Strong confinement limit for the nonlinear Schrödinger equation constrained on a curve
aa r X i v : . [ m a t h - ph ] D ec Strong confinement limit for the nonlinear Schr¨odingerequation constrained on a curve
F. M´ehats and N. RaymondJuly 31, 2018
Abstract
This paper is devoted to the cubic nonlinear Schr¨odinger equation in a two di-mensional waveguide with shrinking cross section of order ε . For a Cauchy data livingessentially on the first mode of the transverse Laplacian, we provide a tensorial approx-imation of the solution ψ ε in the limit ε →
0, with an estimate of the approximationerror, and derive a limiting nonlinear Schr¨odinger equation in dimension one. If theCauchy data ψ ε has a uniformly bounded energy, then it is a bounded sequence in H and we show that the approximation is of order O ( √ ε ). If we assume that ψ ε isbounded in the graph norm of the Hamiltonian, then it is a bounded sequence in H and we show that the approximation error is of order O ( ε ). The Dirichlet realization of the Laplacian on tubes of the Euclidean space plays an im-portant role in the physical description of nanostructures. In the last twenty years, manypapers were concerned by the influence of the geometry of the tube (curvature, torsion)on the spectrum. For instance, in [12], Duclos and Exner proved that bending a waveg-uide in dimension two and three always induces the existence of discrete spectrum belowthe essential spectrum (see also [10]). Another question of interest in their paper is thelimit when the cross section shrinks to a point. In particular they prove that, in somesense, the Dirichlet Laplacian on a bidimensional tube, with cross section ( − ε, ε ) is wellapproximated by Schr¨odinger operator − ∂ x − κ ( x )4 − ε ∂ x , acting on L ( R × ( − , , d x d x ) and where κ denotes the curvature of the center lineof the tube. Such approximations have been recently considered in [15] or in presenceof magnetic fields [14] through a convergence of resolvent method. Concerning this kindof results, one may refer to the memoir by Wachsmuth and Teufel [18] where dynamicalproblems are analyzed in the spirit of adiabatic reductions.In the present paper, we will consider the time dependent Schr¨odinger equation with acubic non linearity in a waveguide and we would especially like to determine if the adiabaticreduction available in the linear framework can be used to reduce the dimension of thenon linear equation and provide an effective dynamics in dimension one. The derivationof nonlinear quantum models in reduced dimensions has been the object of several worksin the past years. For the modeling of the dynamics of electrons in nanostructures, the1imension reduction problem for the Schr¨odinger-Poisson system has been studied in [6, 11]for confinement on the plane, in [4] for confinement on a line, and in [13] for confinement onthe sphere. For the modeling of strongly anisotropic Bose-Einstein condensates, the caseof cubic nonlinear Schr¨odinger equations with an harmonic potential has been consideredin [7, 5, 3, 1, 2]. Let us describe the geometrical context of this paper. With the same formalism, we willconsider the case of unbounded curves and the case of closed curves. Consider a smooth,simple curve Γ in R defined by its normal parametrization γ : x γ ( x ). For ε > ε : S = M × ( − , ∋ ( x , x ) γ ( x ) + εx ν ( x ) = x , (1.1)where ν ( x ) denotes the unit normal vector at the point γ ( x ) such thatdet( γ ′ ( x ) , ν ( x )) = 1 and where M = (cid:26) R for an unbounded curve, T = R / (2 π Z ) for a closed curve.We recall that the curvature at the point γ ( x ), denoted by κ ( x ), is defined by γ ′′ ( x ) = κ ( x ) ν ( x ) . The waveguide is Ω ε = Φ ε ( S ) and we will work under the following assumption whichstates that waveguide does not overlap itself and that Φ ε is a smooth diffeomorphism. Assumption 1.1
We assume that the function κ is bounded, as well as its derivatives κ ′ and κ ′′ . Moreover, we assume that there exists ε ∈ (0 , k κ k L ∞ ) such that, for ε ∈ (0 , ε ) , Φ ε is injective. We will denote by − ∆ Dir Ω ε the Dirichlet Laplacian on Ω ε . We are interested in the followingequation: i∂ t ψ ε = − ∆ Dir Ω ε ψ ε + λε α | ψ ε | ψ ε (1.2)on Ω ε with a Cauchy condition ψ ε (0; · ) = ψ ε and where α ≥ λ ∈ R are parameters.By using the diffeomorphism Φ ε , we may rewrite (1.2) in the space coordinates ( x , x )given by (1.1). For that purpose, let us introduce m ε ( x , x ) = 1 − εx κ ( x ) and considerthe function ψ ε transported by Φ ε , U ε ψ ε ( t ; x , x ) = φ ε ( t ; x , x ) = ε / m ε ( x , x ) / ψ ε ( t ; Φ ε ( x , x )) . Note that U ε is unitary from L (Ω ε , dx ) to L ( S , d x d x ) and maps H (Ω ε ) (resp. H (Ω ε ))to H ( S ) (resp. to H ( S )). Moreover, the operator − ∆ Dir Ω ε is unitarily equivalent to theself-adjoint operator on L ( S , d x d x ), U ε ( − ∆ Ω Dir ε ) U − ε = H ε + V ε , with H ε = P ε, + P ε, , where P ε, = m − / ε D x m − / ε , P ε, = ε − D x and where the effective electric V ε potential is defined by V ε ( x , x ) = − κ ( x ) − εx κ ( x )) .
2e have used the standard notation D = − i∂ . Notice that, for all ε < ε , we have m ε ≥ − ε k κ k L ∞ >
0. The problem (1.2) becomes i∂ t φ ε = H ε φ ε + V ε φ ε + λε α − m − ε | φ ε | φ ε (1.3)with Dirichlet boundary conditions φ ε ( t ; x , ±
1) = 0 and the Cauchy condition φ ε ( · ; 0) = φ ε = U ε ψ ε . We notice that the domains of H ε and H ε + V ε coincide with H ( S ) ∩ H ( S ).In this paper we will analyze the critical case α = 1 where the nonlinear term is of thesame order as the parallel kinetic energy associated to D x . It is well-known that (1.2) (thus(1.3) also) has two conserved quantities: the L norm and the nonlinear energy. Let usintroduce the first eigenvalue µ = π of D x on ( − ,
1) with Dirichlet boundary conditions,associated to the eigenfunction e ( x ) = cos (cid:0) π x (cid:1) and define the energy functional E ε ( φ ) = 12 Z S |P ε, φ | d x d x + 12 Z S |P ε, φ | d x d x + 12 Z S (cid:16) V ε − µ ε (cid:17) | φ | d x d x + λ Z S m − ε | φ | d x d x (1.4)= 12 Z Ω ε |∇ ( U − ε φ ) | d x d x + λε Z Ω ε |U − ε φ | d x d x − µ ε Z Ω ε |U − ε φ | d x d x . Notice that we have substracted the conserved quantity µ ε k φ k L to the usual nonlinearenergy, in order to deal with bounded energies. Indeed, we will consider initial conditionswith bounded mass and energy, which means more precisely the following assumption. Assumption 1.2
There exists two constants M > and M > such that the initialdata φ ε satisfies, for all ε ∈ (0 , ε ) , k φ ε k L ≤ M and E ε ( φ ε ) ≤ M . Let us define the projection Π on e by letting Π u = h u, e i L (( − , e . A consequenceof Assumption 1.2 is that φ ε has a bounded H norm and is close to its projection Π φ ε .Indeed, we will prove the following lemma. Lemma 1.3
Assume that φ ε satisfies Assumption 1.2. Then there exists ε ( M ) ∈ (0 , ε ) and a constant C > independent of ε such that, for all ε ∈ (0 , ε ( M )) , k φ ε k H ( S ) ≤ C and k φ ε − Π φ ε k L ( M , H ( − , ≤ Cε. (1.5)
In the sequel, it will be convenient to consider the following change of temporal gauge φ ε ( t ; x , x ) = e − iµ ε − t ϕ ε ( t ; x , x ). This leads to the equation i∂ t ϕ ε = H ε ϕ ε + ( V ε − ε − µ ) ϕ ε + λm − ε | ϕ ε | ϕ ε (1.6)with conditions ϕ ε ( t ; x , ±
1) = 0, ϕ ε (0; · ) = φ ε .In order to study (1.6), the first natural try is to conjugate the equation by the unitarygroup e it H ε so that the problem (1.6) becomes i∂ t e ϕ ε = e it H ε ( V ε − ε − µ ) e − it H ε e ϕ ε + λW ε ( t ; e ϕ ε ) , e ϕ ε (0; · ) = φ ε , (1.7)3here W ε ( t ; ϕ ) = e it H ε m − ε | e − it H ε ϕ | e − it H ε ϕ (1.8)and where e ϕ ε = e it H ε ϕ ε which satisfies e ϕ ε ( t ; x , ±
1) = 0. Nevertheless this reformulationdoes not take the inhomogeneity with respect to ε into account. In particular it will notbe appropriate to estimate the approximation of the solution.We will see that (1.6) is well approximated in the limit ε → i∂ t θ ε = D x θ ε − κ ( x ) θ ε + λγ | θ ε | θ ε , (1.9)with γ = R − e ( x ) d x = 3 / θ ε (0 , x ) = θ ε ( x ) = R − φ ε ( x , x ) e ( x ) d x for x ∈ M (recall that the notation M stands for R or T ). This last equation can bereformulated as i∂ t e θ ε = e itD x (cid:18) − κ ( x )4 (cid:19) e − itD x e θ ε + λγF ( t ; e θ ε ) , e θ (0; · ) = θ ε , (1.10)where F ( t ; e θ ) = e itD x (cid:12)(cid:12)(cid:12) e − itD x e θ (cid:12)(cid:12)(cid:12) e − itD x e θ, (1.11)and e θ ε = e itD x θ ε .Our main results are the following theorems and their corollary. Theorem 1.4 (Solutions in the energy space)
Consider a sequence of Cauchy data φ ε ∈ H ( S ) satisfying Assumption 1.2. Then:(i) The limit problem (1.9) admits a unique solution θ ε ∈ C ( R + ; H ( M )) ∩ C ( R + ; H − ( M )) .(ii) There exists ε ( M ) ∈ (0 , ε ] such that, for all ε ∈ (0 , ε ( M )) , the two-dimensionalproblem (1.6) admits a unique solution ϕ ε ∈ C ( R + ; H ( S )) ∩ C ( R + ; H − ( S )) .(iii) For all T > there exists C T > such that, for all ε ∈ (0 , ε ( M )) , we have the errorbound sup t ∈ [0 ,T ] k ϕ ε ( t ) − θ ε ( t ) e k L ( S ) ≤ C T ε / . (1.12) Theorem 1.5 ( H solutions) Assume that φ ε ∈ H ∩ H ( S ) and that there exist M > , M > such that, for all ε ∈ (0 , ε ) , k φ ε k L ≤ M , (cid:13)(cid:13)(cid:13) ( H ε − µ ε ) φ ε (cid:13)(cid:13)(cid:13) L ≤ M . (1.13) Then φ ε satisfies Assumption 1.2 and:(i) The limit problem (1.9) admits a unique solution θ ε ∈ C ( R + ; H ( M ) ∩ C ( R + ; L ( M )) .(ii) For all ε ∈ (0 , ε ( M )) , the two-dimensional problem (1.6) admits a unique solution ϕ ε ∈ C ( R + ; H ∩ H ( S )) ∩ C ( R + ; L ( S )) . The constant ε ( M ) is the same as in Theorem1.4.(iii) For all T > there exists C T > such that, for all ε ∈ (0 , ε ( M )) , we have therefined error bound sup t ∈ [0 ,T ] k ϕ ε ( t ) − θ ε ( t ) e k L ( S ) ≤ C T ε. (1.14)4oming back by U − ε to the original equation (1.2), an immediate consequence of our twotheorems is the following corollary. Corollary 1.6
Consider a sequence of Cauchy data ψ ε ∈ H (Ω ε ) with bounded mass andenergy: k ψ ε k L (Ω ε ) ≤ M and k ψ ε k H (Ω ε ) + λε k ψ ε k L (Ω ε ) − µ ε k ψ ε k L (Ω ε ) ≤ M . Then for all ε ∈ (0 , ε ) , the confined NLS equation (1.2) with α = 1 admits a uniquesolution ψ ε ∈ C ( R + ; H (Ω ε )) ∩ C ( R + ; H − (Ω ε )) and, for all T > , we have the estimate sup t ∈ [0 ,T ] k ψ ε ( t ) − e − iµ /ε t U − ε ( θ ε ( t ) e ) k L (Ω ε ) ≤ C T ε / . If, additionally, ψ ε ∈ H (Ω ε ) and k ( − ∆ Dir Ω ε − µ ε ) φ ε k L (Ω ε ) is bounded uniformly with respectto ε , then we have ψ ε ∈ C ( R + ; H ∩ H (Ω ε )) ∩ C ( R + ; L (Ω ε )) with, for all T > , theestimate sup t ∈ [0 ,T ] k ψ ε ( t ) − e − iµ /ε t U − ε ( θ ε ( t ) e ) k L (Ω ε ) ≤ C T ε. This paper is organized as follows. Section 2 is devoted to technical lemmas related tothe well-posedness of our Cauchy problems and to energy estimates. Section 3 deals withthe proof of well-posedness stated in Theorems 1.4 and 1.5. In Section 4 we establish thetensorial approximation announced in Theorems 1.4 and 1.5.
In this section, we give a few technical results that will be useful in the sequel.
Let us first remark that P ε, = (1 − εx κ ( x )) − D x − εx κ ′ ( x )2(1 − εx κ ( x )) . Hence, by Assumption 1.1, there exists three positive constants C , C , C such that, forall ε ∈ (0 , ε ) and for all u ∈ H ( S ),(1 − C ε ) k ∂ x u k L ≤ kP ε, u k L + C ε k u k L ≤ (1 + C ε ) k ∂ x u k L + C ε k u k L . (2.1)Furthermore, the graph norm of H ε is equivalent to the H norm for all ε ∈ (0 , ε ), withconstants depending on ε . More precisely, we have the following result. Lemma 2.1
There exist positive constants C and C such that, for all ε ∈ (0 , ε ) andfor all u ∈ H ∩ H ( S ) , C (cid:18)(cid:13)(cid:13) D x u (cid:13)(cid:13) L + 1 ε (cid:13)(cid:13)(cid:0) D x − µ (cid:1) u (cid:13)(cid:13) L + k u k L (cid:19) ≤ (2.2) ≤ (cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) u (cid:13)(cid:13)(cid:13) L + k u k L ≤ C (cid:18)(cid:13)(cid:13) D x u (cid:13)(cid:13) L + 1 ε (cid:13)(cid:13)(cid:0) D x − µ (cid:1) u (cid:13)(cid:13) L + k u k L (cid:19) . roof. To prove the left inequality in (2.2), we use standard elliptic estimates. For u ∈ H ∩ H ( S ), we let f = (cid:16) H ε − µ ε (cid:17) u = P ε, u + ε − ( D x − µ ) u (2.3)and taking the L scalar product of f with D x u , we get h D x P ε, u, D x u i L + ε − k D x (cid:0) D x − µ (cid:1) / u k L ≤ k f k L k D x u k L . Then we write h D x P ε, u, D x u i L = kP ε, D x u k L + h [ D x , P ε, ] u, P ε, D x u i L − hP ε, u, [ D x , P ε, ] D x u i L and use k [ D x , P ε, ] u k L ≤ Cε ( k D x u k L + k u k L ) , (2.4)together with (2.1) and the interpolation estimate k D x u k L ≤ C k D x u k / L k u k / L , to get h D x P ε, u, D x u i L ≥ (1 − Cε ) k D x u k L − Cε k u k L . It follows that k D x u k L ≤ C k f k L + C k u k L and then, using (2.3) and again (2.1), ε − (cid:13)(cid:13) ( D x − µ ) u (cid:13)(cid:13) L ≤ k f k L + kP ε, u k L ≤ k f k L + C k D x u k L + C k u k L ≤ C k f k L + C k u k L . This proves the left inequality in (2.2). The right inequality can be easily obtained byusing Minkowski inequality, (2.1) and (2.4).In the sequel, we shall denote by C a generic constant independent of ε , and by C ε a generic constant that depends on ε . Moreover, for two positive numbers α ε and β ε ,the notation α ε . β ε means that there exists C > independent of ε such that for all ε ∈ (0 , ε ), α ε ≤ Cβ ε . F and W ε In this subsection, we give some results concerning the two nonlinear functions F and W ε defined in (1.11) and (1.8). Lemma 2.2
The function F is locally Lipschitz continuous on H ( M ) and on H ( M ) :for k = 1 or k = 2 , ∀ u , u ∈ H k ( M ) , sup t ∈ M k F ( t ; u ) − F ( t ; u ) k H k . ( k u k H k + k u k H k ) k u − u k H k (2.5) and, for all ε ∈ (0 , ε ) , the function W ε is locally Lipschitz continuous on H ∩ H ( S ) :there exists C ε > such that ∀ u , u ∈ H ∩ H ( S ) , sup t ∈ M k W ε ( t ; u ) − W ε ( t ; u ) k H ≤ C ε ( k u k H + k u k H ) k u − u k H . (2.6)6 oreover, for all u ∈ H ( M ) , one has sup t ∈ R k F ( t ; u ) k H . k u k H k u k H . (2.7) Moreover, for all
M > and for all ε ∈ (0 , ε ) , there exists a constant C ε ( M ) > suchthat, for all u ∈ H ∩ H ( S ) with k u k H ≤ M , one has sup t ∈ R k W ε ( t ; u ) k H ≤ C ε ( M ) (cid:0) k u k H ) (cid:1) k u k H . (2.8) Proof.
We recall that the group e − iτD x is unitary in L ( M ), H ( M ), H ( M ).Moreover, the group e − iτ H ε is unitary on L ( S ), H ( S ) and H ( S ) ∩ H ( S ), if these twolast spaces are respectively equipped with the norms k ( H ε u ) / k L and kH ε u k L , whichare equivalent to the H and H norms with ε -dependent constants, by (2.2).Let us prove (2.5). We let v j = e − itD x u j . We have e − itD x ( F ( t ; u ) − F ( t ; u )) = | v | v − | v | v = ( | v | + v v )( v − v ) + v ( v − v ) . (2.9)Then we have k F ( t ; u ) − F ( t ; u ) k H k ≤ k| v | ( v − v ) k H k + k v v ( v − v ) k H k + k v ( v − v ) k H k . We are led to estimate products of functions in H k in the form v v v so that, by usingthe Sobolev embedding H ( M ) ֒ → L ∞ ( M ), we get for all k ≥ k v v v k H k . k v k H k k v k H k k v k H k . Let us deal with (2.6). Here we let v j = e − it H ε u j and we estimate k W ε ( t ; u ) − W ε ( t ; u ) k H ≤ C ε k m − ε ( | v | v − | v | v ) k H ≤ C ′ ε k| v | v − | v | v k H where we have used the unitarity of e − it H ε for the graph norm of H ε . Then, the conclusionfollows by using the embeddings H ( S ) ֒ → L ∞ ( S ) and H ( S ) ֒ → W , ( S ). Let us now dealwith (2.7). We notice that, for all u ∈ H ( M ), k F ( t ; u ) k H . k| v | v k H , v = e − itD x u and k| v | v k H . k| v | v k L + k v ′ v k L + k v ′′ v k L . k v k H k v k H + k v ′ k L k v ′ k L ∞ k v k L ∞ + k v k L ∞ k v k H . k v k H k v k H = k u k H k u k H . Let us now deal with (2.8). We first recall the Gagliardo-Nirenberg inequality in dimension2 (see [16, p. 129]): k v k W , . k v k L ∞ k v k H . (2.10)The next Sobolev inequality is due to Br´ezis and Gallouet (see [8, Lemma 2]): there exists C ( M ) > v ∈ H ( R ) with k v k H ( R ) ≤ M , k v k L ∞ ≤ C ( M ) (cid:16) p log(1 + k v k H ) (cid:17) . (2.11)7y using continuous extensions from H ( S ) to H ( R ), one obtains the same inequalityfor u ∈ H ∩ H ( S ). Hence, for all v ∈ H ( S ) with k v k H ≤ M , k| v | v k H . k v k L + k ∆( v ) k L . k v k L + k v ∆ v k L + k v |∇ v | k L . k v k H + k v k L ∞ k ∆ v k L + k v k L ∞ k v k W , . C ( M ) (1 + log(1 + k v k H )) k v k H , where we used the Sobolev embedding H ( S ) ֒ → L ( S ), (2.10) and (2.11). Finally, for all u ∈ H ∩ H ( S ) with k u k H ≤ M , setting v = e − it H ε u we get k v k H ≤ C ε M and k W ε ( t ; u ) k H ≤ C ε k| v | v k H ≤ C ε ( M ) (1 + log(1 + k v k H )) k v k H ≤ C ′ ε ( M ) (1 + log(1 + k u k H )) k u k H . This proves (2.8) and the proof of the lemma is complete.
We will need the following easy lemma.
Lemma 2.3
For all u ∈ H ( M ) , we have k u k L ≤ k u k L k u ′ k L . (2.12) For all u ∈ H ( S ) , we have k u k L ≤ k u k L ( S ) k ∂ x u k L ( S ) k ∂ x u k L ( S ) . (2.13) Proof.
The proof of (2.12) is a consequence of the standard inequality, for f ∈ H ( M ), k f k L ∞ ≤ k f k L k f ′ k L . To prove (2.13), let us recall the following inequality Z S | f | d x d x ≤ k ∂ x f k L ( S ) k ∂ x f k L ( S ) , ∀ f ∈ W , ( S ) . Indeed, by density and extension, we may assume that f ∈ C ∞ ( R ) and we can write f ( x , x ) = Z x −∞ ∂ x f ( u, x ) d u, f ( x , x ) = Z x −∞ ∂ x f ( x , v ) d v. We get | f ( x , x ) | ≤ (cid:18)Z R | ∂ x f ( u, x ) | d u (cid:19) (cid:18)Z − | ∂ x f ( x , v ) | d v (cid:19) and it remains to integrate with respect to x and x . We apply this inequality to f = u ,use the Cauchy-Schwarz inequality and (2.13) follows.Now, we prove a technical lemma on the energy functional. Lemma 2.4
There exists ε ∈ (0 , ε ) such that, for all ε ∈ (0 , ε ) , the energy functionaldefined by (1.4) satisfies the following estimate. For all M > , there exists C > suchthat, for all ϕ ∈ H ( S ) with k ϕ k L ≤ M , one has E ε ( ϕ ) ≥ k ∂ x ϕ k L ( S ) + (cid:18) ε − C M (cid:19) k ∂ x ( Id − Π) ϕ k L ( S ) − C M − C M . (2.14)8 roof. Remark that E ε ( ϕ ) = 12 Z S |P ε, ϕ | d x d x + 12 ε (cid:10)(cid:0) D x − µ (cid:1) ϕ, ϕ (cid:11) L + 12 Z S V ε | ϕ | d x d x + λ Z S m − ε | ϕ | d x d x . Next, recalling that Π denotes the projection on the first eigenfunction e of D x , weeasily get k ϕ k L ( S ) ≤ k Π ϕ k L ( S ) + 8 k ( Id − Π ) ϕ k L ( S ) . We may write Π ϕ ( x , x ) = θ ( x ) e ( x ) so that, with (2.12), k Π ϕ k L ( S ) = γ Z M θ ( x ) d x ≤ γ k θ k L ( M ) k θ ′ k L ( M ) = 2 γ k Π ϕ k L ( S ) k ∂ x (Π ϕ ) k L ( S ) ≤ γ k ϕ k L ( S ) k ∂ x (Π ϕ ) k L ( S ) (2.15)where γ = R − e ( x ) d x , and thus, for all η ∈ (0 , k Π ϕ k L ( S ) ≤ η k Π ∂ x ϕ k L ( S ) + η − γ k ϕ k L ( S ) . Moreover, thanks to (2.13), we have, for all η ∈ (0 , k ( Id − Π ) ϕ k L ( S ) ≤ k ϕ k L ( S ) k ∂ x ( Id − Π ) ϕ k L ( S ) k ∂ x ( Id − Π ) ϕ k L ( S ) ≤ η k ∂ x ( Id − Π ) ϕ k L ( S ) + 4 η − k ϕ k L ( S ) k ∂ x ( Id − Π ) ϕ k L ( S ) . (2.16)Now we remark that, if µ = π denotes the second eigenvalue of D x on ( − ,
1) withDirichlet boundary conditions, we have (cid:10)(cid:0) D x − µ (cid:1) ϕ, ϕ (cid:11) L ( S ) ≥ (cid:18) − µ µ (cid:19) k ∂ x ( Id − Π ) ϕ k L ( S ) = 34 k ∂ x ( Id − Π ) ϕ k L ( S ) . (2.17)Therefore, using (2.1), (2.16), (2.17), using that k V ε k L ∞ ≤ C and that 0 ≤ m − ε ≤ Cε ,we obtain E ε ( ϕ ) ≥
12 (1 − Cε ) k ∂ x ϕ k L ( S ) − C k ϕ k L ( S ) + 38 ε k ∂ x ( Id − Π ) ϕ k L ( S ) − | λ | (1 + Cε ) (cid:16) η k ∂ x ϕ k L ( S ) + 4 η − k ϕ k L ( S ) k ∂ x ( Id − Π ) ϕ k L ( S ) (cid:17) − C k ϕ k L ( S ) ≥ k ∂ x ϕ k L ( S ) + (cid:18) ε − C k ϕ k L ( S ) (cid:19) k ∂ x ( Id − Π ) ϕ k L ( S ) − C k ϕ k L ( S ) − C k ϕ k L ( S ) where we has chosen η = − Cε | λ | (1+ Cε ) , which is positive for ε small enough. Proof of Lemma 1.3.
It is easy now to deduce Lemma 1.3 from Lemma 2.4. Indeed,consider a sequence φ ε satisfying Assumption 1.2 and introduce the constants ε ( M ) = min ε , (cid:18) C M (cid:19) / ! . (2.18)We deduce from (2.14) that, if ε ∈ (0 , ε ( M )), we have316 (cid:18) k ∂ x φ ε k L + 1 ε k ∂ x ( Id − Π ) φ ε k L (cid:19) ≤ k ∂ x φ ε k L + (cid:18) ε − C M (cid:19) k ∂ x ( Id − Π ) φ ε k L ≤ E ε ( φ ε ) + C M + C M ≤ M + C M + C M . (2.19)9he conclusion (1.5) stems from (2.19) by remarking also that k ∂ x Π φ ε k L = kh φ ε , e i L (( − , ∂ x e k L ≤ π k φ ε k L ≤ π M and by using the Poincar´e inequality k ( Id − Π ) φ ε k L ( M , H ( − , ≤ √ π π k ∂ x ( Id − Π ) φ ε k L . The aim of this subsection is to prove briefly the global well-posedness of the limit equation(1.9).
Proposition 3.1
Let θ ∈ H ( M ) . Then (1.9) with the Cauchy data θ admits a uniqueglobal solution θ ∈ C ( R + ; H ( M )) ∩ C ( R + ; H − ( M )) , that satisfies the following conser-vation laws k θ ( t ; · ) k L = k θ k L (mass) , (3.1) E ( θ ( t ; · )) = E ( θ ) (nonlinear energy) , (3.2) where E ( θ ) = 12 Z M (cid:18) | ∂ x θ | − κ ( x ) | θ | (cid:19) d x + λ γ Z M | θ | d x . Moreover, there exists a constant
C > such that ∀ t ∈ [0 , R + ) , k θ ( t ) k H ≤ C ( k θ k H + k θ k H ) (3.3) If θ ∈ H ( M ) then θ ∈ C ( R + ; H ( M )) ∩ C ( R + ; L ( M )) and ∀ t ∈ R + , k θ ( t ) k H ≤ k θ k H exp( C (1 + k θ k H ) t ) . (3.4) Proof.
We introduce F ( θ )( t ) = θ − i Z t (cid:26) e isD x (cid:18) − κ ( x )4 (cid:19) e − isD x θ ( s ; · ) + λγF ( s ; θ ( s ; · )) (cid:27) d s. For
M > , T >
0, we consider the complete space G T,M = { C ([0 , T ]; H ( M )) : ∀ t ∈ [0 , T ] , θ ( t ) ∈ B H ( θ , M ) } , where, for all Banach space X , B X ( θ , M ) denotes the closed ball in X , of radius M ,centered in θ . Let us briefly explain why F is a contraction from G T,M to G T,M as soonas T is small enough. Due to (2.5) and F ( t,
0) = 0, there exists
C >
M, T > , t ∈ [0 , T ] and θ ∈ G T,M , we have kF ( θ )( t ) − θ k H ≤ CT + CM T T ≤ T = M ( C + CM ) − . In the same way, there exists C >
M, T > , t ∈ [0 , T ] and u , u ∈ G T,M , kF ( θ )( t ) − F ( θ )( t ) k H ≤ ( CT + CM T ) sup t ∈ [0 ,T ] k θ ( t ) − θ ( t ) k H so that we choose T < T = ( C + CM ) − . It remains to apply the fixed point theoremfor any T ∈ (0 , (min( T , T )) and the conclusion is standard. By a continuation argument,it is clear moreover that the solution is global in time if it is bounded in H .The conservation of the L -norm (3.1) is obtained by considering the scalar product of(1.9) with θ and then taking the imaginary part. For the conservation of the energy (3.2),we consider the scalar product of the equation with ∂ t θ and take the real part. Let us nowprove (3.3). If λ ≥
0, it is an immediate consequence of the bounds on the energy and L -norm and the Sobolev embedding H ( M ) ֒ → L ( M ). Let us analyze the case λ < | λγ | Z M | θ | d x ≤ | λγ | k θ k L ( M ) k ∂ x θ k L ( M ) = | λγ | k θ k L ( M ) k ∂ x θ k L ( M ) so that, for all η ∈ (0 , | λγ | Z M | θ | d x ≤ | λγ | (cid:16) η − k θ k L ( M ) + η k ∂ x θ k L ( M ) (cid:17) . Choosing η such that η | λγ | < and using the bound on the energy, we get the uniformestimate (3.3). In particular, the solution θ is global in time.The local well-posedness in H ( M ) can be obtained by a similar procedure. To provethat the H solution is global in time, we simply use Assumption 1.1 on κ with (2.7): k θ ( t ) k H ≤ k θ k H + Z t C (1 + k θ ( s ) k H ) k θ ( s ) k H d s and conclude by using the H bound (3.3) and the Gronwall lemma. Let us now analyze the well-posedness of (1.6), but without any ε -control of the solution. Proposition 3.2
Let φ ε ∈ H ( S ) and let ε ∈ (0 , ε ) . Then, the following properties hold:(i) The problem (1.6) admits a unique maximal solution ϕ ε ∈ C ([0 , T ε max ); H ( S )) ∩ C ([0 , T ε max ); H − ( S )) , with T ε max ∈ (0 , + ∞ ] that satisfies the following conservation laws k ϕ ε ( t ; · ) k L = k φ ε k L (mass) , (3.5) E ε ( ϕ ε ( t ; · )) = E ε ( φ ε ) (nonlinear energy) , (3.6) where E ε is defined in (1.4) .(ii) There exists a constant C > such that, if ε < ε (given in Lemma 2.4) and if ε k φ ε k L ≤ C , then T ε max = + ∞ .(iii) If φ ε belongs to H ∩ H ( S ) , then ϕ ε ∈ C ([0 , T ε max ); H ∩ H ( S )) ∩ C ([0 , T ε max ); L ( S )) . roof. Step 1: local well-posedness in H . Let us fix ε ∈ (0 , ε ) and analyze in a firststep the well-posedness in H ∩ H ( S ). For φ ε ∈ H ∩ H ( S ), we consider the conjugateproblem (1.7) in its Duhamel form e ϕ ε ( t ) = φ ε − i Z t (cid:0) e is H ε ( V ε − ε − µ ) e − is H ε e ϕ ε ( s ) + λW ε ( s ; e ϕ ε ( s )) (cid:1) d s = W ε ( e ϕ ε )( t ) . For
M, T >
0, we consider the complete space e G T,M = { C ([0 , T ]; H ∩ H ( S )) : ∀ t ∈ [0 , T ] , θ ( t ) ∈ B H ( θ , M ) } . The application W ε is a contraction from e G T,M to e G T,M for T small enough. Indeed, asin Lemma 3.1 and thanks to (2.6), there exists C ε > T > M > t ∈ [0 , T ] and ϕ , ϕ ∈ e G T,M , kW ε ( ϕ )( t ) − ϕ k H ≤ C ε T + C ε T M , kW ε ( ϕ )( t ) − W ε ( ϕ )( t ) k H ≤ ( C ε T + C ε T M ) sup t ∈ [0 ,T ] k ϕ ( t ) − ϕ ( t ) k H , where we have again used the unitarity of e it H ε with respect to the graph norm of H ε and the equivalence between the graph norm of H ε and the H -norm, for each fixed ε .Therefore the Banach fixed point theorem insures the existence and uniqueness of a localin time solution of (1.7) and thus of (1.6) for each given ε ∈ (0 , ε ). The conservationlaws (3.5) and (3.6) are obtained similarly as (3.1) and (3.2). In fact, it is not difficultto deduce the existence of a maximal existence time T ε max , H ∈ (0 , + ∞ ] such that ϕ ε ∈ C ([0 , T ε max , H ); H ∩ H ( S )) ∩ C ([0 , T ε max , H ); L ( S )) and such that we have the alternative T ε max , H = + ∞ or lim t → T ε max , H k ϕ ε ( t ) k H = + ∞ . (3.7) Step 2: local well-posedness in H . Consider now a Cauchy data φ ε ∈ H ( S ). To prove thelocal well-posedness in H , we can proceed with the usual argument based on Trudinger’sinequality, explained in Section 3.6 of [9] and that we sketch here.We first recall the construction of a local weak solution ϕ ε ∈ L ∞ ([0 , T ); H ( S )) ∩ W , ∞ ((0 , T ); H − ( S )) of (1.6) by a standard regularization method. We approximate theCauchy data φ ε by a sequence φ ε ,n ∈ H ∩ H ( S ) converging to φ ε in H ( S ). Then weapply the well-posedness result in H proved in Step 1 to obtain a sequence of solutions ϕ εn ∈ C ([0 , T n ]; H ∩ H ( S )) ∩ C ([0 , T n ]; L ( S )) of (1.6) with ϕ εn (0; · ) = φ ε ,n , satisfying theconservation of mass and energy and where T n is chosen such that ∀ t ∈ [0 , T n ] , k ϕ εn ( t ) k H ≤ k φ ε k H . From (1.7) and the embedding H ( S ) ֒ → L ( S ), we deduce that for s, t ≤ T n k ϕ εn ( t ) − ϕ εn ( s ) k L = k e ϕ εn ( t ) − e ϕ εn ( s ) k L ≤ C ε | t − s | (cid:0) k φ ε k H + k φ ε k H (cid:1) and then, from the conservation of mass and energy, we get k ϕ εn ( t ) k H ≤ k φ ε ,n k L + k∇ φ ε ,n k L + (cid:13)(cid:13)(cid:13) V ε − µ ε (cid:13)(cid:13)(cid:13) L ∞ (cid:12)(cid:12) k ϕ εn ( t ) k L − k ϕ εn (0) k L (cid:12)(cid:12) + | λ | k m ε k L ∞ (cid:12)(cid:12) k ϕ εn ( t ) k L − k ϕ εn (0) k L (cid:12)(cid:12) ≤ k φ ε ,n ( t ) k H + C ε t (cid:0) k φ ε k H + k φ ε k H (cid:1) . T >
0, independent of n (but ofcourse depending on ε ), such T n ≥ T . The sequence ( ϕ εn ) n ∈ N being bounded in L ∞ ([0 , T ); H ( S )) ∩ W , ∞ ((0 , T ); H − ( S )), we can use the local compactness of H into L to extract a subsequence that converges to a weak solution of (1.3). This weak solutionsatisfies in fact k ϕ ε ( t ) k L = k φ ε k L and the inequality E ε ( ϕ ε ( t )) ≤ E ε ( φ ε ).Next, by using Ogawa’s method [17] (see Theorem 3.6.1 in [9]), we prove the uniquenessof the weak solution ϕ ε ∈ L ∞ ([0 , T ); H ( S )) ∩ W , ∞ ([0 , T ); H − ( S )). This crucial propertyrelies on an L estimate and Trudinger’s inequality.A consequence of the uniqueness property is that the NLS equation (1.3) is time-reversible, so one has E ε ( φ ε ) ≤ E ε ( ϕ ε ( t )) and then the energy is exactly conserved: theweak solution ϕ ε satisfies (3.6). From this, one deduces (see Theorem 3.3.9 of [9]) that ϕ ε ∈ C ([0 , T ]; H ( S )) ∩ C ([0 , T ); H − ( S )), that the solution depends continuously fromthe initial data, and that the exists a maximal existence time T ε max , H ∈ (0 , + ∞ ] with thealternative T ε max , H = + ∞ or lim t → T ε max , H k ϕ ε ( t ) k H = + ∞ . (3.8) Step 3: equality of the maximal existence times . Let φ ε ∈ H ∩ H ( S ). From the previoustwo steps, there exists a maximal existence time T ε max , H (resp. T ε max , H ) of the H (resp. H ) solution of (1.3). Moreover, by (3.7) and (3.8), it is already obvious that T ε max , H ≤ T ε max , H . Let us prove by a contradiction argument that we have in fact the equality ofthese two maximal existence times: T ε max , H = T ε max , H = T ε max . (3.9)Assume that T ε max , H < T ε max , H . Then ϕ ε is bounded by a constant M ε in H norm on[0 , T ε max , H ] and one has lim t → T ε max , H k ϕ ε ( t ) k H = + ∞ . (3.10)From (1.6) and (2.8) we get k ∂ t ϕ ε k H ≤ C ε (cid:0) k ϕ ε ( t ; · ) k H ) (cid:1) k ϕ ε ( t ; · ) k H . It remains to use an argument `a la
Gronwall from [8]. Given a Banach space G , let usconsider a function ϕ ∈ C ([0 , T ∗ ) , G ) such that for, t ∈ [0 , T ∗ ), k ϕ ′ ( t ) k ≤ C (1 + log(1 + k ϕ ( t ) k )) k ϕ ( t ) k . We easily get k ϕ ( t ) k ≤ F ( t ) , with F ( t ) = k ϕ k + C Z t (1 + log(1 + k ϕ ( τ ) k )) k ϕ ( τ ) k d τ and dd t F ( t ) = C (1 + log(1 + k ϕ ( t ) k )) k ϕ ( t ) k ≤ C (1 + log(1 + F ( t ))) F ( t ) , so that dd t log (1 + log(1 + F ( t ))) ≤ C. Consequently, we find an estimate of the form k ϕ ( t ) k ≤ F ( t ) ≤ e ae bt . ϕ ε with G = H ( S ), one gets a bound for the H norm of ϕ ε on the interval [0 , T ε max , H ), which is a contradiction with (3.10). The proof of (3.9) iscomplete. Step 4: global existence for ε small enough. Let M ε = k ϕ ε k L = k φ ε k L . By Lemma 2.4,for ε ∈ (0 , ε ), one has14 k ∂ x ϕ ε k L + (cid:18) ε − C M ε (cid:19) k ∂ x ( Id − Π) ϕ ε k L ≤ E ε ( ϕ ε ) + C M ε + C M ε = E ε ( φ ε ) + C M ε + C M ε . Hence, if εM ε ≤ ( C ) / , this inequality provides an H bound for ϕ ε and, by (3.8), wehave T max = + ∞ . This section is devoted to the proof of our two main theorems. As for the study of theCauchy problem in Subsection 3.2, we shall start with the case of H initial data, which issimpler than the case of data in the energy space H requiring an additional regularizationargument. Consider a sequence of Cauchy data φ ε ∈ H ∩ H ( S ) satisfying (1.13) and let θ ε ( t ) and ϕ ε ( t ) be respectively the solutions of (1.9) and (1.6). Items (i) and (ii) of Theorem 1.5are direct consequences of Propositions 3.1 and 3.2. Notice that ε ( M ) is defined oncefor all by (2.18).Let us prove Item (iii) . To this aim, we first prove that Assumption 1.2 is satisfied,i.e. that the energy of φ ε is bounded from above. From (1.13) and (2.2), one gets (cid:13)(cid:13) D x φ ε (cid:13)(cid:13) L + 1 ε (cid:13)(cid:13)(cid:0) D x − µ (cid:1) φ ε (cid:13)(cid:13) L + k φ ε k L ≤ C (1 + M ) . (4.1)Hence, by using (2.1), we have kP ε, φ ε k L ≤ C, k D x φ ε k L − µ k φ ε k L ≤ Cε . (4.2)Moreover, from the proof of Lemma 2.4, we write k φ ε k L ≤ C k Π φ ε k L + C k ( Id − Π ) φ ε k L ≤ C k φ ε k L k Π ∂ x φ ε k L + C k φ ε k L k ( Id − Π ) ∂ x φ ε k L k ∂ x ( Id − Π ) φ ε k L ≤ C k φ ε k L k ∂ x φ ε k L + Cε k φ ε k L k ∂ x φ ε k L ≤ C (4.3)where we used (4.2) and (2.17). Hence, (4.1), (4.2) and (4.3) yield E ε ( φ ε ) ≤ M , forsome M > ε : the sequence of Cauchy data satisfies Assumption 1.2.Therefore, by conservation of mass and energy, for all t ≥
0, the sequence ϕ ε ( t ) alsosatisfies Assumption 1.2. We can then apply Lemma 1.3 to ϕ ε ( t ): for all t ≥ ε ≤ ε ( M ), we have k ϕ ε ( t ) k H ( S ) ≤ C and k ( Id − Π ) ϕ ε ( t ) k L ( M , H ( − , ≤ C ε. (4.4)14et us now deal with the NLS equation (1.6) projected on e ( x ): setting u ε = h ϕ ε ( t ) , e i L (( − , , we get i∂ t u ε = D x u ε − κ ( x )4 u ε + λγ | u ε | u ε + h R ε ( ϕ ε ) , e i L (( − , + h S ε ( ϕ ε ) , e i L (( − , , (4.5)with u ε (0; · ) = θ ε and where, for all ϕ ∈ H ( S ), we have denoted R ε ( ϕ ) = m − / ε D x (cid:16) m − ε D x ( m − / ε ϕ ) (cid:17) − D x ϕ − κ (cid:0) m − ε − (cid:1) ϕ (4.6) S ε ( ϕ ) = λm − ε | ϕ | ϕ − λ | Π ϕ | Π ϕ. (4.7)Since θ ε = θ ε ( t, x ) and e = Π e , we have k ϕ ε ( t ) − θ ε ( t ) e k L ( S ) = k Π ( ϕ ε ( t ) − θ ε ( t ) e ) k L ( S ) + k ( Id − Π ) ϕ ε ( t ) k L ( S ) ≤ k u ε ( t ) − θ ε ( t ) k L ( M ) + Cε by (4.4). Thus, to deduce (1.14), it is enough to prove the following property: for all T >
0, there exist C T > ε T ∈ (0 , ε ( M )) such that, for all ε < ε T , we have ∀ t ∈ [0 , T ] , k u ε ( t ) − θ ε ( t ) k L ( M ) ≤ C T ε. (4.8)This fact will be a consequence of the following lemmas, that we prove further. Lemma 4.1
For all ϕ ∈ H ( S ) , we have the interpolation estimate k ϕ k L ∞ . k ϕ k / L k ϕ k / H . (4.9) Lemma 4.2
Let ϕ ∈ H ( S ) , then, for all ε ∈ (0 , ε ) , k R ε ( ϕ ) k L . ε k ϕ k H and k S ε ( ϕ ) k L . k ϕ k L k ϕ k H k ( Id − Π ) ϕ k L + ε k ϕ k L k ϕ k H , where R ε and S ε are defined by (4.6) and (4.7) . Lemma 4.3
Let
T > , let ε ∈ (0 , ε ) and let ϕ ε ∈ C ([0 , T ]; H ∩ H ( S )) ∩ C ([0 , T ]; L ( S )) be solution of (1.6) . Assume moreover that we have an L ∞ bound k ϕ ε k L ∞ ([0 ,T ] ×S ) ≤ M ,with M independent of ε . Then there exists C M,T > such that, for all t ∈ [0 , T ] , we have (cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε ( t ) (cid:13)(cid:13)(cid:13) L + k ϕ ε ( t ) k L ≤ C M,T (cid:16)(cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε (0) (cid:13)(cid:13)(cid:13) L + k ϕ ε (0) k L (cid:17) . End of proof of Theorem 1.5.
In this proof, C denotes a generic constant that onlydepends on the two upper bounds M and M in Assumption 1.2. Consider the quantity M = 2 sup ε ∈ (0 ,ε ( M )) sup t ≥ k θ ε ( t ) k L ∞ ( M ) ≤ C sup ε ∈ (0 ,ε ( M )) sup t ≥ k θ ε ( t ) k H ( M ) ≤ C + C sup ε ∈ (0 ,ε ( M )) k θ ε k H ( M ) ≤ C + C sup ε ∈ (0 ,ε ( M )) k φ ε k H ( S ) < + ∞ , H ( M ) ֒ → L ∞ ( M ), the estimate (3.3), Cauchy-Schwarz and the uniform bound (4.4). Next, for ε ∈ (0 , ε ( M )), by (4.4), (4.9) and (4.1)(which yields a uniform H bound on φ ε ), we have k φ ε k L ∞ ≤ k φ ε − θ ε e k L ∞ + k θ ε e k L ∞ = k ( Id − Π ) φ ε k L ∞ + k θ ε k L ∞ ≤ C k ( Id − Π ) φ ε k / L k ( Id − Π ) φ ε k / H + k θ ε k L ∞ ≤ Cε / (1 + M ) / + M . (4.10)Hence, for ε ≤ M / (16 C (1 + M )), one has k φ ε k L ∞ ≤ M/ k ϕ ε ( t ) k L ∞ , we know that (4.11)belongs to (0 , + ∞ ]. By a continuation argument, it is clear moreover thatif T ε < + ∞ then k ϕ ε ( T ε ) k L ∞ = M. (4.12)Let us fix T >
0. For all t ≤ min( T, T ε ), one has k ϕ ε ( t ) k L ∞ ≤ M so, from Lemma 4.3,from (2.2) and from (1.13), we deduce that (cid:13)(cid:13) D x ϕ ε ( t ) (cid:13)(cid:13) L + 1 ε (cid:13)(cid:13)(cid:0) D x − µ (cid:1) ϕ ε ( t ) (cid:13)(cid:13) L + k ϕ ε ( t ) k L ≤≤ C (cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε ( t ) (cid:13)(cid:13)(cid:13) L + C k ϕ ε ( t ) k L ≤ C M,T (cid:16)(cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) φ ε (cid:13)(cid:13)(cid:13) L + k φ ε k L (cid:17) ≤ C M,T (1 + M ) . This yields the H estimate k ϕ ε ( t ) k H ≤ (cid:13)(cid:13) D x ϕ ε ( t ) (cid:13)(cid:13) L + (cid:13)(cid:13)(cid:0) D x − µ (cid:1) ϕ ε ( t ) (cid:13)(cid:13) L + (1 + µ ) k ϕ ε ( t ) k L ≤ CC M,T (1 + M ) . (4.13)We can now apply Lemma 4.2, together with (4.4) and (4.13) and, for all t ≤ min( T, T ε ),obtain k R ε ( ϕ ε ( t )) k L + k S ε ( ϕ ε ( t )) k L ≤ ε CC M,T (1 + M ) . (4.14)Let us define e u ε ( t ) = e itD x u ε ( t ) and e θ ε ( t ) = e itD x θ ε ( t ) and write the equation satisfiedby the difference w ε ( t ) = e u ε ( t ) − e θ ε ( t ): i∂ t w ε = e itD x (cid:18) − κ ( x )4 (cid:19) e − itD x w ε + λγ (cid:16) F ( t ; e u ε ) − F ( t ; e θ ε ) (cid:17) + e itD x (cid:0) h R ε ( ϕ ε ) , e i L (( − , + h S ε ( ϕ ε ) , e i L (( − , (cid:1) (4.15)with w ε (0) = 0. Hence, (2.9) together with the Sobolev embedding H ( M ) ֒ → L ∞ and thebounds (4.4), (3.3), (4.14), yield k ∂ t w ε ( t ) k L ≤ C k w ε ( t ) k L + ε CC M,T (1 + M ) , w ε (0) = 0 . The Gronwall lemma gives then, for all t ≤ min( T, T ε ), k u ε ( t ) − θ ε ( t ) k L = k w ε ( t ) k L ≤ ε CC M,T (1 + M ) e CT . (4.16)16e have proved the estimate (4.8) for all t ≤ min( T, T ε ), and the proof of Theorem 1.5will be complete if we show that there exists ε T > ε ∈ (0 , ε T ), we have T ε ≥ T .Let us proceed by contradiction and assume that T ε < T . Apply as above the interpo-lation estimation (4.9) at time T ε : k ϕ ε ( T ε ) k L ∞ ≤ k ϕ ε ( T ε ) − θ ε ( T ε ) e k L ∞ + k θ ε ( T ε ) e k L ∞ ≤ C k ϕ ε ( T ε ) − θ ε ( T ε ) e k / L k ϕ ε ( T ε ) − θ ε ( T ε ) e k / H + k θ ε ( T ε ) k L ∞ ≤ C ( k u ε ( T ε ) − θ ε ( T ε ) k L + k ( Id − Π ) ϕ ε ( T ε ) k L ) / ( k ϕ ε ( T ε ) k H + k θ ε ( T ε ) e k H ) / + k θ ε ( T ε ) k L ∞ ≤ ε / C (1 + C M,T (1 + M ) e CT ) + M . where we used (4.16), (4.4), (4.13), (3.4) and the definition of M . Now we choose ε T = min ε ( M ) , (cid:0) C (1 + C M,T (1 + M ) e CT ) (cid:1) − (cid:18) M (cid:19) ! and obtain that, for all ε ∈ (0 , ε T ), k ϕ ε ( T ε ) k L ∞ ≤ M < M. Since T ε < + ∞ , this contradicts (4.12). The proof of Theorem 1.5 is complete. Proof of Lemma 4.1.
The interpolation estimate follows from the Sobolev embedding ∀ u ∈ H ( R ) , k u k L ∞ . k u k L + k D x u k L + k D x u k L (4.17)by a simple homogeneity argument. Indeed, for ϕ ∈ H ( R ), non zero, inserting thefunction u λ defined by u λ ( x ) = ϕ ( λx ) with λ = k ϕ k / L k ϕ k − / H in (4.17) yields (4.9). Proof of Lemma 4.2.
The estimate on R ε ( ϕ ) is immediate as soon as one noticesthat, for all ε < ε , we have m ε ≥ − ε k κ k L ∞ > k m ε − k W , ∞ ( S ) ≤ ε k κ k W , ∞ ( M ) ≤ Cε. (4.18)The estimate on S ε follows from (4.18), from | ϕ | ϕ − | Π ϕ | Π ϕ = ( | Π ϕ | + ϕ Π ϕ ) ( Id − Π ) ϕ + ϕ ( Id − Π ) ϕ. (4.19)and from the interpolation estimate (4.9). Proof of Lemma 4.3.
Let us consider the time derivative of (1.6): if χ ε = ∂ t ϕ ε , then i∂ t χ ε = (cid:0) H ε − ε − µ (cid:1) χ ε + V ε χ ε + 2 m − ε λ | ϕ ε | χ ε + λm − ε ( ϕ ε ) χ ε . Take the L ( S ) scalar product with χ ε and then the imaginary part to get,12 dd t k χ ε k L ≤ C Z S | ϕ ε | | χ ε | d x d x ≤ CM k χ ε k L , L ∞ bound of ϕ ε on [0 , T ]. It remains to applythe Gronwall lemma: k ∂ t ϕ ε ( t ) k L ≤ k ∂ t ϕ ε (0) k L e CM t = (cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε (0) + (cid:0) V ε + λm − ε | ϕ ε (0) | (cid:1) ϕ ε (0) (cid:13)(cid:13)(cid:13) L e CM t , ≤ (cid:16)(cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε (0) (cid:13)(cid:13)(cid:13) L + C (1 + M ) k ϕ ε (0) k L (cid:17) e CM t , (4.20)where we used the equation (1.6) to identify and bound ∂ t ϕ ε (0). We can now deduce an H bound of ϕ ε . Indeed, by (1.6), we have (cid:13)(cid:13)(cid:13)(cid:16) H ε − µ ε (cid:17) ϕ ε ( t ) (cid:13)(cid:13)(cid:13) L = (cid:13)(cid:13) i∂ t ϕ ε − V ε ϕ ε − λm − ε | ϕ ε | ϕ ε (cid:13)(cid:13) L ≤ k ∂ t ϕ ε ( t ) k L + C (1 + M ) k φ ε ( t ) k L and the conclusion follows by using (4.20) and k φ ε ( t ) k L = k φ ε (0) k L . Consider a sequence of Cauchy data φ ε ∈ H ( S ) satisfying Assumption 1.2. Let θ ε ( t ) and ϕ ε ( t ) be respectively the global solutions of (1.9) and (1.6) (for ε ∈ (0 , ε ( M )). Items (i) and (ii) of Theorem 1.4 stem from Propositions 3.1 and 3.2.Let us prove Item (iii) . Since E ε ( ϕ ε ( t )) = E ε ( φ ε ) ≤ M and by conservation of the L norm, we deduce as in the proof of Theorem 1.5 that the estimates (4.4) hold true, for all t ≥ φ ε,η = Π (1 + ηεD x ) − / φ ε , (4.21)where η > ε that will be chosen later. It is clear thatwe have φ ε,η ∈ H ∩ H ( S ), with the following estimates: k φ ε,η k L ≤ k φ ε k L , k D x φ ε,η k L ≤ k D x φ ε k L , k D x φ ε,η k L ≤ ( ηε ) − / k φ ε k H , and that ( D x − µ ) φ ε,η = 0. In particular, we deduce from (2.1), (2.2) and (2.15) that k φ ε,η k L ≤ M , E ε ( φ ε,η ) ≤ CM , (cid:13)(cid:13)(cid:13) ( H ε − µ ε ) φ ε,η (cid:13)(cid:13)(cid:13) L ≤ C ( ηε ) − / . (4.22)Let ϕ ε,η ( t ) be the H ∩ H solution of (1.6) with the Cauchy data φ ε,η and let θ ε,η ( t ) bethe solution of (1.9) with the Cauchy data h φ ε,η , e i L (( − , . By Proposition 3.2 and since ε < ε ( M ) and k φ ε,η k L ≤ M , this solution ϕ ε,η ( t ) is defined for all t ∈ R + . Moreover,from E ε ( ϕ ε,η ( t )) = E ε ( φ ε,η ) and the first two inequalities of (4.22), one gets again fromLemma 1.3 that k ϕ ε,η ( t ) k H ( S ) ≤ C and k ( Id − Π ) ϕ ε,η ( t ) k L ( M , H ( − , ≤ Cε. (4.23)
Step 1: estimating ϕ ε,η ( t ) − θ ε,η ( t ) e . Let us reproduce the series of estimates obtainedin the proof of Theorem 1.5. We have to take care to the fact that, here, the H norm of ϕ ε,η ( t ) is not uniformly bounded but is of order ε − / . Defining M by M = 2 sup ε ∈ (0 ,ε ( M )) sup t ≥ k θ ε,η ( t ) k L ∞ ( M ) < + ∞ , φ ε,η ( x , x ) = θ ε,η ( x ) e ( x ), we get k φ ε,η k L ∞ = k θ ε,η k L ∞ ≤ M . This enables to define T ε > T ε = max { t ≥ s ∈ [0 , t ] , k ϕ ε,η ( s ) k L ∞ ≤ M } . Next, Lemma 4.3 yields k ϕ ε,η k H ≤ CC M,T (1 + ( ηε ) − / )and (4.14), (4.16) are now respectively replaced by k R ε ( ϕ ε,η ) k L + k S ε ( ϕ ε,η ) k L ≤ ε CC M,T (1 + ( ηε ) − / )and, if we define u ε,η = h ϕ ε,η ( t ) , e i L (( − , , by k u ε,η ( t ) − θ ε,η ( t ) k L ≤ ε CC M,T (1 + ( ηε ) − / ) e CT ≤ C T ε / . (4.24)This bound (4.24) holds true for all t ≤ min( T, T ε ). To show that T ε ≥ T , we estimateagain the L ∞ norm of ϕ ε,η by interpolation, and obtain k ϕ ε,η ( T ε ) k L ∞ ≤ k θ ε,η ( T ε ) k L ∞ ++ C ( k u ε,η ( T ε ) − θ ε,η ( T ε ) k L + k ( Id − Π ) ϕ ε,η ( T ε ) k L ) / ( k ϕ ε,η ( T ε ) k H + k θ ε,η ( T ε ) e k H ) / ≤ M C ( η − / ε / + ε / )( η − / ε − / + 1) ≤ M Cη − / + Cη − / ε / + Cε / . Hence we first choose η > Cη − / ≤ M . Then we choose ε T > Cη − / ε / T + Cε / T ≤ M . Therefore, for all ε < ε T , we have k ϕ ε,η ( T ε ) k L ∞ ≤ M/
4, whichis sufficient to conclude as above that T ε ≥ T . Finally, by (4.24) and (4.23), we haveobtained that, for all t ≤ T , k ϕ ε,η ( t ) − θ ε,η ( t ) e k L ≤ k u ε,η ( t ) − θ ε,η ( t ) k L + k ( Id − Π ) ϕ ε,η ( t ) k L ≤ Cε / . (4.25) Step 2: stability estimates.
Let us now estimate the two differences θ ε ( t ) − θ ε,η ( t ) and ϕ ε ( t ) − ϕ ε,η ( t ). By (4.21), we have k φ ε − φ ε,η k L ≤ k ( Id − Π ) φ ε k L + k Π (1 − (1 + ηεD x ) − / ) φ ε k L ≤ Cε + ηε k D x (1 + (1 + ηεD x ) / ) − (1 + ηεD x ) − / φ ε k L ≤ Cε + η / ε / k D x φ ε k L ≤ Cε / . (4.26)The two functions θ ε ( t ) and θ ε,η ( t ) satisfy the same equation (1.9), respectively withthe initial data h φ ε , e i L (( − , and h φ ε,η , e i L (( − , . Hence, since θ ε ( t ) and θ ε,η ( t ) areuniformly bounded in L ∞ ( M ) (because they are bounded in H ( M )), a standard stabilityestimate in L yields, with the Gronwall lemma, k θ ε ( t ) − θ ε,η ( t ) k L ≤ k φ ε − φ ε,η k L e Ct ≤ Cε / e Ct . (4.27)To estimate the difference z ( t ) = ϕ ε ( t ) − ϕ ε,η ( t ), we have to proceed in a different way, since ϕ ε ( t ) does not belong to L ∞ ( S ). Recall that ϕ ε ( t ) and ϕ ε,η ( t ) satisfy the same equation191.6), with the initial data φ ε (0) = φ ε and ϕ ε,η (0) = φ ε,η . Hence, the standard L estimateon the difference z ε ( t ) leads to dd t k z ε k L ≤ | λ | (cid:13)(cid:13) | ϕ ε | | ϕ ε | − | ϕ ε,η | | ϕ ε,η | (cid:13)(cid:13) L ≤ k e S ε ( ϕ ε ) k L + k e S ε ( ϕ ε,η ) k L + k| Π ϕ ε | Π ϕ ε − | Π ϕ ε,η | Π ϕ ε,η k L (4.28)where e S ε ( ϕ ) = λ | ϕ | ϕ − λ | Π ϕ | Π ϕ. We now estimate S ε ( ϕ ε ) by coming back to (4.19). By the H¨older and Minkowski inequal-ities and using that k Π ϕ k L p . k ϕ k L p , we get k e S ε ( ϕ ε ) k L ≤ C k ϕ ε k / L k ( Id − Π ) ϕ ε k / L . Then, with a Sobolev embedding and (4.4), we deduce e S ε ( ϕ ε ) k L ≤ C k ϕ ε k / H k ( Id − Π ) ϕ ε k / L ≤ Cε / . Similarly, we also obtain k e S ε ( ϕ ε,η ) k L ≤ Cε / thanks to (4.23). The last term in (4.28)is easy to estimate, since for all ϕ ∈ H ( S ), one has, by Sobolev embedding in dimensionone, k Π ϕ k L ∞ . k ϕ k H . Thus, by using again (4.4) and (4.23), we get, k| Π ϕ ε | Π ϕ ε − | Π ϕ ε,η | Π ϕ ε,η k L ≤ C ( k Π ϕ ε k L ∞ + k Π ϕ ε,η k L ∞ ) k z ε k L ≤ C ( k ϕ ε k H + k ϕ ε,η k H ) k z ε k L ≤ C k z ε k L . Hence, (4.28) and (4.26) yield dd t k z ε ( t ) k L ≤ Cε / + C k z ε ( t ) k L , k z ε (0) k L ≤ Cε / and we conclude by the Gronwall lemma that k ϕ ε ( t ) − ϕ ε,η ( t ) k L = k z ε ( t ) k L ≤ Cε / e Ct . (4.29)Finally, from (4.25), (4.27) and (4.29), one deduces the error estimate (1.12). The proofof Theorem 1.4 is complete. Acknowledgements
We wish to thank Christof Sparber for helpful discussions. F.M. acknowledges support bythe ANR-FWF Project Lodiquas (ANR-11-IS01-0003) and by the ANR Project Moonrise(ANR-14-CE23-0007-01).
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