Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation

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Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation

Philippe Chartier, Loïc Le Treust, Florian Méhats

To cite this version:

Philippe Chartier, Loïc Le Treust, Florian Méhats. Uniformly accurate time-splitting methods for the

semiclassical linear Schrödinger equation. ESAIM: Mathematical Modelling and Numerical Analysis,

EDP Sciences, 2019, 53 (2), pp.443-473. �10.1051/m2an/2018060�. �hal-01257753v3�

https://doi.org/10.1051/m2an/2018060 www.esaim-m2an.org

UNIFORMLY ACCURATE TIME-SPLITTING METHODS FOR THE SEMICLASSICAL LINEAR SCHR ¨ ODINGER EQUATION

Philippe Chartier

, Lo¨ıc Le Treust

and Florian M´ ehats

Abstract.This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr¨odinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.

Mathematics Subject Classification. 35Q55, 35F21, 65M99, 76A02, 76Y05, 81Q20, 82D50.

Received October 2, 2017. Accepted October 3, 2018.

1. Introduction

We are concerned here with the uniformly accurate numerical approximation of the solution Ψ^ε:R⁺×R^d→C, d≥1, of thelinearSchr¨odinger equation in its semiclassical limit

iε∂tΨ^ε=−ε²

2∆Ψ^ε+VΨ^ε (1.1)

whereV is a smooth potential which does not depend on time. The initial datum is assumed to be of the form Ψ^ε(0,·) =A₀(·)e^iS0(·)/ε with kA0k_L2(R^d)= 1. (1.2) Note that theL²-norm, the energy and the momentum of Ψ^ε(0,·), namely

Mass: kΨ^ε(t,·)k²_L2(R^d), (1.3)

Energy:

Z

R^d

ε²|∇Ψ^ε(t, x)|²+V|Ψ^ε(t, x)|²

dx, (1.4)

Momentum: εIm Z

R^d

Ψ^ε(t, x)∇Ψ^ε(t, x)dx, (1.5)

are all preserved by the flow of (1.1), whenever Ψ^ε(0,·)∈H¹(R^d).

Keywords and phrases.Schr¨odinger equation, semiclassical limit, numerical simulation, uniformly accurate, Madelung transform, splitting schemes.

1 Univ Rennes, INRIA, CNRS, IRMAR - UMR 6625, 35000 Rennes, France.

2 Aix Marseille Univ, CNRS, Central Marseille, I2M, Marseille, France.

3 Univ Rennes, CNRS, IRMAR - UMR 6625, 35000 Rennes, France.

∗Corresponding author:[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2019

Owing to its numerous occurrences in a vast number of domains of applications in physics, equation (1.1) has been widely studied (see for instance [27,32] and the references therein). In the semiclassical regime where the rescaled Planck constantεis small, its asymptotic study allows for an appropriate description of the observables of Ψ^εthrough the laws of hydrodynamics. We refer to [12] for a detailed presentation of the semiclassical analysis and to [25] for a review of both theoretical and numerical issues.

Let us also mention that we do not consider the case where the initial datas are Gaussian wave packets for which efficient schemes have already been developed [21–23].

1.1. Motivation

Generally speaking, numerical methods for equation (1.1) exhibit an error of size ∆t^p/ε^r+ ∆x^q/ε^s, where

∆t and ∆x are the time and space steps and p, q, r, s strictly positive numbers. For time-splitting methods for instance, the error on the wave function behaves like ∆x/ε+ ∆t^p/ε [5,17]. Even if we content ourselves with observables¹, the error of a splitting method of Bao et al. [5] grows like ∆x/ε+ ∆t^p. Now, achieving a fixed accuracy for varying values ofεrequires to keep both ratios ∆t/ε^r/p and ∆x/ε^s/q constant, and becomes prohibitively costly when ε → 0. Our aim, in this article, is thus to developnew numerical schemes that are Uniformly Accurate (UA) w.r.t. ε, i.e. whose accuracy does not deteriorate for vanishing ε. In other words, schemes for whichr, s= 0. This seems highly desirable as all available methods with the exception of [9], namely finite difference methods [1,16,26,35], splitting methods [8,17,18,29,31,34], asymptotic splitting methods [3,4], relaxation schemes [7] and symplectic methods [33] fail to be UA.

It is the belief of the authors that, prior to the construction of UA-schemes, it is necessary to reformulate (1.1) as in [9] and we now describe how this can be done.

1.2. Reformulation of the problem

In the spirit of the Wentzel-Kramers-Brillouin (WKB) techniques, we decompose Ψ^ε as the product of a slowly varying amplitude and a fast oscillating factor²

Ψ^ε(t,·) =A^ε(t,·)e^{iSε(t,·)/ε}. (1.7) From this point onwards, various choices are possible, depending on whetherA^εis complex or not³: taking A^ε∈Cleads to the following system [12]

∂tS^ε+|∇S^ε|²

2 +V= 0, (1.8a)

∂_tA^ε+∇S^ε· ∇A^ε+A^ε

2 ∆S^ε= iε∆A^ε

2 (1.8b)

1These authors performed extensive numerical tests in both linear and nonlinear cases [5,6].

2Considering the WKB-ansatz (1.7) transforms the invariants (1.3) into respectively

kA^εk²_L2(

R^d), Z

R^d

|ε∇A^ε+iA^ε∇S^ε|²+V|A^ε|²

dx and Im Z

R^d

A^ε(ε∇A^ε+iA^ε∇S^ε) dx. (1.6)

3The Madelung transform [30] relates the semiclassical limit of (1.1) to hydrodynamic equations

Ψ^ε(t,·) =p

ρ^ε(t,·)e^{iSε(t,·)/ε}

and amounts to choosingA^ε∈R+. However, this formulation leads to both analytical and numerical difficulties in the presence of vacuum,i.e.wheneverρ^εvanishes [14,15].

with S^ε(0,·) = S₀(·) and A^ε(0,·) = A₀(·). Under appropriate smoothness assumptions, (A^ε, S^ε) ∈ C×R converges whenε→0 to the solution (A⁰, S⁰) of

∂tS⁰+|∇S⁰|²

2 +V= 0, (1.9a)

∂tA⁰+∇S⁰· ∇A⁰+A⁰

2 ∆S⁰= 0. (1.9b)

Notice that (ρ, v) = (|A⁰|²,∇S⁰) is then solution of the Euler system

∂tv+v· ∇v+∇V= 0, (1.10a)

∂tρ+ div(ρv) = 0. (1.10b)

Now, an important drawback of (1.8) stems from the formation of caustics in finite time [12]: the solution of (1.8) may indeed cease to be smooth even though Ψ^ε is globally well-defined for ε >0. In order to obtain global existence forε >0, Besse, Carles and Mhats [9] suggested an alternative formulation by introducing an asymptotically-vanishing viscosity term in the eikonal equation (1.8a). Therein, system (1.8) is replaced by

∂tS^ε+|∇S^ε|²

2 +V =ε²∆S^ε, (1.11a)

∂tA^ε+∇S^ε· ∇A^ε+A^ε

2 ∆S^ε=iε∆A^ε

2 −iεA^ε∆S^ε (1.11b)

where S^ε(0, x) =S0(x),A^ε(0, x) =A0(x) and wherex∈R^d. Let us emphasize that both (1.8) and (1.11) are equivalent to (1.1) in the following sense: as long as the solution (S^ε, A^ε) of (1.8) (resp. (1.11)) is smooth, the function Ψ^εdefined by (1.7) solves (1.1). The well-posedeness of (1.11) and the uniform control of the solutions with respect to εare stated in Theorem 2.1below.

The main advantage of the WKB reformulation (1.11) over (1.1) is apparent: the semiclassical parameter ε does not give rise to singular perturbations⁴. Hence, it constitutes a good basis for the development of UA schemes (at least prior to the appearance of caustics), as witnessed by the methods introduced later in this paper.

1.3. Construction of the schemes

First and only (up to our knowledge) UA schemes are based on the formulation (1.11) introduced in [9].

Nevertheless, these schemes are still subject to CFL stability conditions and are of low order in time and space.

In this paper, we consider, in lieu of finite differences as in [9], time-splitting methods, for they enjoy the following favorable features:

(i) they do not suffer from stability restrictions on the time step;

(ii) they are easy to implement;

(iii) they preserve exactly theL²-norm;

(iv) they can be adapted to semilinear Schr¨odinger equations;

(v) they can be composed to attain high-order of convergence in time while remaining spectrally convergent in space.

4The Cole–Hopf transformation ([20], Sect. 4.4.1)

w^ε= exp

−S^ε 2ε²

−1 (1.12)

transforms (1.11a) into∂tw^ε− ^V

2ε²(w^ε+ 1) =ε²∆w^εfor which the regularizing effect of the viscosity term becomes arguably more apparent.

Points (iv) and (v) will be addressed in a forthcoming work using complex time steps (see [10]), while, in this paper, we introduce first and second order in time splitting-schemes and concentrate on the numerical analysis of the first-order one for the sake of clarity.

System (1.11) is split into four pieces as follows:

First flow: We denoteϕ¹_h the approximate flow at timeh∈Rof the system

∂tS+|∇S|²

2 = 0, (1.13a)

∂tA+∇S· ∇A+A

2∆S= i∆A

2 · (1.13b)

The eikonal equation (1.13a) is solved by means of the method of characteristics, while equation (1.13b) is dealt with by noticing thatw=Aexp (iS) satisfies the free Schr¨odinger equationi∂_tw=−¹₂∆w.

Second flow: We defineϕ²_has the exact flow at timeh∈Rof the system

∂tS= 0, (1.14a)

∂_tA=i(ε−1) ∆A

2 , (1.14b)

which is solved in the Fourier space.

Third flow: The third flowϕ³_his defined as the exact flow at time h∈Rof system

∂tS=−V, (1.15a)

∂_tA= 0. (1.15b)

Fourth flow: The fourth flowϕ⁴_his defined as the exact flow at time h∈R+of

∂tS=ε²∆S, (1.16a)

∂_tA=−iεA∆S. (1.16b)

Equation (1.16a) is solved in Fourier space and the solution of (1.16b) is simply obtained through the formula A(h,·) = exp −iε⁻¹(S(h,·)−S(0,·))

A(0,·). Notice thatϕ⁴_hcan thus be viewed as a regularizing flow.

The first-order scheme that we consider for (1.11) is then the concatenation of all previous flows

ϕ¹_h◦ϕ²_h◦ϕ³_h◦ϕ⁴_h (1.17)

while the second-order scheme is given by

ϕ¹_h/2◦ϕ²_h/2◦ϕ³_h/2◦ϕ⁴_h◦ϕ³_h/2◦ϕ²_h/2◦ϕ¹_h/2. (1.18)

1.4. Main result

The main result of this paper is the following theorem: it states thatϕ¹_h◦ϕ²_h◦ϕ³_h◦ϕ⁴_his uniformly accurate w.r.t. the semi-classical parameterε. The proper statement of the result uses the normk · kson the setΣs= H^s+2(R^d)×H^s(R^d) defined for s≥0 andu= (S, A) by

kuk_s=

kSk²_Hs+2(R^d)+kAk²_Hs(R^d)

^1/2 . Theorem 1.1. Let s > d/2 + 1,ε_max>0,u₀∈Σ_s+2 and0< T < T_max where

T_max= sup{t >0 :τ7→φ⁰_τ(u₀)∈L^∞([0, t];Σ_s+2)}

andφ^ε_τ denotes the flow at timeτ of (1.11).

There exists C >0 and h₀ >0 such that the following error estimate holds true for any ε∈(0, ε_max], any h∈[0, h₀] andn∈Nsatisfying nh≤T:

k(ϕ¹_h◦ϕ²_h◦ϕ³_h◦ϕ⁴_h)ⁿ(u₀)−φ^ε_nh(u₀)k_s≤Ch.

The constantsC andh₀ do not depend onε.

Remark 1.2. The constantT_max appearing in Theorem1.1 is well-defined and positive since τ 7→φ⁰_τ exists locally in time (see Thm.2.1).

Remark 1.3. The numerical analysis performed for the proof of Theorem1.1 can immediately be extended after the caustics forTmax≤T andε >0 since the solution of (1.11) (as the one of (1.1)) are global. Nevertheless, the constantsCandh0 appearing in the result will not be independent onεanymore. This point is illustrated in Section4.

Remark 1.4. The proof of Theorem1.1can be adapted to any time-splitting method of order 1 obtained after permutation of the four flows in (1.17).

Our proof is reminiscent of two previous results related to, on the one hand, splitting schemes for equations with Burgers nonlinearity [24] and on the other hand, splitting scheme for NLS in the semiclassical limit with [13]. 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 Lem.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 Schr¨odinger equations with time dependent potentials, to the second-order scheme, to the Schr¨odinger equation with a nonlinearity of Hartree-type and to the weakly nonlinear Schr¨odinger equation (see also [13], Rem. 4.5).

The paper is organized as follows. In Section 2, we first give a theorem of well-posedness of the Cauchy problem for (1.11). Then Theorem 1.1is proved using four technical lemmas. Section3is devoted to the proof of these lemmas. We illustrate the properties of our methods in Section4.

2. Preparatiory results and proof of Theorem 1.1 2.1. Notations

Assume that ε∈(0, ε_max] ands > d/2 + 1. For the sake of simplicity, we keep the notation of all the flows independent ofε. All the constants appearing in the proof depend onV but not onε >0. We denote

ϕ^ij_h =ϕⁱ_h◦ϕ^j_h, ϕ^ijk_h =ϕⁱ_h◦ϕ^jk_h, ϕ¹²³⁴_h =ϕ¹_h◦ϕ²³⁴_h =ϕ¹_h◦ϕ²_h◦ϕ³_h◦ϕ⁴_h, N_i is the possibly nonlinear operator related toϕⁱ_h so that

∂hϕⁱ_h=Niϕⁱ_h.

The quantities∂hϕh(u) and∂2ϕh(u) are the Fr`echet derivatives ofϕwith respect tohandu. The commutator of the nonlinear operatorsNi andNj is given by

[Ni,Nj](u) =DNi(u)· Nj(u)−DNj(u)· Ni(u).

2.2. Existence, uniqueness and uniform boundedness results

The following theorem study some properties of the solutions of equations (1.11).

Theorem 2.1. Let εmax>0,s > d/2 + 1 andu0∈Σs+2. The following two points are true.

(i) The quantity

Tmax= sup{t >0 :φ⁰(u0)∈L^∞([0, t];Σs+2)} (2.1) is well-defined and positive.

(ii) Let 0< T < Tmax. For allε∈[0, εmax], there exists a unique solution φ^ε(u₀)∈C([0, T], Σ_s+2) of the systems of equations (1.11). Moreover, φ^ε(u0)is bounded in

C([0, T], Σs+2) uniformly in ε∈[0, ε_max].

The proof of Theorem2.1is given in Section3.4.

2.3. The main lemmas

In this subsection, we present the main ingredients needed in the proof of Theorem 1.1. Their proof is postponed to Section3.

Lemma 2.2. Let M >0ands > d/2 + 1. There existh1=h1(M)>0such that for any ε∈(0, εmax]and any u0∈Σs satisfying

ku0ks≤M,

we have that the solutionφt(u0)of equation (1.11)is well-defined on[0, h1]and for allt∈[0, h1] kφ_t(u₀)k_s≤2M.

Lemma 2.3. Let M >0 ands > d/2 + 1. There exist C2 =C2(M)>0 such that for any ε∈(0, εmax], any solutions φ_t(u₁)∈L^∞([0, T], Σ_s+1)andφ_t(u₂)∈L^∞([0, T], Σ_s)of equation (1.11), satisfying for all t∈[0, T]

kφt(u1)ks+1+kφt(u2)ks≤M we have

kφt(u1)−φt(u2)ks≤ ku1−u2ksexp(C2t).

Remark 2.4. Let us insist on the fact that in Lemma2.3, we have to controlφt(u1) inΣs+1andφt(u2) inΣs

to get Lipschitz-type stability inΣ_s.

Lemma 2.5. Let M >0ands > d/2 + 1. There existh₃=h₃(M)>0 andC₃=C₃(M)>0such that for any ε∈(0, εmax], any u0∈Σs satisfyingku0ks≤M and any 0≤t≤h3, we have

(a) kϕ¹²³⁴_t (u₀)k_s≤8M.

(b) Furthermore, if u₀∈Σ_s+2, then

kϕ¹²³⁴_t (u₀)k_s+2≤exp (C₃t) (ku₀k_s+2+tkVk_Hs+4).

Lemma 2.6. Let M >0 ands > d/2 + 1. There exist h4=h4(M)>0 and K4 =K4(M)>0 such that for any ε∈(0, εmax]and any u0∈Σs+2 satisfying

ku0ks+2≤M, we have for anyt∈[0, h₄] that

kφt(u0)−ϕ¹²³⁴_t (u0)ks≤K4t².

2.4. Proof of Theorem

M_s^ε(T) := sup{kφ^ε_t(u₀)k_s: 0≤t≤T}. (2.2) forε≥0 andT ≥0.

Let s > d/2 + 1,ε ∈(0, ε_max], u₀ ∈Σ_s+2, n ∈ Nand h >0 be such that nh≤T < T_max (see (2.1)). By Theorem2.1, there existMs,Ms+1 andMs+2 independent ofε∈(0, εmax] such that for allε∈(0, εmax],

M_s^ε≤Ms, M_s+1^ε ≤Ms+1and M_s+2^ε ≤Ms+2

(see (2.2)). We denote

C=C3(2Ms),

c0=ku0ks+2exp (CT) +kVk_Hs+4e^{2T C}/C, C⁰=C2(Ms+1+ 4Ms),

ec=K4(c0)ae^C0T/C⁰. Assume that

0≤h≤min (h3(c0), Ms/ec, h1(2Ms), h4(c0)). (2.3) Here, h1,C2,h3,h4andK4 are defined in Lemmas2.2,2.3,2.5and2.6.

We show by induction on 0≤k≤nthat

(i) (ϕ¹²³⁴_h )^k(u0) is well-defined, belongs toΣs+2 and

k(ϕ¹²³⁴_h )^k(u0)ks+2≤ ku0ks+2exp (Ckh) +hkVk_Hs+4

e^(k+1)hC−e^hC e^hC−1 ≤c0, (ii) kφ_kh(u₀)−(ϕ¹²³⁴_h )^k(u₀)k_s≤h²K₄(c₀)^eC

0hk−1 e^C0h−1 ≤ech, and Theorem1.1follows then from point (ii) with k=n.

The induction hypothesis is true fork= 0. Let us assume points (i) and (ii) true for 0≤k≤n−1.

Lemma2.5, point (i) and (2.3) ensure that

(ϕ¹²³⁴_h )^k+1(u0) is well-defined and belongs toΣs+2. By Point (ii) and (2.3), we have

k(ϕ¹²³⁴_h )^k(u0)ks≤Ms+kφkh(u0)−(ϕ¹²³⁴_h )^k(u0)ks≤2Ms. By Lemma2.5and (2.3), we have

k(ϕ¹²³⁴_h )^k+1ks+2≤exp (Ch) k(ϕ¹²³⁴_h )^k(u0)ks+2+hkVk_Hs+4

and point (i) ensures that

k(ϕ¹²³⁴_h )^k+1ks+2≤ ku0ks+2exp (C(k+ 1)h) +hkVk_Hs+4

e^(k+2)hC−e^hC e^hC−1 ≤c0. By Lemma2.2and (2.3),h⁰ 7→φ_h⁰◦(ϕ¹²³⁴_h )^k(u₀) is well-defined and satisfies for all 0≤h⁰ ≤h

kφh⁰ ◦(ϕ¹²³⁴_h )^k(u₀)ks≤4M_s.

By Lemma2.3, we obtain that

kφ_h(k+1)(u₀)−φ_h◦(ϕ¹²³⁴_h )^k(u₀)k_s≤ kφ_hk(u₀)−(ϕ¹²³⁴_h )^k(u₀)k_sexp(C⁰h).

By Lemma2.6, point (i) and (2.3), we get

kφh◦(ϕ¹²³⁴_h )^k(u0)−ϕ¹²³⁴_h ◦(ϕ¹²³⁴_h )^k(u0)ks≤K4(c0)h², so that

kφ_h(k+1)(u₀)−(ϕ¹²³⁴_h )^k+1(u₀)k_s≤K₄(c₀)h²+kφ_hk(u₀)−(ϕ¹²³⁴_h )^k(u₀)k_sexp(C⁰h).

By point (ii), we have then that

kφ_h(k+1)(u0)−(ϕ¹²³⁴_h )^k+1(u0)ks≤K4(c0)h² e^C0h(k+1)−1 e^C0h−1

!

·

Thus, points (i) and (ii) are true fork+ 1.

3. Proof of the main lemmas 3.1. Auxiliary results

Let us denote byh·,·itheL² scalar product, fors >0 Λ^s= (1−∆)^s/2,

Π1u=S, Π2u=A, foru= S

A

(3.1) and

hu1, u₂i_s=hΠ1u₁,Π₁u₂i +

Λ^s+1∇Π1u1,Λ^s+1∇Π1u2

+ RehΛ^sΠ2u1,Λ^sΠ2u2i. (3.2) We recall two points that will be of constant use in the following: the Sobolev spaceH^s⊂L^∞ is an algebra fors > d/2 and the Kato-Ponce [28] inequality holds true:

Proposition 3.1. Let s0> d/2 + 1. There isc >0 such that for all f ∈H^s0(R^d)andg∈H^s0−1(R^d) kΛ^s0(f g)−fΛ^s0gk_L2≤c(k∇fkL^∞kgk_Hs0−1+kfkH^s0kgkL^∞).

The following lemmas will be used several times in our proof.

Lemma 3.2. Lets0> d/2 + 1. There isC >0such that for allv0,v1andR∈L^∞([0, h0], H^s0(R^d)^d)satisfying

∂tv0+ (v1· ∇)v0=R, we have

∂tkv0k²_Hs0 ≤C kv0k²_Hs0k∇v1kL^∞+kv0kH^s0kv1kH^s0k∇v0kL^∞

+ 2hΛ^s0v0,Λ^s0Ri

≤Ckv0k²_Hs0kv1kH^s0 + 2hΛ^s0v₀,Λ^s0Ri.

Proof. We have by integration by parts that

∂t

kv0k²_Hs0

2 =hΛ^s0v0,Λ^s0∂tv0i=− hΛ^s0v0,Λ^s0(v1· ∇)v0i+hΛ^s0v0,Λ^s0Ri

≤ 1 2 Z

R^d

|Λ^s0v0|²divv1+kv0kH^s0k[Λ^s0,(v1· ∇)]v0k_L2+hΛ^s0v0,Λ^s0Ri. Proposition3.1 ensures that

∂_tkv0k²_Hs0

2 ≤c kv₀k²_Hs0k∇v₁k_L^∞+kv₀k_H^s0kv₁k_H^s0k∇v₀k_L^∞

+hΛ^s0v₀,Λ^s0Ri.

Lemma 3.3. Let s0 > d/2 + 1. There exists C > 0 such that for all A ∈ W^1,∞([0, h0], H^s0(R^d)), v1 ∈ L^∞([0, h0], H^s0+1(R^d)^d) andR∈L^∞([0, h0], H^s0(R^d))satisfying

∂tA+v1· ∇A+Adivv1

2 =R, we have,

∂_tkAk²_Hs0 ≤C kAk²_Hs0kv1kW^2,∞+kAkH^s0kv1k_H^s0 +1kAkW^1,∞

+ 2 RehΛ^s0A,Λ^s0Ri

≤CkAk²_Hs0kv1k_Hs0 +1+ 2 RehΛ^s0A,Λ^s0Ri. Proof. We have by integration by parts that

∂t

kAk²_Hs0

2 = RehΛ^s0A,Λ^s0∂tAi=−Re

Λ^s0A,

v1· ∇+divv₁ 2

Λ^s0A

−Re

Λ^s0A,

Λ^s0,

v₁· ∇+divv₁ 2

A

+ RehΛ^s0A,Λ^s0Ri

≤ kAk_H^s0

Λ^s0,

v₁· ∇+divv₁ 2

A

_L2

+ RehΛ^s0A,Λ^s0Ri. Proposition3.1 ensures that

∂t

kAk²_Hs0

2 ≤CkAkH^s0(k∇v1kL^∞k∇Ak_Hs0−1+kv1kH^s0k∇AkL^∞)

+CkAkH^s0(k∇(divv1)kL^∞kAk_Hs0−1+kdivv1kH^s0kAkL^∞) + RehΛ^s0A,Λ^s0Ri

≤C kAk²_Hs0kv1k_W2,∞+kAkH^s0kv1k_Hs0 +1kAk_W1,∞

+ RehΛ^s0A,Λ^s0Ri

≤CkAk²_Hs0kv1k_Hs0 +1+ RehΛ^s0A,Λ^s0Ri.

3.2. Study of the equation

Let us prove Lemma2.2.

Proof. By the Cole–Hopf transform, we get thatw^ε= exp −_2ε^Sε2

−1 is the solution of

∂tw^ε=ε²∆w^ε+ V

2ε²(w^ε+ 1), w^ε(0) = exp

−S0

2ε²

−1,

Hence, global existence and uniqueness of the solution S^ε of (1.11a) for fixed ε ∈ (0, ε_max], follows from standard semi-group theory. The functionv^ε=∇S^εsolves

∂tv^ε+ (v^ε· ∇)v^ε+∇V=ε²∆v^ε. Sinces > d/2, Lemma3.2and an integration by parts ensure that

∂_tkv^εk²_Hs+1 ≤ckv^εk³_Hs+1+

Λ^s+1v^ε,Λ^s+1 −∇V+ε²∆v^ε

≤ckv^εk³_Hs+1+kv^εk_Hs+1kVk_Hs+2. By (1.11a), we also have that

∂t

kS^εk²_L2

2 ≤ kS^εkL² kVkL²+kv^εk²_L4/2 so that

∂tkS^εk²_Hs+2≤ckVk_Hs+2kS^εk_Hs+2+ckS^εk³_Hs+2.

The global existence and the uniqueness of a solutionA^εof equation (1.11b) follows from the fact that Ψ^ε=A^εexp (iS^ε/ε)

satisfies equation (1.1). By Lemma3.3, recalling thats > d/2 + 1, we also have

∂tkA^εk²_Hs≤ckA^εk²_HskS^εk_Hs+2+ RehΛ^sA^ε,Λ^sRi. whereR=^iε∆A₂ ^ε −iεA^ε∆S^ε so that an integration by parts gives us

∂_tkA^εk²_Hs≤ckA^εk²_HskS^εk_Hs+2. We obtain that

∂tkφt(u0)k²_s≤c1kφt(u0)kskVk_Hs+2+c2kφt(u0)k³_s and

∂_tkφt(u₀)ks≤c₁kVkH^s+2+c₂kφt(u₀)k²_s. We get then that

kφt(u0)ks≤ s

c1kVk_Hs+2

c2

tan

tp

c1c2kVk_Hs+2+ arctan

M

r c2

c1kVk_Hs+2

so that there ish₁=h₁(M)>0 such that for all 0≤t≤h₁ kφt(u0)ks≤2M.

The following result will be used several times and in particular for the proof of the stability of equation (1.11) in Lemma2.3.

Lemma 3.4. Let s₀ > d/2 + 1. Let u₁ = (S₁, A₁) be in L^∞([0, T], Σ_s0+1), u₂ = (S₂, A₂), (R_1,S, R_1,A) and (R2,S, R2,A)be inL^∞([0, T], Σs₀). Assume moreover that fori= 1,2

∂_tS_i+|∇Si|²

2 =R_i,S,

∂tAi+∇Si· ∇Ai+Ai

∆Si

2 =Ri,A. Then, we have

∂_tku1−u₂k²_s0 ≤cku1−u₂k²_s0(ku1ks0+1+ku2ks0) + 2hu1−u₂, R₁−R₂i_s

0

whereRi= (Ri,S, Ri,A)^T.

Proof. Lets₀> d/2 + 1. Let us definev₁=∇S1,v₂=∇S2,w=v₁−v₂,B=A₁−A₂ andu=u₁−u₂. We have that

∂tw=−(v1· ∇)v1+ (v2· ∇)v2+∇(R1,S−R2,S)

=−(v2· ∇)w−(w· ∇)v1+∇(R1,S−R2,S) and Lemma3.2 ensures that

∂tkwk²_Hs0 +1≤ckwk²_Hs0 +1kv2k_Hs0 +1+ 2

Λ^s0+1w,Λ^s0+1R . whereR=−(w· ∇)v1+∇(R1,S−R2,S).We also have that

k(w· ∇)v1k_Hs0 +1 ≤ckwk_Hs0 +1kv1k_Hs0 +2

and

∂tkwk²_Hs0 +1≤ckwk²_Hs0 +1(kS1k_Hs0 +3+kS2k_Hs0 +2) + 2

Λ^s0+1w,Λ^s0+1∇(R1,S−R2,S) . We also have

∂_t(S₁−S₂) =−1

2(v₁+v₂)·w+ (R_1,S−R_2,S) so that

∂tkS1−S2k²_L2 ≤ckS1−S2kL²kwkL²(kS1kW^1,∞+kS2kW^1,∞) + 2hS1−S2, R1,S−R2,Si and then

∂tkS1−S2k²_Hs0 +2≤CkS1−S2k²_Hs0 +2(kS1k_Hs0 +3+kS2k_Hs0 +2) + 2hS1−S2, R1,S−R2,Si+ 2

Λ^s0+1∇(S1−S2),Λ^s0+1∇(R1,S−R2,S) Let us studyB, we have

∂tB+∇S2· ∇B+∆S₂ 2 B =R where

R=−w· ∇A1−div(w)

2 A1+ (R1,A−R2,A)

Hence, we obtain by Lemma3.3

∂_tkBk²_Hs0 ≤ckBk²_Hs0kS₂k_Hs0 +2+ 2 RehΛ^s0B,Λ^s0Ri

≤ckBk²_Hs0kS2k_Hs0 +2+ckBkH^s0kwk_Hs0 +1kA1k_Hs0 +1

+ 2 RehΛ^s0B,Λ^s0(R1,A−R2,A)i and

∂tku1−u2k²_s0 ≤cku1−u2k²_s0(ku1ks₀+1+ku2ks₀) + 2hu1−u2, R1−R2i_s

0

The result follows.

Let us study now the stability of equation (1.11) and prove Lemma2.3.

Proof. Lets > d/2 + 1 andε∈(0, εmax]. Let us define fori= 1,2 Ri,S=−V+ε²∆Si, Ri,A=iε∆Ai

2 −iεAi∆Si. We apply Lemma3.4withs0=s. We have by integrations by parts that

hS1−S2, R1,S−R2,Si+

Λ^s+1∇(S1−S2),Λ^s+1∇(R1,S−R2,S)

≤0, and

RehΛ^s(A1−A2),Λ^s(R1,A−R2,A)i ≤ckA1−A2k²_HskS1kH^s+2

+ckA₁−A₂k_HskS₁−S₂k_Hs+2kA₂k_Hs. so that

∂tkφt(u1)−φt(u2)k²_s≤cku1−u2k²_s(kφt(u1)ks+1+kφt(u2)ks)

and the result follows.

3.3. Study of the numerical flow

The following lemma is inspired by the work of Holdenet al.[24].

Lemma 3.5. Let s0> d/2 + 1andM >0. There existsh5=h5(M)>0such that for anyu0∈Σs₀ satisfying ku0ks₀ ≤M and any 0≤t≤h5, the following two points are true.

(i) We have that kϕ¹_t(u0)ks₀ ≤2M.

(ii) Let s1≥s0. There isC5=C5(M)>0 such that ifu0∈Σs₁, then kϕ¹_t(u0)ks₁ ≤exp (C5t)ku0ks₁.

Proof. The existence of the solutionSof (1.13a) follows for instance from the method of characteristics. Lemma 3.2ensures that fors > d/2 + 1

∂_tk∇Sk²_Hs+1≤ck∇Sk²_Hs+1k∇(∇S)kL^∞ ≤Ck∇Sk²_Hs+1kS(t)kW^2,∞. We also have

∂_tkSk²_L2 ≤ckS(t)kL²k∇S(t)k²_L4

so that

∂tkSk²_Hs+2≤CkS(t)k²_Hs+2kS(t)kW^2,∞.

The remaining of the proof follows exactly the same lines as the one of Lemma2.2. By Lemma3.3 and an integration by parts, we have

∂_tkAk²_Hs ≤C kAk²_HskSkW^3,∞+kAkH^skSkH^s+2kAkW^1,∞

≤Ckϕ¹_t(u0)k²_skϕ¹_t(u0)k_W3,∞×W^1,∞

and

∂_tkϕ¹_t(u₀)k²_s≤Ckϕ¹_t(u₀)k²_skϕ¹_t(u₀)k_W3,∞×W^1,∞

≤Ckϕ¹_t(u0)k³_s. Takings=s₀, we get that

kϕ¹_t(u0)ks₀≤ M 1−cM t and there ish5=h5(M)>0 such that for allt∈[0, h9]

kϕ¹_t(u₀)ks0≤2M.

We also obtain fors=s1≥s0> d/2 + 1 andt∈[0, h9] that

∂tkϕ¹_t(u0)k²_s1 ≤Ckϕ¹_t(u0)k²_s1kϕ¹_t(u0)ks₀

≤2CMkϕ¹_t(u₀)k²_s

1.

and the result follows from Gronwall’s Lemma.

We immediately get the following result for the second and the third flows.

Lemma 3.6. Let s0>0 and M >0. There ish6=h6(M) such that for any u0∈Σs₀ satisfying ku0ks₀ ≤M any 0≤t≤h₆, the following two points holds true.

(i) kϕ²_t(u0)ks₀ ≤M andkϕ³_t(u0)ks₀ ≤2M,

(ii) Let s₁≥0. If moreoveru₀∈Σ_s1, then, we have

kϕ²_t(u0)ks₁ ≤ ku0ks₁ andkϕ³_t(u0)ks₁ ≤ ku0ks₁+tkVk_Hs1 +2. The following lemma study the fourth flow.

Lemma 3.7. Let s0> d/2 + 1andM >0. There existsh7=h7(M)>0such that for anyu0∈Σs₀ satisfying ku0ks₀ ≤M and any 0≤t≤h7, the following two points holds true.

(i) kϕ⁴_t(u0)ks₀ ≤2M,

(ii) Let s1≥s0. There isC7=C7(M)>0 such that ifu0∈Σs₁, kϕ⁴_t(u₀)ks1 ≤exp (C₇t)ku0ks1. Proof. Lets > d/2 + 1. By integration by parts, we have∂tkS(h)k²_Hs+2 ≤0 and

∂_tkAk²_Hs

2 = RehΛ^sA,Λ^s(−iεA∆S)i= RehΛ^sA,[Λ^s,−iε∆S]Ai

≤ckAk_Hs(k∇(∆S)k_L^∞kAk_Hs−1+k∆Sk_HskAk_L^∞)

≤ckAk²_HskSk_W^3,∞+ckAkH^skSk_Hs+2kAkL^∞.

We obtain fors=s₀ that

∂tkϕ⁴_t(u0)k²_s0 ≤ckϕ⁴_t(u0)k³_s0, fors=s1 that

∂_tkϕ⁴_t(u₀)k²_s

1 ≤ckϕ⁴_t(u₀)k²_s

1kϕ⁴_t(u₀)k_s0

and the result follows from the arguments of the end of the proof of Lemma3.5.

Takings0=sand s1=s0+ 2, we immediately get Lemma2.5combining Lemmas3.5–3.7.

3.4. Proof of Theorem

LetM >0. Lemma2.2ensures that there ish₁=h₁(M)>0 such that for anyε∈(0, ε_max] and anyu₀∈Σ_s+2 satisfyingku0ks+2 ≤M, the solutions t 7→φ^ε_t(u0) of equation (1.11) are well-defined in L^∞([0, h1], Σs+2) and uniformly bounded with respect toε.

Letε, ε⁰ ∈(0, εmax],u0, u⁰₀∈Σs+2 such thatku0ks+2≤M andku⁰₀ks+2≤M. We define (S^ε, A^ε)^T =φ^ε(u0), (S^ε0, A^ε0)^T =φ^ε0(u⁰₀) and

R_1,S=−V+ε²∆S^ε, R_2,S=−V+ε⁰²∆S^ε0, R_1,A=iε∆A^ε

2 −iεA^ε∆S^ε, R_2,A=iε⁰∆A^ε0

2 −iε⁰A^ε0∆S^ε0.

We apply Lemma3.4withs0=s,u1=φ^ε(u0) andu2=φ^ε0(u0). We have by integrations by parts that D

S^ε−S^ε0, R1,S−R2,S

E +D

Λ^s+1∇

S^ε−S^ε0

,Λ^s+1∇(R1,S−R2,S)E

≤c|ε−ε⁰|kS^ε−S^ε0k_Hs+2kS^εk_Hs+4, and

ReD Λ^s

A^ε−A^ε0

,Λ^s(R1,A−R2,A)E

≤c|ε−ε⁰|kA^ε−A^ε0kH^skA^εk_Hs+2

+ckA^ε−A^ε0k²_HskS^ε0k_Hs+2+ckA^εk_HskA^ε

−A^ε0kH^skS^ε−S^ε0k_Hs+2

+c|ε−ε⁰|kA^ε−A^ε0kH^skA^εkH^skS^εk_Hs+2. so that

∂tkφ^ε_t(u0)−φ^ε_t⁰(u⁰₀)k²_s≤ckφ^ε_t(u0)−φ^ε_t⁰(u⁰₀)k²_s

kφ^ε_t(u0)ks+1+kφ^ε_t⁰(u⁰₀)ks

+c|ε−ε⁰|kφ^ε_t(u0)−φ^ε_t⁰(u⁰₀)ks(kφ^ε_t(u0)ks+2+kφ^ε_t(u0)k²_s).

Gronwall’s Lemma ensures that for allt∈[0, h1]

kφ^ε_t(u0)−φ^ε_t⁰(u⁰₀)ks≤C(ku0−u⁰₀ks+|ε−ε⁰|) (3.3) where

C=C(kφ^ε(u0)k_L∞([0,h1],Σs+2),kφ^ε0(u⁰₀)k_L∞([0,h1],Σs))>0.

Thus, (φ^ε_t(u0))_t∈[0,h1] is a Cauchy sequence of ε of L^∞([0, h1], Σs). The limit φ⁰(u0) is solution of (1.11) with ε = 0. Uniqueness follows from (3.3). We get immediately that Lemma 2.2 is also true for ε = 0 and φ⁰(u0)∈L^∞([0, h1], Σs+2).

Let

T_max= sup{t >0 :φ⁰(u₀)∈L^∞([0, t];Σ_s+2)}>0,