A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations

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by  Borys Alvarez-Samaniego

A NASH-MOSER THEOREM FOR SINGULAR EVOLUTION
EQUATIONS. APPLICATION TO THE SERRE AND
7
0 GREEN-NAGHDI EQUATIONS
0
2
BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
n
a
J Abstract. We study the well-posedness of the initial value problem for a

4 wideclassofsingularevolution equations. Weproveageneralwell-posedness
2 theoremunderthreeassumptionseasytocheck: thefirstcontrolsthesingular
part of the equation, the second the behavior of the nonlinearities, and the

] third one assumes that an energy estimate can be found for the linearized
P system. We allow losses of derivatives in this energy estimate and therefore
A construct a solution by a Nash-Moser iterative scheme. As an application
to this general theorem, weprove the well-posedness ofthe Serreand Green-
.
h Naghdiequationanddiscusstheproblemoftheirvalidityasasymptoticmodels
t forthewater-wavesequations.
a
m
[

1 1. Introduction
v
1 1.1. Generalsetting. Weinvestigateinthispaperthelocalintimewell-posedness
8 of singular evolution equations of the form
6
1
1 ∂ uε+ ε(t)uε+ ε[t,uε]=hε
0 (1) t εL F
 uε =uε,
7  |t=0 0
0
whereε (0,ε )isaparameter, ε(t)isalinearoperator,while ε[t, ]isnonlinear.
h/ Under a∈ppropr0iate assumptionsL, we prove that the initial valuFe pr·oblems (IVP)
t
a (1)0<ε<ε0 admit a solution on a time interval [0,T], with T >0 independent of ε.
m Such a result is known in the case of quasilinear symmetric hyperbolic systems,
: andprovidedthatthelinear(andsingular)part 1 ε(t)is,say,aconstantcoefficient
v εL
anti-adjoint differential operator (see e.g. [19] for the case of classical symmetric
i
X system, and [6] for an extension of these results).
r Inthequasilinearcaseforinstance,anessentialstepisthestudyoftheIVPassoci-
a
ated to the linearization of (1) around any reference function u belonging to some
functionalspace X: if a solutionv to this IVP canbe found in X,and if anenergy
estimate controls the norm of v in terms of the norm of u, then a solution to (1)
can be constructed by a standard Picard iterative scheme.
Our goal here is to investigate situations where this general approach fails. In
particular, it sometimes happens that the energy estimate associated to the lin-
earizedproblem only controls v in a space strictly larger than X; when such a loss
of information occurs, the standard Picard iterative scheme cannot converge. It is
howeverpossible,undercertainassumptions,tousethe iterativeschemedeveloped
by Nash and Moser and used for the first time to solve the embedding problem for
ThisworkwassupportedbytheACIJeunesChercheusesetJeunesChercheurs“Dispersionet
nonlin´earit´es”.
1
2 BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
Riemannian manifolds [15]. There exists now an extensive literature (e.g. [7, 1])
showing that the technique of Nash and Moser can be used to prove an abstract
implicit function theorem.
The implementation of a Nash-Moser iterative scheme is however very technical,
and is only used as a last recourse to solve nonlinear evolution equations, though
some recent works show that it is a useful tool (e.g. [16, 17, 14, 13, 10, 8]). We
develophereaNash-MosertheoremspecifictothegeneralclassofIVP(1),whichal-
lowsustogreatlysimplifythegeneraltheory(atthecost,sometimes,ofoptimality
– see also [18] for a simplified generalNash-Moser implicit function theorem). The
interest of these simplifications is twofold: i) we can state a general well-posedness
theorem for (1) under three assumptions easy to check on ε(t), ε and the lin-
L F
earizationof (1); ii) we can also handle the presence in the equation of parameters
andsingularterms. Wealsoshowhowtheseresultscanbeusedforthejustification
of asymptotic systems.
As an illustration, we solve the Serre and Green-Naghdi equations which are two
ofthe mostwidely usedmodels incoastaloceanography([4,5,3]and,forinstance,
[20,9]). Wealsoaddresstheproblemoftherelevanceofthesemodelsasasymptotic
models for the exact water-wavesequations.
1.2. Organization of the paper. We start by giving the three assumptions of
our general well-posedness theorem for (1) in Section 1.4. Section 2 is devoted to
the main theorem: it is stated in Section 2.1 and proved in Sections 2.2 and 2.3.
In Section 3, we give some generalizations and a corollary of the theorem. The
three main assumptions are weakened in Section 3.1 where we allow a more com-
plex dependence of the energy estimate on time derivatives. In Section 3.2, some
useful and easy generalizations are given: a slight weakening of the three main as-
sumptions(3.2.2),thepossibilityofhandlingotherparametersthanε(3.2.1)andof
replacing the linearization of (1) by an approximate linearization (3.2.3). Finally,
a corollary is given in Section 3.3, which gives a stability property very useful for
the justification of asymptotics to (1).
An application of the main theorem is given in Section 4 where the Serre and
Green-Naghdiequationsaresolveduniformly withrespecttothe so-calledshallow-
ness parameter (Section 4.2). The results of Section 4 are then used in Section 4.3
to address the justification of the Serre and Green-Naghdi models as asymptotic
models for the full water-waves equations.
1.3. Notations. - We generically denote by C(λ ,λ ,...) a constant depending
1 2
on the parameters λ ,λ ,...; the dependence on the λ is always assumed to be
1 2 j
nondecreasing.
- If X and X are two Banach spaces, we denote by L(X ,X ) the set of all
1 2 1 2
continuous linear mappings defined on X and with values in X .
1 2
-IfX isaBanachspaceandT >0,thenX standsforC([0,T];X),andwedenote
T
by its canonical norm.
|·|XT
- If X and X are two Banach spaces and C([0,T];Cj(X ;X )), we denote
1 2 1 2
F ∈
by , and the first, second and j-th order derivatives of the mapping
u uu (j)
F F F
u [,u].
7→F ·
- If X and X are two Banach spaces and Cj([0,T];C(X ;X )), we denote
1 2 1 2
F ∈
by (j) the j-th order derivative of the mapping t [t, ].
-WFedenoteΛ:=(1 ∆)1/2andHs(Rd)(s R)the7→usFual·SobolevspaceHs(Rd)=
− ∈
A NASH-MOSER THEOREM FOR SINGULAR EVOLUTION EQUATIONS 3
u (Rd), u < , where u = Λsu . We keep this notation if u is a
′ Hs Hs L2
{vect∈orSor mat|rix| with∞co}efficients|in|Hs(R|d). |
- We use the condensed notation
(2) A =B + C
s s h sis>s
to say that A =B if s s and A =B +C if s>s.
s s s s s
≤
0 0
- By convention, we take =0 and =1.
j=1 j=1
X Y
1.4. Main assumptions. We state here three assumptions which imply the well-
posednessof(1). Thefirstonedealswiththelinearoperator ε,thesecondonewith
L
thenonlinearterm ε,andthelastonewiththewell-posednessofthelinearization
F
of (1). Throughout this article, we assume that (Xs)s R is a Banach scale in the
∈
following sense:
Definition 1. We say that a family of Banach spaces ((Xs), s)s R is a Banach
|·| ∈
scale if:
For all s s′, one has Xs′ Xs and s s′;
• ≤ ⊂ |·| ≤|·|
There exists a family of smoothing operators (θ 1) such that
θ
• S ≥
s<s′, u Xs′, (1 θ)us Cs,s′θs−s′ us′
∀ ∀ ∈ | −S | ≤ | |
and
s s′, u Xs, θu Xs′ and θus′ Cs,s′θs′−s us;
∀ ≤ ∀ ∈ S ∈ |S | ≤ | |
The norms satisfy a convexity property:
•
∀s≤s′′ ≤s′, ∀u∈Xs′, |u|s′′ ≤Cs,s′,s′′|u|µs|u|1s′−µ,
where µ is given by the relation µs+(1 µ)s =s .
′ ′′
−
The assumption made on the linear operator ε is the following:
L
Assumption 1. There exist T >0, s R and m 0 such that:
0
∈ ≥
(1) For all s s , one has ε C(R;L(Xs+m;Xs)) and ( ε()) is
bounded in≥C([00,T];L(Xs+Lm;X∈s)); L · 0<ε<ε0
(2) One can define an evolution operator Uε() C(R;L(Xs,Xs)) (s s ) as
0
· ∈ ≥
1
g Xs, Uε(t)g :=uε(t), where ∂ uε+ ε(t)uε =0, uε =g,
t
∀ ∈ εL |t=0
and (Uε()) is bounded in C([ T,T];L(Xs,Xs)).
· 0<ε<ε0 −
We can now state our assumption on the nonlinear operator ε:
F
Assumption 2. There exist m 0, T > 0, and s R such that for all s s ,
0 0
≥ ∈ ≥
C([0,T];C2(Xs+m,Xs)) and:
F ∈
(1) For all u Xs+m,
∈
sup ε[t,u] C(s,T, u )u ;
|F |s ≤ | |s0+m | |s+m
t [0,T]
∈
(2) For all u,v Xs+m one has
∈
sup ε[t,u]v C(s,T, u ) v + u v ;
|Fu |s ≤ | |s0+m | |s+m | |s+m| |s0+m
t [0,T]
∈ (cid:0) (cid:1)
4 BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
(3) For all u,v ,v Xs+m one has
1 2
∈
sup ε [t,u](v ,v ) C(s,T, u ) v v + v v
|Fuu 1 2 |s ≤ | |s0+m | 1|s+m| 2|s0+m | 1|s0+m| 2|s+m
t [0,T]
∈
(cid:0)
+u v v .
| |s+m| 1|s0+m| 2|s0+m
Remark 1. Theestimates ofthe assumptionareuniform(cid:1)withrespecttoε (0,ε )
0
∈
and called tame estimates after Hamilton [7]: the dependence of the r.h.s. on the
norms involving the index s is linear.
Before stating the assumption made on the linearizationof (1), let us define the
space Xs (j N) and Fs as
(j) ∈
j j
(3) X(sj) := Ck([0,T];Xs−km), |u|X(sj) := |(ε∂t)ku|XTs−km,
k=0 k=0
\ X
(4) Fs := C([0,T];Xs) Xs+m, (f,g)Fs := f Xs + g s+m
× | | | | T | |
and, for all (f,g) Fs and t [0,T],
∈ ∈
t
(5) s(t,f,g):= g + sup f(t ) dt.
s ′′ s ′
I | | Z0 0≤t′′≤t′| |
Assumption 3. Let s ,m and T be as in Assumption 2. There exist d ,d 0
such that for all s≥s00+m, uε ∈X(s1+)d1 and (fε,gε)∈Fs+d′1, the IVP 1 ′1 ≥
1
(6) ∂ vε+ ε(t)vε+ ε[t,uε]vε =fε, vε =gε,
t εL Fu |t=0
admits a unique solution vε C([0,T];Xs) for all ε (0,ε ), and
0
∈ ∈
|vε|XTs ≤ C(ε0,s,T,|uε|X(s10)+m+d1)
× Is+d′1(t,fε,gε)+|uε|Xs+d1Is0+m+d′1(t,fε,gε) .
(1)
(cid:0) (cid:1)
Remark 2. Theaboveenergyestimateexhibits alossofd derivativeswithrespect
1
to the reference state uε (and of d derivatives with respect to the source term
′1
and initial data) in the sense that a control of vε in Xs requires a control of uε in
T
Xs+d1. This loss of information makes a standard Picard iterative scheme useless
T
to find a solution to (1). However, since the energy estimate is tame, one can
perform a Nash-Moser type iterative scheme. The fact that the energy estimate is
also uniform with respect to ε (0,ε ) is essential to obtain an existence time T
0
∈
independent of ε.
2. A Nash-Moser type theorem
2.1. Statement of the theorem. We state here the main theorem of this article
(a generalization is also given in Theorem 1’ below). In the following statement,
we use the notations
δ :=max d ,d +m , q :=D m d and P :=δ+D √δ+ 2(δ+q) 2,
{ 1 ′1 } − − ′1 min q
(cid:0) p (cid:1)
and we also recall that Fs+P =C([0,T];Xs+P) Xs+P+m.
×
A NASH-MOSER THEOREM FOR SINGULAR EVOLUTION EQUATIONS 5
Theorem 1. Let T >0, s , m, d and d be such that Assumptions 1-3 are satis-
0 1 ′1
fied. LetalsoD >δ, P >P , s s +mand(hε,uε) beboundedinFs+P.
min ≥ 0 0 0<ε<ε0
Thenthereexists0<T T andauniquefamily(uε) boundedinC([0,T];Xs+D)
≤ 0<ε<ε0
and solving the IVPs (1) .
0<ε<ε0
2.2. Proof of the theorem. With the evolution operator Uε() defined in As-
·
sumption 1, one can define a nonlinear operator ε[t, ] as
G ·
t [ T,T], u Xs0+m, ε[t,u]:=Uε( t) ε[t,Uε(t)u].
∀ ∈ − ∀ ∈ G − F
The next lemma shows that one can reduce the study of (1) to the study of
∂ uε+ ε[t,uε]=Uε( t)hε
t
(7) G −
uε =uε
0
(cid:26) |t=0
and also states that ε has theesame proeperties as ε.
G e F
Lemma 1. i. If uε C([0,T];Xs) solves (7) then uε C([0,T];Xs) solves (1),
∈ ∈
where uε(t):=Uε(t)uε(t).
ii. Assumption 2 still holds if one replaces ε by ε.
e
F G
Proof. Assumption 1eshows that if uε C([0,T];Xs) then uε C([0,T];Xs).
∈ ∈
Remark now that if uε solves (7), then
1 e
∂ Uε(t)uε = ε(t)Uε(t)uε Uε(t) ε[t,uε]+hε;
t e −εL − G
since Uε(t) ε[t,u(cid:0)ε] = ε(cid:1)[t,Uε(t)uε], the first point of the lemma follows. The
G eF e e
second point is a direct consequence of Assumptions 1 and 2. (cid:3)
Defining the sepace Fs as in (4)eand Es as C([0,T];Xs) C1([0,T];Xs m) en-
−
∩
dowed with its canonical norm (which makes Es different from Xs ), we can use
(1)
Lemma 1, to check that finding a solution uε to (1) is equivalent to finding a root
uε of the equation Φε(uε)=0, where
Es Fs m
−
→
e Φε : eu 7→ (∂tu+Gε[·,u]−hε,u|t=0 −uε0),
:=Φ1(u)
e
for all s s +m and ε (0,ε ),|and wit{hzhε(t):=} Uε( t)hε(t).
0 0
≥ ∈ −
We seek a root uε to the equation Φε(uε)=0 as the limit of a Nash-Mosertype
iterative scheme, namely, e
(8) e uε =ueε +S vε,
k+1 k k k
with S := , for some θ >0 to be determined, and where vε solves
k Sθk k k
∂ vε+ ε[t,uε]vε = Φ (uε),
(9) t k Gu k k − 1 k
vε =uε uε .
(cid:26) k|t=0 0− k|t=0
The following lemma shows that the above IVP can be solved and that the knowl-
edge of uε thus determines vε.
k k
Lemma2. SupposethatAssumptions1-3aresatisfied, andlets s +m. Assume
0
also that uεk ∈Es+d1 and Φε(uεk)∈Fs+d′1. ≥
Then there exists a unique solution vε Es to (9) and
k ∈
|vkε|Es ≤C(ε0,s,T,|uεk|Es0+m+d1) |Φε(uεk)|Fs+d′1 +|Φε(uεk)|Fs0+m+d′1|uεk|Es+d1 .
(cid:0) (cid:1)
6 BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
Proof. FromAssumption 3, we know that there is a unique solutionwε ofthe IVP
k
∂ wε + 1 ε(t)wε + ε[t,Uε(t)uε]wε = Uε(t)Φ (uε),
t k εL k Fu k k − 1 k
wε =uε uε ;
(cid:26) k|t=0 0− k|t=0
as in the proof of Lemma 1, it is easy to check that vε :=Uε( t)wε solves (9).
k − k
SinceAssumption1implies that|Uε(·)uεk|X(r1) ≤C(ε0)|uεk|Er (r ≥s0+m),onecan
deduce from the estimate of Assumption 3 and Assumption 1 that
(10)
|vkε|XTr ≤C(ε0,r,T,|uεk|Es0+m+d1) |Φε(uεk)|Fr+d′1 +|Φε(uεk)|Fs0+m+d′1|uεk|Er+d1 ;
with r = s, this is the control we n(cid:0)eed on |vkε|XTs; to conclude the proof, we m(cid:1)ust
therefore show that the same bound holds for |∂tvkε|XTs−m. From the equation one
has ∂ vε = ε[t,uε]vε Φ (uε), so that using Lemma 1.ii, one gets
t k −Gu k k− 1 k
|∂tvkε|XTs−m ≤C(s,T,|uεk|XTs0+m) |vkε|XTs +|vkε|XTs0+m|uεk|XTs +|Φ1(uεk)|XTs−m.
and one can conclude with (10) (wi(cid:0)th r=s and r =s +m).(cid:1) (cid:3)
0
Letusnowstatethe threelemmaswhichformthe heartofthe proof,andwhose
proof is postponed to the next subsections for the sake of clarity.
Lemma 3. Let D m + d and s s + m. If, for some M > 0, one has
≥ ′1 ≥ 0
uε M (j =k,k+1), then
| j|Es+D ≤
|Φε(uεk+1)|Fs+d′1 ≤C(s,T,M) θkm+d′1−D+|vkε|Es+D |vkε|Es+D,
with C(s,T,M) independent of ε. (cid:0) (cid:1)
Lemma 4. Let D d and s s +m. If, for some M >0, one has uε
≥ 1 ≥ 0 | k+1|Es+D ≤
M, then
|vkε+1|Es ≤C(ε0,s,T,M)|Φε(uεk+1)|Fs+d′1,
with C(ε ,s,T,M) independent of ε.
0
Lemma 5. Let δ :=max d ,(d +m) , P D δ and s s +m. If, for some
{ 1 ′1 } ≥ ≥ ≥ 0
M >0, one has |uεk|Es+D ≤M and |(hε,uε0)|Fs+P−m ≤M, then
uε C(ε ,s,T,M)(1+θδ)(1+ uε ).
| k+1|Es+P ≤ 0 k | k|Es+P
If moreover uε M, then one also has
| k+1|Es+D ≤
|vkε+1|Es+P−δ ≤C(ε0,s,T,M)(1+|uεk+1|Es+P).
Wecannowproceedwiththeproofofthetheorem,whichisatypicalNash-Moser
iterativescheme: Lemmas3and4provideacontrolof vε intermsof vε ,
| k+1|Es | k|Es+D
thus exhibiting a loss of D derivatives but providing a rapid decay of vε ,
| k+1|Es
while Lemma5 controlthe growthof|vkε+1|Es+P−δ. A controlof|vkε+1|Es+D is then
recoveredby the interpolation formula
(11) |vkε+1|Es+D ≤Cst |vkε+1|µEs|vkε+1|1E−s+µP−δ,
with µ=1 D .
− P δ
Before enteri−ng the heart of the proof, let us define the sequence (θ ) used for
k k
the smoothing operators as θ =θr (k N), for some r >1 defined below.
k+1 k ∈
Remark 3. One has θk = θ0rk, so that (if r > 1), k Nθk−q =: θ converges if and
onlyifθ >1. Moreover,θ canbemadearbitrarilysm∈allprovidedthatθ ischosen
0 0
P
large enough.
A NASH-MOSER THEOREM FOR SINGULAR EVOLUTION EQUATIONS 7
We are now set to control the sequences (uεk)k N and (vkε)k N by induction. For
∈ ∈
some M >0 such that
(12) |(hε,uε0)|Fs+P−m ≤M and |uε0|Es+D ≤M/2,
we define the properties (i) -(iii) as
k k
(i) : uε θα;
• k | k|Es+P ≤ k
(ii) : uε M;
• k | k|Es+D ≤
• (iii)k: |vkε|Es+D ≤θk−q, with q =D−m−d′1 >0.
Proof of (i) -(iii) assuming (i) -(iii) . Sinceonehas uε M by(ii) ,
k+1 k+1 k k | k|Es+D ≤ k
|uεk|Es+P ≤θkα by (i)kand |(hε,uε0)|Fs+P−m ≤M by definition of M, one can apply
Lemma 5 to obtain
uε C(ε ,s,T,M)(1+θδ)(1+θα)
| k+1|Es+P ≤ 0 k k
= f(ε ,s,T,M,k)θα ,
0 k+1
with
f(ε0,s,T,M,k)=C(ε0,s,T,M)(1+θkδ)(1+θkα)θk−αr.
Assuming that
(13) δ α(r 1)<0,
− −
it follows from the explicit expression of f(ε ,s,T,M,k) that f(ε ,s,T,M,k) 1
0 0
for all k N provided that θ is chosen large enough. This proves (i) . ≤
0 k+1
∈
Recalling that uε = uε + S vε, one has uε = uε + k S vε, and thus
k+1 k k k k+1 0 j=0 j j
|uεk+1|Es+D ≤M/2+ kj=0θk−q. AsseeninRemark3,onethenPgets(ii)k+1provided
that θ is chosen large enough.
0
P
In order to prove (iii) , remark first that it follows from Lemmas 3 and 4 and
k+1
the choice of the sequence (θk)k N that
∈
(14) |vkε+1|Es ≤C(ε0,s,T,M)θk−+2q1/r.
We can also use the second assertion of Lemma 5 to obtain
(15) |vkε+1|Es+P−δ ≤C(ε0,s,T,M)(1+θkα+1).
It follows therefore from (11), (14) and (15) that
|vkε+1|Es+D ≤ C(ε0,s,T,M)θk−+2µ1q/r(1+θkα+1)1−µ
= g(ε0,s,T,M,k)θk−+q1,
with g(ε0,s,T,M,k) := C(ε0,s,T,M)θk−+2µ1q/r(1+θkα+1)1−µθkq+1. Choosing r such
that
2µq
(16) 1<r < ,
q+α(1 µ)
−
one gets that g(ε ,s,T,M,k) 1 for all k N, provided that θ is chosen large
0 0
≤ ∈
enough.
Itfollowsfromthelinesabovethatinordertocompletetheproofoftheheredityof
theinductionproperty,we justhavetotakeθ largeenough,andtoprovethatone
0
can choose α, r and P such that the conditions (13) and (16) are satisfied. This is
done in the following lemma:
Lemma 6. Let α=δ+ 2δ(δ+q); if P >δ+ D(√δ+ 2(δ+q))2, there exists
q
r>1 such that conditions (13) and (16) are satisfied.
p p
8 BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
Proof. Let us denote r := 2µq . Quite obviously, (13) and (16) are satisfied
q+α(1 µ)
−
with r =r ǫ (ǫ>0 small enough), provided that r 1>δ/α, that is,
− −
(2q+α)µ (q+α) δ
− > ,
q+α(1 µ) α
−
or equivalently, if
q(1 δ/α)
µ>1 − =:µ (α).
min
− 2q+α+δ
The value of α given in the statement of the lemma corresponds to the minimum
ofµ (α). One then computesthat µ (α)=1 q ,andthe lemma
min min −(√δ+√2(δ+q))2
then follows from the observation that µ > µ (α) is equivalent to P > δ +
min
D . (cid:3)
1−µmin
Proof of (i) -(iii) . We have to construct here the first term of the sequence uε in
0 0 0
such a way that (i) -(iii) and (12) are satisfied for some M > 0 and θ > 0. We
0 0 0
need the following lemma:
Lemma 7. For all s s +m and (hε,uε) Fs+P, there exists uε Es+P such
≥ 0 0 ∈ 0 ∈
that uε =uε and such that
0|t=0 0
uε C(s,T, (hε,uε) ) and uε C(s,T, (hε,uε) ),
| 0|Es+D ≤ | 0 |Fs+D | 0|Es+P ≤ | 0 |Fs+P
and
|Φε(uε0)|Fs+D+d′1 ≤TC(s,T,|(hε,uε0)|Fs+D+d′1+m).
Proof. Let us define uε C([0,T];Xs+P) as
0 ∈
t
uε(t)=uε+ hε(t) ε[t,uε] dt.
0 0 ′ −G ′ 0 ′
Z0
(cid:0) (cid:1)
From Lemma 1 and the definition of eEs, one gets for all r 0,
|·| ≥
(17) uε uε +C(s,T, uε ) hε + uε ;
| 0|Es+r ≤| 0|s+r | 0|s0+m | |XTs+r | 0|s+r+m
the estimates onuε givenin the lemma are thus(cid:0)a consequenceof (17),(cid:1)with r =D
0
and r =P.
By definition of Φε, one also has
Φε(uε) = ε[,uε] ε[,uε],0
0 G · 0 −G · 0
1
= (cid:0) ε[,uε+z(uε (cid:1)uε)](uε uε)dz,0 ,
Gu · 0 0− 0 0− 0
Z0
(cid:0) (cid:1)
so that one deduces from Assumptions 1 and 2 that
|Φε(uε0)|Fs+D+d′1 ≤C(s,T,|uε0|Xs+D+d′1+m,|uε0|s+D+d′1+m)|uε0−uε0|Xs+D+d′1+m,
T T
and the estimate on Φε(uε) of the lemma follows easily. (cid:3)
0
Thankstothelemma,takingM =M(s,T, (hε,uε) )largeenough,onegets
| 0 |Fs+D
uε M/2, which proves (ii) . Choosing θ = θ (s,T, (hε,uε) ) large
| 0|Es+D ≤ 0 0 0 | 0 |Fs+P
enough, one also gets (i) from Lemma 7. In order to prove (iii) , remark first
0 0
that Lemma 2 yields |v0ε|Es+D ≤ C(ε0,s,T,M)|Φε(uε0)|Fs+D+d′1(1+θ0α). It follows
therefore from the lemma that, taking a smaller T if necessary, (iii) is satisfied,
0
which ends the induction proof of properties (i) , (ii) and (iii) .
k k k
A NASH-MOSER THEOREM FOR SINGULAR EVOLUTION EQUATIONS 9
Theendoftheexistencepartoftheproofofthetheoremisnowstraightforward:
it follows from (i) , (ii) and (iii) that the series uε+ S vε convergesto some
k k k 0 k k k
uε Es+D and taking the limit k in Lemma 3 shows that Φε(uε)=0.
∈ →∞ P
In order to conclude the proof of the theorem, we must now prove that the
solution constructed above is unique. Assuming that uε,j Es+D (j = 1,2) are
∈
both solutions to (1), we show that w :=uε,2 uε,1 is identically 0. Let us remark
−
that w solves the IVP
∂ w+ 1 ε(t)w+ ε[t,uε,2]w =H,
t εL Fu
w =0,
(cid:26) |t=0
with H := ε[t,uε,1] ε[t,uε,2] ε[t,uε,2](uε,1 uε,2).
F −F −Fu −
A direct application of Assumption 3 yields
t
|w(t)|s0+m ≤C(ε0,T,|uε,1|X(s10)+m+δ)Z0 0≤stu′′p≤t′|H(t′′)|s0+m+d′1dt′,
and since |H(t′)|s0+m+d′1 ≤ C(s,T,|uε,2|XTs0+m+δ)|w(t′)|s0+m by Assumption 2(3),
a Gronwall argument shows that w=0.
2.3. Proof of Lemmas 3, 4, 5.
2.3.1. Proof of Lemma 3. In order to give an upper bound for |Φε(uεk+1)|Fs+d′1, we
nFeiresdttroemcoanrktrothlaΦt1a(usεke+c1o)ndinoXrdTse+rdT′1aaynldor|uexεk+pa1n|ts=i0on−oufε0Φ|s+(du′1ε+m.) yields
1 k+1
Φ (uε ) = Φ (uε)+Φ (uε)(uε uε)
1 k+1 1 k ′1 k k+1− k
1
+ (1 z)Φ (uε +z(uε uε))(uε uε,uε uε)dz.
− ′1′ k k+1− k k+1− k k+1− k
Z0
Since by (8), one has uε uε = S vε, and since by definition Φ (uε) = ∂ uε +
k+1− k k k 1 k t k
ε[,uε] hε, it follows that
G · k −
(18) Φ (uε )=E +E ,
e 1 k+1 1 2
with
E = Φ (uε)+∂ vε + ε[,uε]vε
1 1 k t k Gu · k k
1
+ (1 z) ε [,uε +z(uε uε)](S vε,S vε)dz
− Guu · k k+1− k k k k k
Z0
1
(19) = (1 z) ε [,uε +z(uε uε)](S vε,S vε)dz
− Guu · k k+1− k k k k k
Z0
(the last equality stemming from the fact that vε solves (9)), and
k
(20) E =(S 1)∂ vε + ε[,uε] (S 1)vε .
2 k− t k Gu · k k− k
Since by Lemma 1.ii, Gε satisfies Assumption 2(3(cid:0)), and since(cid:1)s+d′1+m ≤s+D,
one can control E as
1
|E1|XTs+d′1 ≤ C(s,T,|uεk|XTs+D,|uεk+1|XTs+D)|Skvkε|2XTs+D
(21) C(s,T,M)vε 2 .
≤ | k|XTs+D
10 BORYSALVAREZ-SAMANIEGOANDDAVIDLANNES
Since Lemma 1.ii also ensures that Gε satisfies Assumption 2(2), one gets
|Guε[·,uεk] (Sk−1)vkε |XTs+d′1 ≤C(s,T,M)ts[u0p,T]|(Sk−1)vkε(t)|s+m+d′1.
(cid:0) (cid:1) ∈
It is then a consequence of the properties of the regularizing operators (recall that
S = ), that
k Sθk
|E2|XTs+d′1 ≤ Cst θkm+d′1−D |∂tvkε|XTs+D−m +C(s,T,M)|vkε|XTs+D
(22) ≤ C(s,T,M)θkm(cid:0)+d′1−D|vkε|Es+D. (cid:1)
It is then a simple consequence of (18), (21) and (22) to conclude that
(23) |Φ1(uεk+1)|Xs+d′1 ≤C(s,T,M) θkm+d′1−D+|vkε|Es+D |vkε|Es+D.
T
(cid:0) (cid:1)
Wenowturntocontrol|uεk+1|t=0−uε0|s+d′1+m. Sinceuεk+1|t=0−uε0 =(Sk−1)vkε|t=0,
one gets
|uεk+1|t=0 −uε0|s+d′1+m ≤ Cst θkm+d′1−Dts[u0p,T]|vkε(t)|s+D
∈
(24) ≤ Cst θkm+d′1−D|vkε|Es+D.
The lemma follows directly from (23) and (24).
2.3.2. Proof of Lemma 4. Since uε M, one gets therefore from Lemma
| k+1|Es+d1 ≤
2 (at step k+1),
(25) |vkε+1|Es ≤C(ε0,s,T,M)|Φε(uεk+1)|Fs+d′1,
and the lemma is proved.
2.3.3. Proof of Lemma 5. Thanks to Lemma 1.ii, one has, for all r s ,
0
≥
(26) Φε(u) C(u )u +Cst (hε,uε) ;
| |Fr ≤ | |Es0+m | |Er+m | 0 |Fr
remark also that since uε = uε +S vε, one can use the properties of the regu-
k+1 k k k
larizing operator S = to obtain
k Sθk
(27) |uεk+1|Es+P ≤|uεk|Es+P +Cst θkδ|vkε|Es+P−δ.
From Lemma 2, one deduces
|vkε|Es+P−δ ≤C(ε0,s,T,M) |Φε(uεk)|Fs0+m+d′1|uεk|Es+P +|Φε(uεk)|Fs+P−m
(cid:16) (cid:17)
so that, using (26) with r = s +m+d and r = s+P m, and the assumption
0 ′1 −
made on (hε,uε), one obtains
0
(28) |vkε|Es+P−δ ≤C(ε0,s,T,M)(1+|uεk|Es+P).
Together with (27), this last estimate shows that
uε C(ε ,s,T,M)(1+θδ)(1+ uε ),
| k+1|Es+P ≤ 0 k | k|Es+P
so that the proof of the first assertion is complete.
The last part of the lemma is exactly (28) with the index k replaced by k+1.