermore, by
aresultofConstantinandWu[5]thistheoremimpliesthatintherange1/4<α<1/2solutionsof(QG) aresmooth.
α
Let us say a few words about the proof of this theorem. A classical result of harmonic analysis states that Ho¨lder
spaces Cγ(Rn) can be paired with local Hardy spaces hσ(Rn). Therefore, if we prove that the duality bracket
hθ(·,t),ψ i= θ(x,t)ψ (x)dx (2)
0 0
ZRn
is bounded for every ψ ∈hσ(Rn), with t∈]0,T[, we obtain that θ(·,t)∈Cγ(Rn). One of the main features of Hardy
0
spacesisthatthey admitacharacterizationbymolecules (seedefinition1.1below),whicharerathersimplefunctions,
and this allows us to study the quantity (2) only for such molecules.
This dual approach was originally given in the torus Tn by A. Kiselev & F. Nazarov in [10], but only for the
critical case α = 1/2 and with a very special family of test functions. Thus, the main novelty of this paper besides
the generalization to Rn and the use of Hardy spaces is the treatment of the super-critical case 0 < α < 1/2 for the
equation (T) with initial data in L (Rn).
α ∞
Let us stress that the study of super-critical case is made possible by carefully choosing the molecules used in the
decomposition of Hardy spaces. Broadly speaking and following [17] p. 130, a molecule is a function ψ so that
(i) ψ(x)dx=0
Rn
R
(ii) |ψ(x)|≤r n/σmin{1;rβn/|x−x |βn},
− 0
with βσ >1 and x ∈Rn, where the parameter r ∈]0,+∞[ stands for the size of the molecule ψ.
0
Since we are going to work with local Hardy spaces, we will introduce a size treshold in order to distinguish small
molecules from big ones in the following way:
Definition 1.1 (r-molecules) Let α∈]0,1/2], set n <σ <1, define γ =n(1 −1) and fix a real number ω such
n+2α σ
that γ <ω <2α. An integrable function ψ is an r-molecule if we have
2
• Small molecules (0<r<1):
|ψ(x)||x−x |ωdx≤rω γ, for x ∈Rn (concentration condition) (3)
0 − 0
ZRn
1
kψkL∞ ≤ rn+γ (height condition) (4)
ψ(x)dx=0 (moment condition) (5)
ZRn
• Big molecules (1≤r <+∞):
Inthis caseweonlyrequireconditions (3)and(4)for ther-moleculeψ while themomentcondition (5)is dropped.
Itisinterestingtocomparethisdefinitionofmoleculestotheoneusedin[10]. Inourmoleculestheparameterγ reflects
explicitlytherelationshipbetweenHardyandHo¨lderspaces(seetheorem4below). However,themostimportantfact
relies in the parameter ω which gives us the additional flexibility that will be crucial in the following calculations.
Remark 1.1
1) Note that the point x ∈Rn can be considered as the “center” of the molecule.
0
2) Conditions (3) and (4) are an easy consequence of condition (ii) and they both imply the estimate
kψk ≤Cr γ
L1 −
thus every r-molecule belongs to Lp(Rn) for 1<p<+∞.
The main interest of using molecules relies in the possibility of transfering the regularity problem to the evolution of
such molecules:
Proposition 1.1 (Transfer property) Let α∈]0,1/2]. Let ψ(x,s) be a solution of the backward problem
∂ ψ(x,s)= −∇·[v(x,t−s)ψ(x,s)]−Λ2αψ(x,s)
s
 ψ(x,0)= ψ (x)∈L1∩L (Rn) (6)
 0 ∞
div(v)= 0 and v ∈L ([0,T];bmo(Rn))
∞
If θ(x,t) is a solution of (1) with θ ∈L (Rn) then we have the identity
0 ∞
θ(x,t)ψ(x,0)dx = θ(x,0)ψ(x,t)dx. (7)
ZRn ZRn
Proof. We first consider the expression
∂ θ(x,t−s)ψ(x,s)dx= −∂ θ(x,t−s)ψ(x,s)+∂ ψ(x,s)θ(x,t−s)dx.
s s s
ZRn ZRn
Using equations (1) and (6) we obtain
∂ θ(x,t−s)ψ(x,s)dx = −∇·[(v(x,t−s)θ(x,t−s)]ψ(x,s)+Λ2αθ(x,t−s)ψ(x,s)
s
ZRn ZRn
− ∇·[(v(x,t−s)ψ(x,s))]θ(x,t−s)−Λ2αψ(x,s)θ(x,t−s)dx.
Now, using the fact that v is divergence free we have that expression above is equal to zero, so the quantity
θ(x,t−s)ψ(x,s)dx
ZRn
remains constant in time. We only have to set s=0 and s=t to conclude.
(cid:4)
3
This proposition says, that in order to control hθ(·,t),ψ i, it is enough (and much simpler) to study the bracket
0
hθ ,ψ(·,t)i. Let us explain in which sense this transfer property is useful: in the bracket hθ ,ψ(·,t)i we have much
0 0
more informations than in the bracket (2) since the initial data ψ is a molecule which satisfies conditions (3)-(5).
0
Proof of the theorem 1. Once we have the transfer property proven above, the proof of theorem 1 is quite
direct and it reduces to a L1 estimate for molecules. Indeed, assume that for all molecular initial data ψ we have
0
a L1 control for ψ(·,t) a solution of (6), then theorem 1 follows easily: applying proposition 1.1 with the fact that
θ ∈L (Rn) we have
0 ∞
|hθ(·,t),ψ0i|= θ(x,t)ψ0(x)dx = θ(x,0)ψ(x,t)dx ≤kθ0kL∞kψ(·,t)kL1 <+∞. (8)
(cid:12)ZRn (cid:12) (cid:12)ZRn (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
From this, we obtain that θ(·,(cid:12)(cid:12)t) belongs to the H(cid:12)(cid:12)o¨lde(cid:12)(cid:12)r space Cγ(Rn). (cid:12)(cid:12)
Now we need to study the control of the L1 norm of ψ(·,t) and we divide our proof in two steps following the
molecule’s size. For the initial big molecules, i.e. if r ≥1, the needed control is straightforward: apply the maximum
principle (10) below and the remark 1.1-2) to obtain
1
kθ0kL∞kψ(·,t)kL1 ≤kθ0kL∞kψ0kL1 ≤Crγkθ0kL∞,
but, since r ≥1, we have that |hθ(·,t),ψ i|<+∞ for all big molecules.
0
In orderto finish the proofof the theorem, it only remains to treatthe L1 controlfor small molecules. This is the
most complex part of the proof and we will present it in section 3 where we will prove the next theorem:
Theorem 2 For all small initial molecular data ψ , there existe an small constant δ >0 such that
0
−ω
kψ(·,t)kL1 ≤Cδ2α for all 0<t<T (0<γ <ω <2α).
We obtain in this way a good control over the quantity kψ(·,t)k for all 0<r <1.
L1
Finally, getting back to (8) we obtain that |hθ(·,t),ψ i| is always bounded for 0<t<T and for any molecule ψ :
0 0
we have proven by a duality argument the theorem 1.
(cid:4)
Letusrecallnowthatfor(smooth)solutionsof(QG) and(T) abovewecanusetheremarkablepropertyofmaximum
α α
principle mentioned before. This was proven in [7] and in [15] and it gives us the following inequalities.
t
kθ(·,t)k +p |θ(x,s)|p 2θ(x,s)Λ2αθ(x,s)dxds ≤ kθ k (2≤p<+∞) (9)
Lp − 0 Lp
Z0 ZRn
or more generally kθ(·,t)k ≤ kθ k (1≤p≤+∞) (10)
Lp 0 Lp
These estimates are extremely useful and they are the starting point of severalworks. Indeed, the study of inequality
(9) helps us incidentally to solve a question pointed out by F. Marchand in [15] concerning weak solution’s global
regularity:
Theorem 3 (Weak solution’s regularity) Let 2≤ p <+∞ and α∈]0,1/2]. If θ ∈Lp(Rn) is an initial data for
0
(QG) or(T) equations,thentheassociatedweaksolutionθ(x,t)belongstoL ([0,T];Lp(Rn))∩Lp([0,T];B˙2α/p,p(Rn)).
α α ∞ p
Theorem 1, theorem 3 and the L1 control for small molecules are the core of the paper, however, for the sake of
completness, we will prove some other interesting results concerning the equation (1).
The plan of the article is the following: in the section 2 we recall some facts concerning the molecular char-
acterization of local Hardy spaces and some other facts about H¨older and bmo spaces. In section 3 we study the
L1-normcontrolformolecules andin section4 westudy existence andunicity ofsolutionswith initialdatainLp with
2≤p<+∞andweprovetheorem3. Section5isdevotedtoapositivity principle thatwillbeusefulinourproofsand
section6studiesexistenceofsolutionwithθ ∈L . Finally,section7appliestheseresultstothe2D-quasi-geostrophic
0 ∞
equation (QG) .
α
4
2 Molecular Hardy spaces, H¨older spaces and bmo
Hardy spaces have several equivalent characterizations (see [3], [8] and [17] for a detailed treatment). In this paper
we are interested mainly in the molecular approachthat defines local Hardy spaces hσ with 0<σ <1:
Definition 2.1 (Local Hardy spaces hσ) Let 0< σ < 1. The local Hardy space hσ(Rn) is the set of distributions
f that admits the following molecular decomposition:
f = λ ψ (11)
j j
j N
X∈
where (λj)j N is a sequence of complex numbers such that j N|λj|σ < +∞ and (ψj)j N is a family of r-molecules
in the sense∈of definition 1.1 above. The hσ-norm1 is then fixe∈d by the formula ∈
P
1/σ
kfk =inf |λ |σ : f = λ ψ
hσ  j j j
 
 Xj∈N Xj∈N 
 
where the infimum runs over all possible decompositions (11). 
LocalHardyspaceshavemanyremarquablepropertiesandwewillonlystresshere,beforepassingtodualityresults
concerninghσ spaces,thefactthatSchwartzclassS(Rn)isdenseinhσ(Rn). Forfurtherdetailssee[9],[17],[8]and[5].
Now,letustakeacloserlookatthedualspaceoflocalHardyspaces. In[8]D.Goldbergprovedthenextimportant
theorem:
Theorem 4 (Hardy-Ho¨lder duality) Let n <σ <1 and fix γ =n(1 −1). Then the dual of local Hardy space
n+2α σ
hσ(Rn) is the Ho¨lder space Cγ(Rn) fixed by the norm
|f(x)−f(y)|
kfkCγ =kfkL∞ +xsu=py |x−y|γ .
6
This result allows us to study the Ho¨lder regularity of functions in terms of Hardy spaces and it will be applied to
solutions of the n-dimensional fractional diffusion transport equation (1).
1/σ
Remark 2.1 Since 0<σ <1, we have |λ |≤ |λ |σ thus for testing Ho¨lder continuity of a function
j N j j N j
∈ ∈
f it is enough to study the quantities |hfP,ψji| where(cid:16)ψPj is an r-m(cid:17)olecule.
We finish this section by recalling some useful facts about the bmo space used to characterize velocity v. This space
is defined as locally integrable functions f such that
1 1
sup |f(x)−f |dx<M and sup |f(x)|dx<M for a constant M;
B
|B| |B|
|B|≤1 ZB |B|>1 ZB
wherewenotedB(R)aballofradiusR>0andf = 1 f(x)dx. The normk·k isthenfixedasthesmallest
B B bmo
| |ZB(R)
constant M satisfying these two conditions.
We will use the next properties for a function belonging to bmo:
Proposition 2.1 Let f ∈bmo, then
1) for all 1<p<+∞, f is locally in Lp and 1 |f(x)−f |pdx≤Ckfkp
B B bmo
| |ZB
2) for all k ∈N, we have |f −f |≤Ckkfk where 2kB(R)=B(2kR).
2kB B bmo
Proposition 2.2 Let f be a function in bmo(Rn). For k ∈N, define f by
k
−k if f(x)≤−k
f (x)= f(x) if −k ≤f(x)≤k (12)
k 
k if k ≤f(x).
Then (fk)k N converges weakly to f in bmo(Rn).
∈
For a proof of these results and more details on Hardy, Ho¨lder and bmo spaces see [3], [8], [12], [9] and [17].
1itisnotactuallyanormsince0<σ<1. LocalHardyspacesare,however,completemetricspacesforthedistanced(f,g)=kf−gkσ .
hσ
Moredetailscanbefoundin[8]and[17].
5
3 L1 control for small molecules: proof of theorem 2
As said in the introduction, we need to construct a suitable control in time for the L1-norm of the solutions ψ(·,t) of
the backward problem (6) where the inital data ψ is a small r-molecule. This will be achieved by iteration in two
0
different steps. The first step explains the molecules’ deformation after a very small time s > 0, which is related to
0
the size r by the bounds 0 < s ≤ ǫr with ǫ a small constant. In order to obtain a control of the L1 norm for larger
0
times we need to perform a second step which takes as a starting point the results of the first step and gives us the
deformation for another small time s , which is also related to the original size r. Once this is achieved it is enough
1
to iterate the second step as many times as necessary to get rid of the dependence of the times s ,s ,... from the
0 1
molecule’s size. This way we obtain the L1 control needed for all time 0<t<T.
It is important to explain why we need two steps and why the iteration is performed over the second step: the
firststepbelow givesussmalltime molecule’s evolution,howeverwearenotgoingtorecovermoleculesinthe senseof
definition 1.1, but deformed ones: we can not simply reapply these calculations since the input is quite different from
the output. The second step is thus an attempt to recover some of the previous molecular properties taking as initial
data these molecules after deformation. We will see how to control the deformed molecules to obtain similar profiles
in the inputs and the outputs in section 3.2 and only then we will perform the iteration in section 3.3.
3.1 Small time molecule’s evolution: First step
The following theorem shows how the molecular properties are deformed with the evolution for a small time s .
0
Theorem 5 Let α ∈]0,1/2]. Set σ, γ and ω three real numbers such that n < σ < 1, γ = n(1 − 1) and
n+2α σ
0<γ <ω <2α. Let ψ(x,s ) be a solution of the problem
0
∂ ψ(x,s )= −∇·(vψ)(x,s )−Λ2αψ(x,s )
s0 0 0 0

 ψ(x,0)= ψ0(x) (13)
div(v)= 0 and v ∈L ([0,T];bmo(Rn)) with sup kv(·,s )k ≤µ
∞ 0 bmo
If ψ is a small r-molecule in the sense of definition 1.1 for the local Hardyss0p∈a[0c,eT]hσ(Rn), then there exists a positive
0
constant K = K(µ) big enough and a positive constant ǫ such that for all 0 < s ≤ ǫr small we have the following
0
estimates
|ψ(x,s )||x−x(s )|ωdx ≤ G (r,s )ω γ (14)
0 0 0 0 −
ZRn
1
kψ(·,s0)kL∞ ≤ r2αnn++ωγ +c0s0 n2+αω (15)
v
n
kψ(·,s0)kL1 ≤ (cid:0)r2αnn++ωγ +c0s0(cid:1)2ωα (16)
where G (r,s ) = (r +Ks ) is a target function, 0 < c < 1 i(cid:0)s a constant an(cid:1)d v denotes the volume of the n-
0 0 0 0 n
dimensional unit ball.
The new molecule’s center x(s ) used in formula (14) is fixed by
0
x(s )= v = 1 v(y,s )dy
 ′ 0 Br |Br|ZBr 0 (17)
 x(0)= x .
0
where B =B(x(s ),r). 
r 0
Remark 3.1
1) The definition of the point x(s ) given by (17) reflects the molecule’s center transport using velocity v.
0
2) Estimates (14)-(16) explain the molecules’ deformation following the evolution of the system. Note in particular
that if s −→0 in (14) and (15) we recover the initial molecular conditions (3) and (4).
0
3) Remark that it is enough to treat the case 0 < G (r,s ) < 1. Indeed, if G (r,s ) = 1 and if s0 ≈ ǫ, we have
0 0 0 0 r
r ≈ 1 and the right-hand side of (16) will be bounded by v 1+Kǫ 2ωα. The L1 control will be trivial then
1+Kǫ n 1+c0ǫ
for time s and beyond: we only need to apply the maximum principle.
0
(cid:0) (cid:1)
6
The proof of this theorem follows the next scheme: the small concentration condition (14), which is proven in the
proposition3.1, implies the height condition (15) (provedin proposition3.2). Once we have these two conditions, the
L1 estimate (16) will follow easily and this is proven in proposition 3.3.
Proposition 3.1 (Small time Concentration condition) Under the hypothesis of theorem 5, if ψ is a small r-
0
molecule, then the solution ψ(x,s) of (13) satisfies
|ψ(x,s0)||x−x(s0)|ωdx≤G0(r,s0)ω−γ
ZRn
for x(s )∈Rn fixed by formula (17), with 0≤s ≤ǫr and G (r,s )=(r+Ks ).
0 0 0 0 0
Proof. Let us write Ω(x−x(s ))=|x−x(s )|ω and ψ(x)=ψ (x)−ψ (x) where the functions ψ (x) ≥0 have
0 0 +
− ±
disjoint support. We will note ψ (x,s ) solutions of (13) with ψ (x,0)=ψ (x).
0
± ± ±
At this point, we assume the following positivity principle which is proven in section 5:
Theorem 6 Let 1 ≤ p ≤ +∞, if initial data ψ ∈ Lp(Rn) is such that 0 ≤ ψ (x) ≤ M, then the associated solution
0 0
ψ(x,s ) of (13) satisfies 0≤ψ(x,s )≤M for all s ∈[0,T].
0 0 0
Thus, by linearity and using the above theorem we have that
|ψ(x,s )|=|ψ (x,s )−ψ (x,s )|≤ψ (x,s )+ψ (x,s )
0 + 0 0 + 0 0
− −
and we can write
|ψ(x,s )|Ω(x−x(s ))dx≤ ψ (x,s )Ω(x−x(s ))dx+ ψ (x,s )Ω(x−x(s ))dx
0 0 + 0 0 0 0
ZRn ZRn ZRn −
so we only have to treat one of the integrals on the right side above. We have:
I = ∂ Ω(x−x(s ))ψ (x,s )dx
s0 0 + 0
(cid:12) ZRn (cid:12)
(cid:12) (cid:12)
= (cid:12)(cid:12) ∂s0Ω(x−x(s0))ψ+(x,s0)+Ω(cid:12)(cid:12) (x−x(s0)) −∇·(vψ+(x,s0))−Λ2αψ+(x,s0) dx
(cid:12)ZRn (cid:12)
(cid:12) (cid:2) (cid:3) (cid:12)
= (cid:12)(cid:12) −∇Ω(x−x(s0))·x′(s0)ψ+(x,s0)+Ω(x−x(s0)) −∇·(vψ+(x,s0))−Λ2αψ+(x(cid:12)(cid:12),s0) dx
(cid:12)ZRn (cid:12)
(cid:12) (cid:2) (cid:3) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Using the fact that v is divergence free, we obtain
I = ∇Ω(x−x(s0))·(v−x′(s0))ψ+(x,s0)−Ω(x−x(s0))Λ2αψ+(x,s0)dx .
(cid:12)ZRn (cid:12)
(cid:12) (cid:12)
Finally, using the defini(cid:12)tion of x(s ) given in (17) and replacing Ω(x−x(s )) by |x−x(s )|ω we(cid:12) obtain
(cid:12) ′ 0 0 0 (cid:12)
I ≤c |x−x(s )|ω 1|v−v ||ψ (x,s )|dx+c |x−x(s )|ω 2α|ψ (x,s )|dx. (18)
0 − Br + 0 0 − + 0
ZRn ZRn
I1 I2
| {z } | {z }
Remark 3.2 Note that when 0<α<1/2the power ofthe factor |x−x(s )| is different in I andI , so calculations
0 1 2
are a slightly easier when α=1/2. We will see that the other case is not too complicated either.
We will study separately each of the integrals I and I in the next lemmas:
1 2
Lemma 3.1 For integral I above we have the estimate
1
I ≤Cµrω 1 γ.
1 − −
Proof. We begin by considering the space Rn as the union of a ball with dyadic coronas centered on x(s ), more
0
precisely we set Rn =B ∪ E where
r k 1 k
≥
SB = {x∈Rn :|x−x(s )|≤r}. (19)
r 0
Ek = {x∈Rn :r2k−1 <|x−x(s0)|≤r2k} for k ≥1,
7
(i) Estimations over the ball B .
r
Applying Ho¨lder inequality on integral I we obtain
1
|x−x(s )|ω 1|v−v ||ψ (x,s )|dx ≤ k|x−x(s )|ω 1k (20)
0 − Br + 0 0 − Lp(Br)
ZBr
(1)
× k|v−vBrkL{zz(Br)kψ+(·},s0)kLq(Br)
(2) (3)
where 1 + 1 + 1 =1 and p,z,q>1. We treat each of the prev|ious ter{mzs sepa}r|ately: {z }
p z q
• First observe that for 1<p<n/(1−ω) we have for the term (1) above:
k|x−x(s )|ω 1k ≤Crn/p+ω 1.
0 − Lp(Br) −
• By hypothesis we have v(·,s )∈bmo, thus
0
kv−v k ≤C|B |1/zkv(·,s )k .
Br Lz(Br) r 0 bmo
since sup kv(·,s )k ≤µ we write for the term (2)
0 bmo
s0∈[0,T]
kv−v k ≤Cµrn/z.
Br Lz(Br)
• Finally for (3) by the maximum principle (10) for Lq norms we have kψ (·,s )k ≤kψ (·,0)k ; hence
+ 0 Lq(Br) + Lq
using the fact that ψ is an r-molecule and remark 1.1-2) we obtain
0
1/q 1 1/q
1 −
kψ (·,s )k ≤C r γ .
+ 0 Lq(Br) − rn+γ
(cid:20) (cid:21) (cid:20) (cid:21)
We gather all these inequalities together in order to obtain the following estimation for (20):
|x−x(s )|ω 1|v−v ||ψ (x,s )|dx≤Cµrω 1 γ. (21)
0 − Br + 0 − −
ZBr
(ii) Estimations for the dyadic corona E .
k
Let us note I the integral
k
I = |x−x(s )|ω 1|v−v ||ψ (x,s )|dx.
k 0 − Br + 0
ZEk
Since over E we have2 |x−x(s )|ω 1 ≤C2k(ω 1)rω 1 we write
k 0 − − −
Ik ≤ C2k(ω−1)rω−1 |v−vBr2k||ψ+(x,s0)|dx+ |vBr −vBr2k||ψ+(x,s0)|dx
(cid:18)ZEk ZEk (cid:19)
where we noted B =B(x(s ),r2k), then
r2k 0
I ≤ C2k(ω 1)rω 1 |v−v ||ψ (x,s )|dx
− − ZBr2k Br2k + 0
+ |v −v ||ψ (x,s )|dx .
ZBr2k Br Br2k + 0 !
Now, since v(·,s )∈bmo, using proposition 2.1 we have
0
|v −v |≤Ckkv(·,s )k ≤Ckµ
Br Br2k 0 bmo
and we can write
I ≤ C2k(ω 1)rω 1 |v−v ||ψ (x,s )|dx+Ckµkψ (·,s )k
k − − ZBr2k Br2k + 0 + 0 L1!
≤ C2k(ω−1)rω−1 kψ+(·,s0)kLa0kv−vBr2kkLa0a−01 +Ckµr−γ
(cid:16) (cid:17)
2recallthat0<γ<ω<2α≤1.
8
where we used Ho¨lder inequality with 1<a < n and maximum principle for the last term above. Using
0 n+(ω 1)
again the properties of bmo spaces we have −
Ik ≤C2k(ω−1)rω−1 kψ+(·,0)kL1/1a0kψ+(·,0)kL1−∞1/a0|Br2k|1−1/a0kv(·,s)kbmo+Ckµr−γ .
(cid:16) (cid:17)
Let us now apply estimates given by hypothesis over kψ+(·,0)kL1, kψ+(·,0)kL∞ and kv(·,s0)kbmo to obtain
Ik ≤C2k(n−n/a0+ω−1)rω−1−γµ+C2k(ω−1)kµrω−1−γ.
Since 1<a < n , we have n−n/a +(ω−1)<0, so that, summing over eachdyadic coronaE , we have
0 n+(ω 1) 0 k
−
I ≤Cµrω 1 γ. (22)
k − −
k 1
X≥
Finally, gathering together estimations (21) and (22) we obtain the desired conclusion.
(cid:4)
Lemma 3.2 For integral I in inequality (18) we have the following estimate
2
I2 ≤Crω−1−γ.
Proof. As for lemma 3.1, we consider Rn as the union of a ball with dyadic coronas centered on x(s ) (cf. (19)).
0
(i) Estimations over the ball B .
r
We apply now the Ho¨lder inequality in the integral I above with 1<a <n/(2α−ω) and 1 + 1 =1 in order
2 1 a1 b1
to obtain
|x−x(s )|ω 2α|ψ (x,s )|dx ≤ k|x−x(s )|ω 2αk kψ (·,s )k
0 − + 0 0 − La1(Br) + 0 Lb1(Br)
ZBr
≤ Crn/a1+ω−2αkψ+(·,0)k1L/1b1kψ+(·,0)k1L−∞1/b1.
Using hypothesis over kψ+(·,0)kL1 and kψ+(·,0)kL∞ we obtain
|x−x(s )|ω 2α|ψ (x,s )|dx≤Crω 2α γ.
0 − + 0 − −
ZBr
We havetwocases: ifα=1/2weobtainthe estimateneededovertheballB ; butfor0<α<1/2recallthatwe
r
are working with small molecules so we have 0<r <1 and rω 2α γ ≤rω 1 γ. Thus, in any case we can write:
− − − −
|x−x(s )|ω 2α|ψ (x,s )|dx≤Crω 1 γ. (23)
0 − + 0 − −
ZBr
(ii) Estimations for the dyadic corona E .
k
Here we have
|x−x(s )|ω 2α|ψ (x,s )|dx ≤ C2k(ω 2α)rω 2α |ψ (x,s )|dx≤C2k(ω 2α)rω 2αkψ (·,s )k
0 − + 0 − − + 0 − − + 0 L1
ZEk ZEk
≤ C2k(ω 2α)rω 2αkψ (·,0)k ≤C2k(ω 2α)rω 2α γ
− − + L1 − − −
It is at this step that the flexibility of molecules is essential. Indeed, in the definition 1.1 we have fixed 0<γ <
ω <2α so we have ω−2α<0 and thus, summing over k ≥1, we obtain
|x−x(s )|ω 2α|ψ (x,s )|dx≤Crω 2α γ.
0 − + 0 − −
k 1ZEk
X≥
Repeating the same argument used before (i.e. the fact that 0<r <1), we finally obtain
|x−x(s )|ω 2α|ψ (x,s )|dx≤Crω 1 γ. (24)
0 − + 0 − −
k 1ZEk
X≥
9
In order to finish the proof of lemma 3.2 we glue together estimates (23) and (24).
(cid:4)
Now we continue the proof of the proposition 3.1. Using lemmas 3.1 and 3.2 and getting back to estimate (18) we
have
∂ Ω(x−x(s ))ψ (x,s )dx ≤C(µ+1)rω 1 γ
s0 0 + 0 − −
(cid:12) ZRn (cid:12)
(cid:12) (cid:12)
This last estimation is compa(cid:12)tible with the estimate (14) for(cid:12)0≤s0 ≤ǫr small enough: just fix K such that
(cid:12) (cid:12)
C(µ+1)≤K(ω−γ). (25)
Indeed, since the time s is very small, we can linearize the target function in the right-hand side of (14) in order
0
to obtain
s
φ=G (r,s )ω γ =(r+Ks )ω γ ≈rω γ 1+[K(ω−γ)] 0 . (26)
0 0 − 0 − −
r
(cid:16) (cid:17)
Finally, taking the derivative with respect to s in the above expression we have φ ≈ rω 1 γK(ω −γ) and with
0 ′ − −
condition (25) proposition 3.1 follows.
(cid:4)
Now we will give a sligthly different proof of the maximum principle of A. C´ordoba & D. C´ordoba. Indeed, the
following proof only relies on the concentration condition proved in the lines above.
Proposition 3.2 (Small time Height condition) Under the hypothesis of theorem 5, if ψ(x,s ) satisfies concen-
0
tration condition (14), then we have the next height condition
1
kψ(·,s0)kL∞ ≤ n+ω . (27)
r2αnn++ωγ +Cs0 2α
(cid:16) (cid:17)
Proof. To begin with, we observe that this inequality is easily verified if
1
kψ(·,0)kL∞ ≤ 2rn+γ.
Indeed, since time s is small we can obtain (27) from this estimate. So, without loss of generality, we can assume
0
that
1 1
2rn+γ ≤kψ(·,0)kL∞ ≤ rn+γ. (28)
Assume now that molecules we are working with are smooth enough. Following an idea of [7] (section 4 p.522-523),
we will note x the point of Rn such that ψ(x,s0)=kψ(·,s0)kL∞. Thus we can write
d |ψ(x,s )−ψ(y,s )|
0 0
ds0kψ(·,s0)kL∞ ≤−ZRn |x−y|n+2α dy ≤0. (29)
For simplicity, we will assume that ψ(x,s ) is positive.
0
Let us consider the corona centered in x defined by
C(R ,R )={y ∈Rn :R ≤|x−y|≤R }
1 2 1 2
where R =ρR with ρ>2 and where R will be fixed later. Then:
2 1 1
ψ(x,s )−ψ(y,s ) ψ(x,s )−ψ(y,s )
0 0 0 0
dy ≥ dy.
ZRn |x−y|n+2α ZC(R1,R2) |x−y|n+2α
Define the sets B and B by B = {y ∈ C(R ,R ) : ψ(x,s )−ψ(y,s ) ≥ 1ψ(x,s )} and B = {y ∈ C(R ,R ) :
1 2 1 1 2 0 0 2 0 2 1 2
ψ(x,s )−ψ(y,s )< 1ψ(x,s )} such that C(R ,R )=B ∪B .
0 0 2 0 1 2 1 2
We obtain the inequalities
ψ(x,s )−ψ(y,s ) ψ(x,s )−ψ(y,s )
0 0 0 0
dy ≥ dy
|x−y|n+2α |x−y|n+2α
ZC(R1,R2) ZB1
ψ(x,s ) ψ(x,s )
0 0
≥ |B |= (|C(R ,R )|−|B |).
2Rn+2α 1 2Rn+2α 1 2 2
2 2
10