Uniformly accurate time-splitting methods for the semiclassical Schr\"odinger equationPart 2 : Numerical analysis of the linear case

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by  Philippe Chartier

UNIFORMLY ACCURATE TIME-SPLITTING METHODS FOR
THE SEMICLASSICAL SCHRÖDINGER EQUATION
PART 2 : NUMERICAL ANALYSIS OF THE LINEAR CASE
PHILIPPE CHARTIER,LOÏCLE TREUST, ANDFLORIANMÉHATS
6 Abstract. This article is second part of a twofold paper, devoted to the con-
1
struction of numerical methodswhich remain insensitiveto thesmallness of the
0
semiclassical parameter for the Schrödinger equation in the semiclassical limit.
2
Here,wespecificallyanalysetheconvergencebehaviorofthefirst-ordersplitting

n introduced in Part I, for a linear equation with smooth potential. Our main
a result is a proof of uniform accuracy.
J

9
1

]
P 1. Introduction
A
This paper is the follow-up of a first part which introduces high-order uniformly
.
h
accurate schemes in the non-linear case. We are concerned here with the numerical
t
a approximation of the solution Ψε : R ×Rd → C, d ≥ 1, of the linear Schrödinger
+
m
equation in its semiclassical limit
[
ε2
iε∂ Ψε = − ∆Ψε+VΨε (1.1)
1 t
2
v
5 where V is a smooth potential. The initial datum is assumed to be of the form
2 Ψε(0,·) = A (·)eiS0(·)/ε with kA k = 1. (1.2)
8 0 0 L2(Rd)
4
AsdescribedinPartI,theproblemisreformulatedaccordingtothestrategyadopted
0
in [2]. This is achieved by decomposing Ψε as the product of a slowly varying
.
1
amplitude and a fast oscillating factor
0
6 Ψε(t,·) = Aε(t,·)eiSε(t,·)/ε, (1.3)
1
: where (Sε,Aε) satisfies
v
i |∇Sε|2
X ∂ Sε+ +V = ε2∆Sε, (1.4a)
t
2
r
a Aε iε∆Aε
∂ Aε+∇Sε·∇Aε+ ∆Sε = −iεAε∆Sε (1.4b)
t
2 2
with Sε(0,x) = S (x), Aε(0,x) = A (x) and x ∈ Rd. Recall that system (1.4) is
0 0
equivalent to the original equation (1.1) (see Part I). The existence and uniqueness
of the solution of equation (1.1) is proved for instance in [4]. The corresponding
result for equations 1.4 will be derived in Section 3.4.
1991 Mathematics Subject Classification. 35Q55,35F21,65M99, 76A02,76Y05,81Q20,82D50.
Key words and phrases. Schrödinger equation, semiclassical limit, numerical simulation, uni-
formly accurate, Madelung transform, splitting schemes.
ThisworkwassupportedbytheANR-FWFProjectLodiquasANR-11-IS01-0003andtheANR-
10-BLAN-0101 Grant (L.L.T.) and by theANR project Moonrise ANR-14-CE23-0007-01.
1
2 P.CHARTIER,L.LETREUST,ANDF.MÉHATS
In this second part, we concentrate on the numerical analysis of the first-order
splitting scheme introduced in the first part, which, we believe, is of interest for
its own sake. For the sake of clarity, we now recall it in this specific case (linear).
System (1.4) is split into four pieces as follows:
First flow: We denote ϕ1 the approximate flow at time h ∈ R of the system
h
|∇S|2
∂ S + = 0, (1.5a)
t
2
A i∆A
∂ A+∇S ·∇A+ ∆S = . (1.5b)
t
2 2
The eikonal equation (1.5a) is solved by means of the method of characteristics,
while equation (1.5b) is dealt with by noticing that w = Aexp(iS) satisfies the free
Schrödinger equation i∂ w =−1∆w.
t 2
Second flow: We define ϕ2 as the exact flow at time h ∈ R of the system
h
∂ S = 0, (1.6a)
t
i(ε−1)∆A
∂ A = , (1.6b)
t
2
which is solved in the Fourier space.
Third flow: The third flow ϕ3 is defined as the exact flow at time h ∈ R of system
h
∂ S = −V, (1.7a)
t
∂ A= 0. (1.7b)
t
Fourth flow: The fourth flow ϕ4 is defined as the exact flow at time h ∈ R of
h +
∂ S = ε2∆S, (1.8a)
t
∂ A= −iεA∆S. (1.8b)
t
Equation (1.8a) is solved in the Fourier space and the solution of (1.8b) is simply
obtained through the formula A(h,·) = exp −iε−1(S(h,·)−S(0,·)) A(0,·). No-
tice that ϕ4 can thus be viewed as a regularizing flow.
h (cid:0) (cid:1)
The first-order scheme that we consider for (1.4) is then the concatenation of all
previous flows
ϕ1 ◦ϕ2 ◦ϕ3 ◦ϕ4. (1.9)
h h h h
Themainresultofthispartisthefollowingtheorem: itstatesthatϕ1◦ϕ2◦ϕ3◦ϕ4
h h h h
is uniformly accurate w.r.t. the semi-classical parameter ε. The proper statement
of the result uses the norm k·k on the set Σ = Hs+2(Rd)×Hs(Rd) defined for
s s
s ≥ 0 and u = (S,A) by
1/2
kuk = kSk2 +kAk2 .
s Hs+2(Rd) Hs(Rd)
(cid:16) (cid:17)
Theorem 1.1. Let s > d/2+1, ε > 0, u ∈Σ and 0 < T < T where
max 0 s+2 max
T = sup{t > 0 : τ 7→ φ0(u ) ∈ L∞([0,t];Σ )}
max τ 0 s+2
3
and φε denotes the flow at time τ of (1.4). There exists C > 0 and h > 0 such
τ 0
that the following error estimate holds true for any ε ∈ (0,ε ], any h ∈ [0,h ]
max 0
and n ∈ N satisfying nh≤ T:
k(ϕ1 ◦ϕ2 ◦ϕ3 ◦ϕ4)n(u )−φε (u )k ≤ Ch.
h h h h 0 nh 0 s
The constants C and h do not depend on ε.
0
Remark 1.2. The constant T appearing in Theorem 1.1 is well-defined and
max
positive (see Theorem 2.1).
Our proof is reminiscent of two previous results related to, on the one hand,
splitting schemes forequations with Burgers nonlinearity [5]and on the other hand,
splitting scheme for NLS in the semiclassical limit with [3]. Nonetheless, due to the
finite-time existence of both exact and approximate flows, and to the peculiarity of
the Lipschitz-type stability of the exact flows (see Lemma 2.3), our proof follows a
different path. In particular, we lean the approximate solutions on the exact one to
ensure that they do not blow up. Besides, the application of Lady Windermere’s
fan argument is somehow hidden in an induction procedure. Finally, let us mention
that, in spite of the fact that we do not specifically address this case, it is our belief
that this result can be extended to the Schrödinger equation with a nonlinearity of
Hartree-type (see also [3, Remark 4.5]).
2. Numerical study of the scheme
2.1. Notations. Assume that ε ∈ (0,ε ] and s > d/2 + 1. For the sake of
max
simplicity, we keep the notation of all the flows independent of ε. All the constants
appearing in the proof depend on V but not on ε > 0. We denote
ϕij = ϕi ◦ϕj, ϕijk = ϕi ◦ϕjk, ϕ1234 = ϕ1 ◦ϕ234,
h h h h h h h h h
N is the possibly nonlinear operator related to ϕi. The quantities ∂ ϕ (u) and
i h h h
∂ ϕ (u) are the Fréchet derivatives of ϕ with respect to h and u. The commutator
2 h
of the nonlinear operators N and N is given by
i j
[N ,N ](u) = DN (u)·N (u)−DN (u)·N (u).
i j i j j i
2.2. Existence, uniqueness and uniformboundedness results. Thefollowing
theorem study some properties of the solutions of equations (1.4).
Theorem 2.1. Let ε > 0, s > d/2+1 and u ∈ Σ . The following two points
max 0 s+2
are true.
(i) The quantity
T = sup{t > 0 :φ0(u )∈ L∞([0,t];Σ )} (2.1)
max 0 s+2
is well-defined and positive.
(ii) Let 0 < T < T . For all ε ∈ [0,ε ], there exists a unique solution
max max
φε(u ) ∈ C([0,T],Σ )
0 s+2
of system of equations (1.4). Moreover, φε(u ) is bounded in
0
C([0,T],Σ )
s+2
uniformly in ε ∈ [0,ε ].
max
The proof of Theorem 2.1 is given in Section 3.4.
4 P.CHARTIER,L.LETREUST,ANDF.MÉHATS
2.3. Themainlemmas. Inthissubsection,wepresentthemainingredientsneeded
in the proof of Theorem 1.1. Their proof is postponed to Section 3.
Lemma 2.2. Let M > 0 and s > d/2+1. There exist h = h (M) > 0 such that
1 1
for any ε ∈ (0,ε ] and any u ∈ Σ satisfying
max 0 s
ku k ≤ M,
0 s
we have that the solution φ (u ) of equation (1.4) is well-defined on [0,h ] and for
t 0 1
all t ∈ [0,h ]
1
kφ (u )k ≤ 2M.
t 0 s
Lemma 2.3. Let M > 0 and s > d/2 + 1. There exist C = C (M) > 0 such
2 2
∞
that for any ε ∈ (0,ε ], any solutions φ (u ) ∈ L ([0,T],Σ ) and φ (u ) ∈
max t 1 s+1 t 2
∞
L ([0,T],Σ ) of equation (1.4), satisfying for all t ∈ [0,T]
s
kφ (u )k +kφ (u )k ≤ M
t 1 s+1 t 2 s
we have
kφ (u )−φ (u )k ≤ ku −u k exp(C t).
t 1 t 2 s 1 2 s 2
Remark 2.4. Let us insist on the fact that in Lemma 2.3, we have to control φ (u )
t 1
in Σ and φ (u ) in Σ to get Lipschitz-type stability in Σ .
s+1 t 2 s s
Lemma 2.5. Let M > 0 and s > d/2 + 1. There exist h = h (M) > 0 and
3 3
C = C (M) > 0 such that for any ε∈ (0,ε ], any u ∈Σ satisfying ku k ≤ M
3 3 max 0 s 0 s
and any 0 ≤ t ≤ h , we have
3
(a) kϕ1234(u )k ≤ 8M.
t 0 s
(b) Furthermore, if u ∈ Σ , then
0 s+2
kϕ1234(u )k ≤ exp(C t)(ku k +tkVk ).
t 0 s+2 3 0 s+2 Hs+4
Lemma 2.6. Let M > 0 and s > d/2 + 1. There exist h = h (M) > 0 and
4 4
K = K (M) > 0 such that for any ε ∈(0,ε ] and any u ∈ Σ satisfying
4 4 max 0 s+2
ku k ≤ M,
0 s+2
we have for any t ∈ [0,h ] that
4
kφ (u )−ϕ1234(u )k ≤ K t2.
t 0 t 0 s 4
2.4. Proof of Theorem 1.1. Let us denote
Mε(T):= sup{kφε(u )k : 0≤ t ≤ T}. (2.2)
s t 0 s
for ε ≥ 0 and T ≥ 0.
Let s > d/2 + 1, ε ∈ (0,ε ], u ∈ Σ , n ∈ N and h > 0 be such that
max 0 s+2
nh ≤ T < T (see (2.1)). By Theorem 2.1, there exist M , M and M
max s s+1 s+2
independent of ε ∈ (0,ε ] such that for all ε ∈ (0,ε ],
max max
Mε ≤ M , Mε ≤ M and Mε ≤ M
s s s+1 s+1 s+2 s+2
5
(see (2.2)). We denote
t
a = sup , t ≥ 0 ,
et−1
(cid:26) (cid:27)
C = C (2M ),
3 s
c = ku k exp(CT)+akVk e2TC/C,
0 0 s+2 Hs+4
′
C = C (M +4M ),
2 s+1 s
c = K (c )aeC′T/C′.
4 0
Assume that
e
0 ≤ h≤ min(h (c ),M /c,h (2M ),h (c )). (2.3)
3 0 s 1 s 4 0
Here, h , C , h , h and K are defined in Lemmas 2.2, 2.3, 2.5 and 2.6.
1 2 3 4 4
We show by induction on 0 ≤ k ≤ n that e
(i) (ϕ1234)k(u ) is well-defined, belongs to Σ and
h 0 s+2
e(k+1)hC −ehC
k(ϕ1234)k(u )k ≤ ku k exp(Ckh)+hkVk ≤ c ,
h 0 s+2 0 s+2 Hs+4 ehC −1 0
′
(ii) kφ (u )−(ϕ1234)k(u )k ≤ h2K (c )eC hk−1 ≤ ch,
kh 0 h 0 s 4 0 eC′h−1
and Theorem 1.1 follows then from point (ii) with k = n.
The induction hypothesis are true for k = 0. Let eus assume points (i) and (ii)
true for 0 ≤ k ≤ n−1.
Lemma 2.5, point (i) and (2.3) ensure that
(ϕ1234)k+1(u )
h 0
is well-defined and belongs to Σ . By Point (ii) and (2.3), we have
s+2
k(ϕ1234)k(u )k ≤ M +kφ (u )−(ϕ1234)k(u )k ≤ 2M .
h 0 s s kh 0 h 0 s s
By Lemma 2.5 and (2.3), we have
k(ϕ1234)k+1k ≤ exp(Ch) k(ϕ1234)k(u )k +hkVk
h s+2 h 0 s+2 Hs+4
(cid:16) (cid:17)
and point (i) ensures that
e(k+2)hC −ehC
k(ϕ1234)k+1k ≤ ku k exp(C(k+1)h)+hkVk ≤ c .
h s+2 0 s+2 Hs+4 ehC −1 0
By Lemma 2.2 and (2.3), h′ 7→ φh′ ◦(ϕ1h234)k(u0) is well-defined and satisfies for all
′
0 ≤ h ≤ h
kφh′ ◦(ϕ1h234)k(u0)ks ≤ 4Ms.
By Lemma 2.3, we obtain that
kφ (u )−φ ◦(ϕ1234)k(u )k ≤ kφ (u )−(ϕ1234)k(u )k exp(C′h).
h(k+1) 0 h h 0 s hk 0 h 0 s
By Lemma 2.6, point (i) and (2.3), we get
kφ ◦(ϕ1234)k(u )−ϕ1234 ◦(ϕ1234)k(u )k ≤ K (c )h2,
h h 0 h h 0 s 4 0
so that
kφ (u )−(ϕ1234)k+1(u )k ≤ K (c )h2 +kφ (u )−(ϕ1234)k(u )k exp(C′h).
h(k+1) 0 h 0 s 4 0 hk 0 h 0 s
6 P.CHARTIER,L.LETREUST,ANDF.MÉHATS
By point (ii), we have then that
eC′h(k+1)−1
kφ (u )−(ϕ1234)k+1(u )k ≤ K (c )h2 .
h(k+1) 0 h 0 s 4 0 eC′h−1
!
Thus, points (i) and (ii) are true for k+1.
3. Proof of the main lemmas
3.1. Auxiliary results. Let us denote by h·,·i the L2 scalar product, for s> 0
Λs = (1−∆)s/2,
S
Π u= S, Π u= A, for u = (3.1)
1 2 A
(cid:18) (cid:19)
and
hu ,u i = hΠ u ,Π u i
1 2 s 1 1 1 2
(3.2)
+ Λs+1∇Π u ,Λs+1∇Π u +RehΛsΠ u ,ΛsΠ u i.
1 1 1 2 2 1 2 2
We recall two poi(cid:10)nts that will be of consta(cid:11)nt use in the following: the Sobolev
space Hs ⊂ L∞ is an algebra for s > d/2 and the Kato-Ponce [6] inequality holds
true:
Proposition 3.1. Let s > d/2+1. There is c > 0 such that for all f ∈ Hs0(Rd)
0
and g ∈ Hs0−1(Rd)
kΛs0(fg)−fΛs0gkL2 ≤ c(k∇fkL∞kgkHs0−1 +kfkHs0kgkL∞).
The following lemmas will be used several times in our proof.
Lemma 3.2. Let s > d/2 + 1. There is C > 0 such that for all v , v and
0 0 1
R ∈ L∞([0,h ],Hs0(Rd)d) satisfying
0
∂ v +(v ·∇)v = R,
t 0 1 0
we have
∂tkv0k2Hs0 ≤ C kv0k2Hs0k∇v1kL∞ +kv0kHs0kv1kHs0k∇v0kL∞ +hΛs0v0,Λs0Ri
≤ Ck(cid:0)v0k2Hs0kv1kHs0 +hΛs0v0,Λs0Ri. (cid:1)
Proof. We have by integration by parts that
kv k2
∂ 0 Hs0 = hΛs0v ,Λs0∂ v i= −hΛs0v ,Λs0(v ·∇)v i+hΛs0v ,Λs0Ri
t 0 t 0 0 1 0 0
2
1
≤ 2 Rd|Λs0v0|2divv1+kv0kHs0k[Λs0,(v1 ·∇)]v0kL2 +hΛs0v0,Λs0Ri.
Z
Proposition 3.1 ensures that
kv k2
∂t 02Hs0 ≤ c kv0k2Hs0k∇v1kL∞ +kv0kHs0kv1kHs0k∇v0kL∞ +hΛs0v0,Λs0Ri.
(cid:0) (cid:1) (cid:3)
7
Lemma 3.3. Let s > d/2 + 1. There exists C > 0 such that for all v ∈
0 1
L∞([0,h ],Hs0+1(Rd)d) and R ∈ L∞([0,h ],Hs0(Rd)) satisfying
0 0
divv
1
∂ A+v ·∇A+A = R,
t 1
2
we have,
∂tkAk2Hs0 ≤C kAk2Hs0kv1kW2,∞ +kAkHs0kv1kHs0+1kAkW1,∞ +RehΛs0A,Λs0Ri
≤Ck(cid:0)Ak2Hs0kv1kHs0+1 +RehΛs0A,Λs0Ri. (cid:1)
Proof. We have by integration by parts that
kAk2 divv
∂ Hs0 = RehΛs0A,Λs0∂ Ai= −Re Λs0A, v ·∇+ 1 Λs0A
t t 1
2 2
(cid:28) (cid:18) (cid:19) (cid:29)
divv
−Re Λs0A, Λs0, v ·∇+ 1 A +RehΛs0A,Λs0Ri
1
2
(cid:28) (cid:20) (cid:18) (cid:19)(cid:21) (cid:29)
divv
≤ kAkHs0 Λs0, v1·∇+ 1 A +RehΛs0A,Λs0Ri.
2
(cid:13)(cid:20) (cid:18) (cid:19)(cid:21) (cid:13)L2
(cid:13) (cid:13)
Proposition 3.1 ensures th(cid:13)at (cid:13)
(cid:13) (cid:13)
kAk2
∂t 2Hs0 ≤ CkAkHs0(k∇v1kL∞k∇AkHs0−1 +kv1kHs0k∇AkL∞)
+CkAkHs0(k∇(divv1)kL∞kAkHs0−1 +kdivv1kHs0kAkL∞)
+RehΛs0A,Λs0Ri
≤ C kAk2Hs0kv1kW2,∞ +kAkHs0kv1kHs0+1kAkW1,∞ +RehΛs0A,Λs0Ri
≤ Ck(cid:0)Ak2Hs0kv1kHs0+1 +RehΛs0A,Λs0Ri. (cid:1)
(cid:3)
3.2. Study of the equation (1.4). Let us prove Lemma 2.2.
Proof. By the Cole-Hopf transform, we get that wε = exp −Sε −1 is the solution
2ε2
of
(cid:0) (cid:1)
V S
∂ wε = ε2∆wε+ (wε+1), wε(0) = exp − 0 −1,
t 2ε2 2ε2
(cid:18) (cid:19)
Hence, global existence and uniqueness of the solution Sε of (1.4a) for fixed ε ∈
(0,ε ], follows from standard semi-group theory. The function vε = ∇Sε solves
max
∂ vε +(vε·∇)vε+∇V = ε2∆vε.
t
Since s > d/2, Lemma 3.2 and an integration by parts ensure that
∂ kvεk2 ≤ ckvεk3 + Λs+1vε,Λs+1 −∇V +ε2∆vε
t Hs+1 Hs+1
≤ ckvεk3Hs+1 +k(cid:10)vεkHs+1kVkH(cid:0)s+2. (cid:1)(cid:11)
By (1.4a), we also have that
kSεk2
∂ L2 ≤ kSεk kVk +kvεk2 /2
t 2 L2 L2 L4
so that (cid:0) (cid:1)
∂ kSεk2 ≤ ckVk kSεk +ckSεk3 .
t Hs+2 Hs+2 Hs+2 Hs+2
8 P.CHARTIER,L.LETREUST,ANDF.MÉHATS
The global existence and the uniqueness of a solution Aε of equation (1.4b) follows
from the fact that
Ψε = Aεexp(iSε/ε)
satisfies equation (1.1). By Lemma 3.3, recalling that s > d/2+1, we also have
∂ kAεk2 ≤ ckAεk2 kSεk +RehΛsAε,ΛsRi.
t Hs Hs Hs+2
where R = iε∆Aε −iεAε∆Sε so that an integration by parts gives us
2
∂ kAεk2 ≤ckAεk2 kSεk .
t Hs Hs Hs+2
We obtain that
∂ kφ (u )k2 ≤ c kφ (u )k kVk +c kφ (u )k3
t t 0 s 1 t 0 s Hs+2 2 t 0 s
and
∂ kφ (u )k ≤ c kVk +c kφ (u )k2.
t t 0 s 1 Hs+2 2 t 0 s
We get then that
c kVk c
kφ (u )k ≤ 1 Hs+2 tan t c c kVk +arctan M 2
t 0 s s c2 (cid:18) 1 2 Hs+2 (cid:18) rc1kVkHs+2(cid:19)(cid:19)
p
so that there is h = h (M) > 0 such that for all 0 ≤ t ≤ h
1 1 1
kφ (u )k ≤ 2M.
t 0 s
(cid:3)
The following result will be used several times and in particular for the proof of
the stability of equation (1.4) in Lemma 2.3.
Lemma 3.4. Let s > d/2+1. Let u = (S ,A ) be in L∞([0,T],Σ ), u =
0 1 1 1 s0+1 2
∞
(S ,A ), (R ,R ) and (R ,R ) be in L ([0,T],Σ ). Assume moreover that
2 2 1,S 1,A 2,S 2,A s0
for i = 1,2
|∇S |2
i
∂ S + = R ,
t i i,S
2
∆S
i
∂ A +∇S ·∇A +A = R .
t i i i i i,A
2
Then, we have
∂ ku −u k2 ≤ cku −u k2 (ku k +ku k )+hu −u ,R −R i
t 1 2 s0 1 2 s0 1 s0+1 2 s0 1 2 1 2 s0
where R = (R ,R )T.
i i,S i,A
Proof. Lets > d/2+1. Letusdefinev = ∇S ,v =∇S ,w = v −v ,B = A −A
0 1 1 2 2 1 2 1 2
and u= u −u .
1 2
We have that
∂ w = −(v ·∇)v +(v ·∇)v +∇(R −R )
t 1 1 2 2 1,S 2,S
= −(v ·∇)w−(w·∇)v +∇(R −R )
2 1 1,S 2,S
and Lemma 3.2 ensures that
∂ kwk2 ≤ ckwk2 kv k + Λs0+1w,Λs0+1R .
t Hs0+1 Hs0+1 2 Hs0+1
where R = −(w·∇)v1+∇(R1,S −R2,S). We also(cid:10)have that (cid:11)
kRk ≤ ckwk kv k + Λs0+1w,Λs0+1∇(R −R )
Hs0+1 Hs0+1 1 Hs0+2 1,S 2,S
(cid:10) (cid:11)
9
and
∂ kwk2 ≤ ckwk2 (kS k +kS k )+ Λs0+1w,Λs0+1∇(R −R ) .
t Hs0+1 Hs0+1 1 Hs0+3 2 Hs0+2 1,S 2,S
We also have (cid:10) (cid:11)
1
∂ (S −S ) = − (v +v )·w+(R −R )
t 1 2 1 2 1,S 2,S
2
so that
∂tkS1−S2k2L2 ≤ ckS1−S2kL2kwkL2(kS1kW1,∞ +kS2kW1,∞)+hS1−S2,R1,S −R2,Si
and then
∂ kS −S k2 ≤ CkS −S k2 (kS k +kS k )
t 1 2 Hs0+2 1 2 Hs0+2 1 Hs0+3 2 Hs0+2
+hS −S ,R −R i+ Λs0+1∇(S −S ),Λs0+1∇(R −R )
1 2 1,S 2,S 1 2 1,S 2,S
Let us study B, we have (cid:10) (cid:11)
∆S
2
∂ B +∇S ·∇B+ B = R
t 2
2
where
div(w)
R = −w·∇A − A +(R −R )
1 1 1,A 2,A
2
Hence, we obtain by Lemma 3.3
∂ kBk2 ≤ ckBk2 kS k +RehΛs0B,Λs0Ri
t Hs0 Hs0 2 Hs0+2
≤ ckBk2Hs0kS2kHs0+2 +ckBkHs0kwkHs0+1kA1kHs0+1
+RehΛs0B,Λs0(R −R )i
1,A 2,A
and
∂ ku −u k2 ≤ cku −u k2 (ku k +ku k )+hu −u ,R −R i
t 1 2 s0 1 2 s0 1 s0+1 2 s0 1 2 1 2 s0
The result follows. (cid:3)
Let us study now the stability of equation (1.4) and prove Lemma 2.3.
Proof. Let s > d/2+1 and ε∈ (0,ε ]. Let us define for i = 1,2
max
R = −V +ε2∆S ,
i,S i
∆A
i
R = iε −iεA ∆S .
i,A i i
2
We apply Lemma 3.4 with s = s. We have by integrations by parts that
0
hS −S ,R −R i+ Λs+1∇(S s−S ),Λs+1∇(R −R ) ≤ 0,
1 2 1,S 2,S 1 2 1,S 2,S
and (cid:10) (cid:11)
RehΛs(A −A ),Λs(R −R )i ≤ ckA −A k2 kS k
1 2 1,A 2,A 1 2 Hs 1 Hs+2
+ckA1−A2kHskS1−S2kHs+2kA2kHs.
so that
∂ kφ (u )−φ (u )k2 ≤ cku −u k2(kφ (u )k +kφ (u )k )
t t 1 t 2 s 1 2 s t 1 s+1 t 2 s
and the result follows. (cid:3)
10 P.CHARTIER,L.LETREUST,ANDF.MÉHATS
3.3. Study of the numerical flow ϕ1234. The following lemma is inspired by the
work of Holden, Lubich and Risebro [5].
Lemma 3.5. Let s > d/2+1 and M > 0. There exists h = h (M) > 0 such
0 5 5
that for any u ∈ Σ satisfying ku k ≤ M and any 0 ≤ t ≤ h , the following two
0 s0 0 s0 5
points are true.
(i) We have that kϕ1(u )k ≤ 2M.
t 0 s0
(ii) Let s ≥ s . There is C = C (M) > 0 such that if u ∈ Σ , then
1 0 5 5 0 s1
kϕ1(u )k ≤ exp(C t)ku k .
t 0 s1 5 0 s1
Proof. TheexistenceofthesolutionS of (1.5a)followsforinstancefromthemethod
of characteristics. Lemma 3.2 ensures that for s > d/2+1
∂tk∇Sk2Hs+1 ≤ ck∇Sk2Hs+1k∇(∇S)kL∞ ≤ Ck∇Sk2Hs+1kS(t)kW2,∞.
We also have
∂ kSk2 ≤ckS(t)k k∇S(t)k2
t L2 L2 L4
so that
∂tkSk2Hs+2 ≤ CkS(t)k2Hs+2kS(t)kW2,∞.
The remaining of the proof follows exactly the same lines as the one of Lemma 2.2.
By Lemma 3.3 and an integration by parts, we have
∂tkAk2Hs ≤ C kAk2HskSkW3,∞ +kAkHskSkHs+2kAkW1,∞
≤ Ck(cid:0)ϕ1t(u0)k2skϕ1t(u0)kW3,∞×W1,∞ (cid:1)
and
∂tkϕ1t(u0)k2s ≤ Ckϕ1t(u0)k2skϕ1t(u0)kW3,∞×W1,∞
≤ Ckϕ1(u )k3.
t 0 s
Taking s= s , we get that
0
M
kϕ1(u )k ≤
t 0 s0 1−cMt
and there is h = h (M) > 0 such that for all t ∈ [0,h ]
5 5 9
kϕ1(u )k ≤ 2M.
t 0 s0
We also obtain for s= s ≥ s > d/2+1 and t ∈ [0,h ] that
1 0 9
∂ kϕ1(u )k2 ≤ Ckϕ1(u )k2 kϕ1(u )k
t t 0 s1 t 0 s1 t 0 s0
≤ 2CMkϕ1(u )k2 .
t 0 s1
and the result follows from Grönwall’s Lemma. (cid:3)
We immediately get the following result for the second and the third flows.
Lemma 3.6. Let s > 0 and M > 0. There is h = h (M) such that for any
0 6 6
u ∈ Σ satisfying ku k ≤ M any 0 ≤t ≤ h , the following two points holds true.
0 s0 0 s0 6
(i) kϕ2(u )k ≤ M and kϕ3(u )k ≤ 2M,
t 0 s0 t 0 s0
(ii) Let s ≥ 0. If moreover u ∈ Σ , then, we have
1 0 s1
kϕ2(u )k ≤ ku k and kϕ3(u )k ≤ ku k +tkVk .
t 0 s1 0 s1 t 0 s1 0 s1 Hs1+2
The following lemma study the fourth flow.