Damped wave dynamics for a complex Ginzburg-Landau equation with low dissipation
aa r X i v : . [ m a t h . A P ] M a r DAMPED WAVE DYNAMICS FOR A COMPLEX GINZBURG-LANDAUEQUATION WITH LOW DISSIPATION
EVELYNE MIOT
Abstract.
We consider a complex Ginzburg-Landau equation on R N , corresponding to aGross-Pitaevskii equation with a small dissipation term. We study an asymptotic regime forlong-wave perturbations of constant maps of modulus one. We show that such solutions nevervanish on R N and we derive a damped wave dynamics for the perturbation. Our results areobtained in the same spirit as those by Bethuel, Danchin and Smets for the Gross-Pitaevskiiequation [2]. Introduction
We consider a complex Ginzburg-Landau equation ∂ t Ψ = ( κ + i )[∆Ψ + Ψ(1 − | Ψ | )] , (C)where Ψ = Ψ( t, x ) : R + × R N → C , with N ≥
1, is a complex-valued map and where 0 < κ < R N , so that wemay write them into the form Ψ = r exp( iφ ) . Secondly, we assume that ( r , ∇ φ ) is a long-wave perturbation of (1 , ε > r , ∇ φ ) through the change of variables r ( t, x ) = 1 + ε √ a ε ( εt, εx )2 ∇ φ ( t, x ) = εu ε ( εt, εx ) , (1.1)where ( a ε , u ε ) belongs to C ( R + , H s +1 × H s ), with s ≥
2, and satisfies suitable bounds.Our objective is two-fold. First, to define ( a ε , u ε ) we wish to determine how long a solutioninitially given by (1.1) does not vanish on R N . Our second purpose is to investigate thedynamics of ( a ε , u ε ) when ε vanishes and κ is small. This asymptotic dynamics depends onthe balance between the amount κ of dissipation in Eq. (C) and the size ε of the perturbation;to characterize this balance we introduce the ratio ν ε = κε . According to (C) we obtain the equations for the perturbation ( a ε , u ε ) ( ∂ t a ε + √ u ε + 2 ν ε − κε ∆ a ε = f ε ( a ε , u ε ) ∂ t u ε + √ ∇ a ε − κε ∆ u ε = g ε ( a ε , u ε ) , (1.2) Date : November 23, 2018. where f ε and g ε are given by f ε ( a ε , u ε ) = √ κ (cid:16) − |∇ ρ a | − ρ a | u ε | − a ε (cid:17) − ε div( a ε u ε )g ε ( a ε , u ε ) = κε ∇ (cid:18) ∇ ρ a ρ a · u ε (cid:19) + 2 ε ∇ ∆ ρ a ρ a − εu ε · ∇ u ε , (1.3)with ρ a ( t, x ) = 1 + ε √ a ε ( t, x ) . Our first result establishes that if the initial perturbation is not too large, the solution Ψnever exhibits a zero so that (1.1) does hold for all time.
Theorem 1.1.
Let s be an integer such that s > N/ . There exist positive numbers K ( s, N ) , K ( s, N ) and < κ ( s, N ) < , depending only on s and N , satisfying the followingproperty.Let < κ ≤ κ ( s, N ) . For < ε ≤ , let ( a ε , ϕ ε ) ∈ H s +1 ( R N ) such that M := k ( a ε , u ε ) k H s + ε k a ε k H s +1 + k ϕ ε k L ≤ min( ν ε , κ − , ε − ) K ( s, N ) , where u ε = 2 ∇ ϕ ε .Then Eq. (1.2) - (1.3) has a unique global solution ( a ε , u ε ) in C ( R + , H s +1 × H s ) such that ( a ε , u ε )(0) = ( a ε , u ε ) . Moreover k ( a ε , u ε ) k L ∞ ( H s ) + ε k a ε k L ∞ ( H s +1 ) ≤ K ( s, N ) M . Finally, if Ψ denotes the corresponding solution to Eq. (C) , we have for all t ≥ (cid:13)(cid:13) | Ψ( t ) | − (cid:13)(cid:13) ∞ < . Remark 1.1.
Fixing κ = κ and ε = ε , Theorem 1.1 entails that for initial data Ψ ( x ) = (cid:0) a ( x ) (cid:1) / exp( i ˜ ϕ ( x )) , with k (˜ a , ˜ ϕ ) k H s +1 ≤ C , where C only depends on s and N , the corresponding solution Ψ to Eq. (C) remains bounded and bounded away from zero for all time. Remark 1.2.
For all < ε ≤ ε and < κ ≤ κ satisfying ε ≤ κ , so that ν ε ≥ , Theorem1.1 allows to handle initial data Ψ ε ( x ) = (cid:18) ε √ a ( εx ) (cid:19) / exp( iϕ ( εx )) , (1.4) where ( a , ϕ ) ∈ H s +1 ( R N ) does not depend on ε , so that M is constant, and where M issmaller than a number depending only on s and N . Once the question of existence for ( a ε , u ε ) has been settled, our next task is to determinea simplified system of equations to describe its asymptotic dynamics. From now on we focuson a regime with low dissipation, namely we further assume that κ = κ ( ε ) and lim ε → κ ( ε ) = 0 . Given by Theorem 3.1 below.
AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 3
In view of (1.3), this is a natural ansatz in order to treat the second members f ε and g ε asperturbations in the limit ε →
0. Eq. (1.2) then formally reduces to a damped wave equation ( ∂ t a + √ u + 2 ν ε a = 0 ∂ t u + √ ∇ a = 0 , (1.5)with propagation speed equal to √ ν ε .As a consequence of Theorem 1.1 we can compare the solution ( a ε , u ε ) to the one of thelinear damped wave equation (1.5) with loss of three derivatives. Theorem 1.2.
Let s be an integer such that s > N/ . Let ( a ε , ϕ ε ) ∈ H s +1 ( R N ) satisfythe assumptions of Theorem 1.1. Let u ε = 2 ∇ ϕ ε .We denote by ( a ℓ , u ℓ ) ∈ C ( R + , H s +1 × H s ) the solution of Eq. (1.5) with initial datum ( a ε , u ε ) .There exists a constant K ( s, N ) depending only on s and N such that for all t ≥ k ( a ε − a ℓ , u ε − u ℓ )( t ) k H s − ≤ K ( s, N )( εκt ) / max(1 , ν − ε )( M + M ) , where M is defined in Theorem 1.1. In particular, for initial data given by (1.4), the approximation by the damped wave equa-tion is optimal when κ and ε are comparable. Moreover, Theorem 1.2 yields a correct approx-imation up to times of order C ( κε ) − . In order to handle larger times, it is helpful to takeinto account the linear parabolic terms in (1.2): ( ∂ t a + √ u + 2 ν ε a − κε ∆ a = 0 ∂ t u + √ ∇ a − κε ∆ u = 0 . (1.6)Our next result presents uniform in time comparison estimates with the solution of Eq. (1.6)for high order derivatives. Theorem 1.3.
Let s be an integer such that s > N/ . Let ( a ε , ϕ ε ) ∈ H s +1 ( R N ) satisfythe assumptions of Theorem 1.1.We denote by ( a ℓ , u ℓ ) ∈ C ( R + , H s +1 × H s ) the solution of Eq. (1.6) with initial datum ( a ε , u ε ) .There exists a constant K ( s, N ) depending only on s and N such that • k ( a ε − a ℓ , u ε − u ℓ ) k L ∞ ( H s − ) ≤ K ( s, N ) (cid:0) κ max(1 , ν − ε ) M + ε max(1 , ν − ε ) M (cid:1) , • k ( a ε − a ℓ , u ε − u ℓ ) k L ∞ ( H s − ) ≤ K ( s, N ) (cid:16) max(1 , ν − ε ) (cid:0) max( κ, ε ) + ν − ε (cid:1) M + ν − ε M (cid:17) , • k ( a ε − a ℓ , u ε − u ℓ ) k L ∞ ( H s ) ≤ K ( s, N ) (cid:16) ( ν − ε max(1 , ν − ε ) + κ − ) M + κ − M (cid:17) . Finally, for all t ≥ • k ( a ε − a ℓ , u ε − u ℓ )( t ) k H s − ≤ K ( s, N )( εκt ) / (cid:0) max(1 , ν − ε ) M + ν − ε M (cid:1) , • k ( a ε − a ℓ , u ε − u ℓ )( t ) k H s − ≤ K ( s, N )( εκ − t ) / M . We come back to initial data given by (1.4). Since κ − diverges when ε →
0, Theorem1.3 does not provide a correct approximation for s -order derivatives. However, Eq. (1.6)yields a satisfactory large in time approximation for the derivatives of order s − ν − ε vanishes with ε . In fact, the corresponding comparison estimate is optimal whenever κ and √ ε are proportional. This is due to the fact that the regularizing properties of the paraboliccontributions in (1.6) become less efficient when κ is small. On the other hand, as in Theorem EVELYNE MIOT s − κ and ε are proportional.The complex Ginzburg-Landau equations are widely used in the physical literature as amodel for various phenomena such as superfluidity, Bose-Einstein condensation or supercon-ductivity, see [1]. In the specific form considered here, Eq. (C) corresponds to a dissipativeextension of the purely dispersive Gross-Pitaevskii equation ∂ t Ψ = i [∆Ψ + Ψ(1 − | Ψ | )] . (GP)A similar asymptotic regime for (GP) has been recently investigated by Bethuel, Danchin andSmets [2]. The analysis of [2] exhibits a lower bound for the first time T ε where the solutionvanishes and shows that ( a ε , u ε ) essentially behaves according to the free wave equation ( ν ε ≡ N = 2, there exists a formal analogy between Eq. (C) andthe Landau-Lifschitz-Gilbert equation for sphere-valued magnetizations in three-dimensionalferromagnetics, see [3, 7]. We mention that a thin-film regime leading to a damped wavedynamics for the in-plane components of the magnetization has been studied by Capella,Melcher and Otto [3].Finally, still in the two-dimensional case N = 2, Eq. (C) presents another remarkableregime in which the solutions exhibit zeros (vortices). This regime has been investigated byKurtzke, Melcher, Moser and Spirn [6] and the author [9] when κ is proportional to | ln ε | − .In this setting, Eq. (C) is considered under the form ∂ t Ψ ε = ( κ + i )[∆Ψ ε + 1 ε Ψ ε (1 − | Ψ ε | )] , (C ε )which is obtained from the original equation via the parabolic scalingΨ ε ( t, x ) = Ψ (cid:18) tε , xε (cid:19) . (1.7)A natural extension of the results in [6, 9] would consist in allowing for superpositions ofvortices and oscillating phases in the initial data. This difficult issue was a strong motivationto analyze the behavior of the phase in the regime (1.1), excluding vortices, as a first attemptto tackle the general situation where it is coupled with vortices.2. General strategy
We now present our approach for proving Theorems 1.1, 1.2 and 1.3, which will be partlyborrowed from the analysis in [2] for the Gross-Pitaevskii equation.First, we handle Eq. (C) in its parabolic scaling (1.7) yielding Eq. (C ε ). We define thevariables b ε ( t, x ) = a ε (cid:18) tε , x (cid:19) v ε ( t, x ) = u ε (cid:18) tε , x (cid:19) , so that in the regime (1.1) we haveΨ ε ( t, x ) = ρ ε ( t, x ) exp( iϕ ε ( t, x )) on R + × R N , (2.1) AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 5 where ρ ε ( t, x ) = 1 + ε √ b ε ( t, x )2 ∇ ϕ ε ( t, x ) = εv ε ( t, x ) . (2.2)The system for ( b ε , v ε ) translates into ∂ t b ε + √ ε div v ε + 2 ν ε ε b ε − κ ∆ b ε = ˜ f ε ( b ε , v ε ) ∂ t v ε + √ ε ∇ b ε − κ ∆ v ε = ˜ g ε ( b ε , v ε ) , (2.3)where ˜ f ε ( b ε , v ε ) = √ ν ε (cid:18) − |∇ ρ ε | − ρ ε | v ε | − b ε (cid:19) − div( b ε v ε )˜ g ε ( b ε , v ε ) = κ ∇ (cid:18) ∇ ρ ε ρ ε · v ε (cid:19) + 2 ∇ (cid:18) ∆ ρ ε ρ ε (cid:19) − v ε · ∇ v ε . (2.4)For a map Ψ ∈ H , the Ginzburg-Landau energy of Ψ is defined by E ε (Ψ) = Z R N (cid:16) |∇ Ψ | − | Ψ | ) ε (cid:17) dx, and E denotes the corresponding space of finite energy fields. For the Gross-Pitaevskii equationthe Ginzburg-Landau energy is an Hamiltonian, whereas for solutions to Eq. (C ε ) it decreasesin time. Note that, in the regime (2.1)-(2.2), the solution Ψ ε belongs to E since ( b ε , v ε ) ∈ H × L . In fact, one has E ε (Ψ ε ) ≃ C ( k ( b ε , v ε ) k L + ε k∇ b ε k L )provided that k| Ψ ε | − k ∞ < ε ) so that ( b ε , v ε ) being defined by(2.2), as long as Ψ ε does not vanish, does belong to C ( H s +1 × H s ). As mentioned, the initialfield Ψ ε has finite Ginzburg-Landau energy. In [4] (see also [5]) it has been shown that E ⊂ W + H ( R N ) . Here the space W , which will be defined in Section 3 below, contains in particular all constantmaps of modulus one. It is therefore natural to determine the solution Ψ ε in C ( W + H s +1 ).This is done in Section 3.In Theorems 1.1, 1.2 and 1.3 one assumes that k b ε k ∞ is bounded in such a way that | Ψ ε | isbounded and bounded away from zero. More precisely, the constant K ( s, N ) can be adjustedso that c ( s, N ) ε √ k b ε k H s < . (2.5)Here the constant c ( s, N ) corresponds to the Sobolev embedding H s ( R N ) ⊂ L ∞ ( R N ) for s > N/
2. Hence (2.5) guarantees that k| Ψ ε | − k ∞ < / R N | Ψ ε ( t ) | >
0, one may define ( b ε , v ε )( t ) explicitely as a function of Ψ ε ( t ). Infact, to prove that Ψ ε and ( b ε , v ε ) are globally defined, and to establish Theorems 1.2 and 1.3it suffices to show that k ( b ε , v ε ) k H s +1 × H s remains bounded. Moreover, to obtain the bound k| Ψ ε ( t ) | − k ∞ < /
2, it suffices to show that (2.5) holds as long as b ε is defined.Due to the presence of higher order derivatives in the right-hand sides in (2.3), controlling k ( b ε , v ε ) k H s +1 × H s is however a difficult issue. As in [2], this control will be carried out by EVELYNE MIOT incorporating the equation satisfied by ∇ ln( ρ ε ). More precisely, we focus on the new variable( b ε , z ε ), where z ε = v ε − i ∇ ln( ρ ε ) = ∇ (cid:0) ϕ ε − i ln( ρ ε ) (cid:1) ∈ C N . We remark that ( b ε , z ε ) is well-suited to our analysis since E ε (Ψ ε ) = 18 (cid:16) k b ε k L + k z ε k L ((1+ εb/ √ dx ) (cid:17) . Moreover, there exists a constant C = C ( s, N ) such that C − k ( b ε , z ε ) k H s ≤ k ( b ε , v ε ) k H s + ε k b ε k H s +1 ≤ C k ( b ε , z ε ) k H s . From now on we will sometimes omit the subscript ε for more clarity in the notations.The equations for ( b, z ) are given in the following Proposition 2.1.
Let s ≥ , T > and Ψ be a solution to (C ε ) on [0 , T ] satisfying inf ( t,x ) ∈ [0 ,T ] × R N | Ψ( t, x ) | ≥ m > and such that ( b, v ) ∈ C ([0 , T ] , H s +1 × H s ) . Then ∂ t b + √ ε divRe z = κ (cid:16) − ( √ ε + b )div(Im z ) −
12 ( √ ε + b )Re h z, z i− √ ε ( √ ε + b ) b (cid:17) − div( b Re z ) ∂ t z + √ ε ∇ b = ( κ + i )∆ z + − κi ∇h z, z i + κ √ ε i ∇ b. Dealing with ( b, z ) instead of ( b, v ) presents many advantages when computing energyestimates. Indeed, in contrast with System (2.3) for ( b, v ), the equations for ( b, z ) involveonly non linear first-order quadratic terms and a linear second-order operator ( κ + i )∆ z . Thisis due to the identity ε √ ∇ b = − (1 + ε √ b )Im z, which enables to save one derivative.For the Gross-Pitaevskii equation (GP), the energy estimates performed in [2] for ( b, z )involve a family of semi-norms with a suitable weightΓ k ( b, z ) := Z R N | D k b | + Z R N (1 + ε √ b ) | D k z | , k = 0 , . . . , s. In particular, we have the remarkable identityΓ ( b, z ) = 8 E ε (Ψ) , which in fact was the principal motivation to add the imaginary part of z . Moreover weremark that Γ k ( b, z ) and k ( D k b, D k z ) k L are comparable as long as | Ψ | is close to one.For the complex Ginzburg-Landau equation (C ε ) we will partly rely on the estimates alreadystated in [2] to establish the following See (5.4) below. Here h z, z i = N X i =1 z i , where z = ( z , . . . , z N ) ∈ C N . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 7
Proposition 2.2.
Let s > N/ and T > . Let Ψ be a solution to (C ε ) on [0 , T ] such that k| Ψ | − k L ∞ ([0 ,T ] × R N ) < and such that ( b, z ) ∈ C ([0 , T ] , H s +1 ) . There exists a constant K = K ( s, N ) depending onlyon s and N such that for ≤ k ≤ s and t ∈ [0 , T ] ddt (cid:0) Γ k ( b, z ) + E ε (Ψ) (cid:1) + κ (cid:0) Γ k +1 ( b, z ) + 1 ε Γ k ( b, (cid:1) ≤ K (cid:0) ν ε k b k ∞ + κ k ( b, z ) k ∞ + k ( Db, Dz ) k ∞ (cid:1)(cid:0) Γ k ( b, z ) + E ε (Ψ) (cid:1) . We further assume that s > N/
2. Combining Proposition 2.2 and Sobolev embeddingwe readily find k ( b, z )( t ) k H s ≤ C k ( b, z )(0) k H s + C ( ε ) Z t k ( b, z )( τ ) k H s dτ. This provides a first control of the norm k ( b, z )( t ) k H s up to times of order C ( ε ) − k ( b, z )(0) k − H s .However, we need to refine this control since C ( ε ) diverges as ε tends to zero. In fact, onemay also apply Cauchy-Schwarz inequality and Sobolev imbedding together with Proposition2.2 to infer an estimate for k ( b, z ) k L ∞ t ( H s ) in terms of the norms k ( b, z ) k L t ( H s ) and k b k L t ( L ∞ ) . Proposition 2.3.
Under the assumptions of Proposition 2.2, we assume moreover that s > N/ . There exists a constant K = K ( s, N ) depending only on s and N such that for [0 , T ] K − k ( b, z ) k L ∞ t ( H s ) ≤ k ( b, z )(0) k H s + ν ε k ( b, z ) k L t ( H s ) k b k L t ( L ∞ ) + (cid:0) κ k ( b, z ) k L ∞ t ( H s ) + 1 (cid:1) k ( b, z ) k L t ( H s ) and K − κ k ( Db, Dz ) k L t ( H s ) ≤ k ( b, z )(0) k H s + k ( b, z ) k L ∞ t ( H s ) (cid:0) ν ε k ( b, z ) k L t ( H s ) k b k L t ( L ∞ ) + (cid:0) κ k ( b, z ) k L ∞ t ( H s ) + 1 (cid:1) k ( b, z ) k L t ( H s ) (cid:1) . In the second step of the proofs, we will exploit the decreasing properties of the semi-group operator associated to System (2.3) to derive estimates for the norms k ( b, z ) k L t ( H s ) and k b k L t ( L ∞ ) in terms of k ( b, z ) k L ∞ t ( H s ) . These estimates are summarized in the following Proposition 2.4.
Under the assumptions of Proposition 2.3, there exists a constant K = K ( s, N ) depending only on s and N such that for t ∈ [0 , T ] K − k ( b, z ) k L t ( H s ) ≤ κ / max(1 , ν − ε ) M + (cid:0) ε k ( b, z ) k L ∞ t ( H s ) (cid:1) k ( b, z ) k L t ( H s ) (cid:0) κ / k ( b, z ) k L t ( H s ) + ( ε + ν − ε ) k ( b, z ) k L ∞ t ( H s ) (cid:1) and K − k b k L t ( L ∞ ) ≤ ( εν − ε ) / M + (cid:0) ε k ( b, z ) k L ∞ t ( H s ) (cid:1) k ( b, z ) k L t ( H s ) ε max(1 , ν − ε ) k ( b, z ) k L ∞ t ( H s ) , where M is defined in Theorem 1.1. Combining Propositions 2.3 and 2.4 yields an improved estimate for k ( b, z ) k L ∞ t ( H s ) which,in turn, leads to Theorems 1.1, 1.2 and 1.3.The remainder of this work is organized in the following way. In Section 3 we study theCauchy problem for (C ε ) and prove local well-posedness for ( b, z ). Propositions 2.1, 2.2 and EVELYNE MIOT
The Cauchy problem for the complex Ginzburg-Landau equation
In this section, we address the Cauchy problem for (C ε ) in a space including the fieldsΨ = (1 + a ) / exp( iϕ ), where ( a, ϕ ) ∈ H s +1 ( R N ) and s + 1 ≥ N/
2. We consider the set W = (cid:8) U ∈ L ∞ ( R N ) , ∇ U ∈ H ∞ ( R N ) and 1 − | U | ∈ L ( R N ) (cid:9) . Applying a standard fixed point argument (see, e.g., the proof of Theorem 1 in [9]) andusing the Sobolev embedding H s +1 ⊂ L ∞ if s + 1 > N/
2, it can be shown the following
Theorem 3.1.
Let s + 1 > N/ and U ∈ W . For any ω ∈ H s +1 ( R N ) there exists T ∗ = T ( U , ω ) > and a unique maximal solution Ψ ∈ { U } + C ([0 , T ∗ ) , H s +1 ( R N )) to Eq. (C ε ) such that Ψ(0) = U + ω .The Ginzburg-Landau energy of Ψ is finite and satisfies E ε (Ψ( t )) ≤ E ε (Ψ(0)) , ∀ t ∈ [0 , T ∗ ) . Moreover, there exists a number C depending only on E ε (Ψ(0)) such that k Ψ( t ) − Ψ(0) k L ( R N ) ≤ C exp( Ct ) , ∀ t ∈ [0 , T ∗ ) . Finally, either T ∗ = + ∞ or lim sup t → T ∗ k∇ Ψ( t ) k H s = + ∞ . We recall that E denotes the space of finite energy fields. Thanks to the already mentionedinclusion (see [4]) E ⊂ W + H ( R N ) , a consequence of Theorem 3.1 is the Corollary 3.1.
Let s + 1 > N/ . Let ( a , ϕ ) ∈ H s +1 ( R N ) . We assume that ε √ k a k ∞ < . There exists T > and a unique solution ( b, v ) ∈ C ([0 , T ] , H s +1 × H s ) to System (2.3) withinitial datum ( a , u = 2 ∇ ϕ ) . Moreover, there exists ϕ ∈ C ([0 , T ] , H ) such that v = 2 ∇ ϕ .Proof. Set Ψ ( x ) = (cid:0) ε √ a ( x ) (cid:1) / exp( iϕ ( x )) . By assumption on ( a , ϕ ), Ψ belongs to E and k| Ψ | − k ∞ < . (3.1)Since E ⊂ W + H ( R N ), we have Ψ ∈ { U } + H ( R N ) for some U ∈ W . Using the embedding H s +1 ( R N ) ⊂ L ∞ ( R N ), we check that k∇ Ψ k H s ≤ C (1 + k ( a , u ) k H s +1 × H s ) . This shows that actually Ψ ∈ { U } + H s +1 ( R N ). Hence, by virtue of Theorem 3.1 thereexists T ∗ > ∈ { U } + C ([0 , T ∗ ) , H s +1 ) to (C ε ) such thatΨ(0) = Ψ . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 9
Next, thanks to (3.1) and to the inclusion H s +1 ( R N ) ⊂ L ∞ ( R N ), there exists by timecontinuity a non trivial interval [0 , T ] ⊂ [0 , T ∗ ) for whichinf ( t,x ) ∈ [0 ,T ] × R N | Ψ( t, x ) | ≥ m > . Consequently, we may find a lifting for Ψ on [0 , T ] :Ψ( t, x ) = (cid:0) ε √ b ( t, x ) (cid:1) / exp( iϕ ( t, x )) , where ϕ ∈ L . Setting then v = 2 ∇ ϕ , we determine b and v in a unique way through the identities b = √ ε ( | Ψ | −
1) and v = 2 | Ψ | (Ψ × ∇ Ψ) . In view of the regularity of Ψ we have ( b, v ) ∈ C ([0 , T ] , H s +1 × H s ). In addition, ( b, v ) is asolution to System (2.3) on [0 , T ], and the conclusion follows. (cid:3) Proofs of Propositions 2.1, 2.2 and 2.3.
Notations.
We use this paragraph to fix some notations. The notation a · b denotes thestandard scalar product on R N or R N , which we extend to complex vectors by setting z · ζ = (Re z, Im z ) · (Re ζ, Im ζ ) ∈ R , ∀ z, ζ ∈ C N . We define the complex product of z = ( z , . . . , z N ) and ζ = ( ζ , . . . , ζ N ) ∈ C N by h z, ζ i = N X j =1 z j ζ j ∈ C . Therefore when z = a + ib ∈ C N and ζ = x + iy ∈ C N with a, b, x, y ∈ R N we have h z, ζ i = a · x − b · y + i ( a · y + b · x ) and z · ζ = a · x + b · y. With the same notations as above we finally introduce ∇ z = ∇ a + i ∇ b ∈ C N × N and ∇ z : ∇ ζ = ∇ a : ∇ x + ∇ b : ∇ y ∈ R , where for A, B ∈ R N × N we have set A : B = tr( A t B ).4.2. Proof of Proposition 2.1.
Since Ψ = ρ exp( iϕ ) is a solution to (C ε ), we have, with v = 2 ∇ ϕ , ∂ t ρ ρ = 2 κ (cid:18) ∆ ρρ − | v | − ρ ε (cid:19) − div( ρ v ) ρ ∂ t (2 ϕ ) = 2 (cid:18) ∆ ρρ − | v | − ρ ε (cid:19) + κ div( ρ v ) ρ . Taking the gradient in both equations we obtain ∇ ∂ t ρ ρ = 2 κ ∇ ∆ ρρ − κ ∇ | v | κ ∇ − ρ ε − ∇ div( ρ v ) ρ ∂ t v = 2 ∇ ∆ ρρ − ∇ | v | ∇ − ρ ε + κ ∇ div( ρ v ) ρ . Since ∂ t z = ∂ t v − i ∇ ∂ t ρ ρ , we have ∂ t z = (1 − κi )2 ∇ ∆ ρρ − (1 − κi ) ∇ | v | − κi ) ∇ − ρ ε + ( κ + i ) ∇ div( ρ v ) ρ . Next, expanding ∆ ln ρ = ∆ ρρ − |∇ ρ | ρ , we obtain 2 ∇ ∆ ρρ = ∇ ∆ ln ρ + 2 ∇|∇ ln ρ | = − ∆Im z + 12 ∇| Im z | . On the other hand, since v is a gradient we have ∇ div( ρ v ) ρ = ∇ div v + ∇ (cid:16) v · ∇ ρ ρ (cid:17) = ∆Re z − ∇ (cid:16) Im z · v (cid:17) . Finally, using the fact that 2 ∇ − ρ ε = − √ ε ∇ b, we are led to the equation for z∂ t z = ( κ + i )∆ z − − κi ∇h z, z i − √ ε (1 − κi ) ∇ b. We next turn to the equation for b , recalling that ρ verifies ∂ t ρ = κ (cid:18) ρ ∆ ρ − ρ | v | ρ (1 − ρ ) ε (cid:19) − div( ρ v ) . Expanding the expression2 ρ ∆ ρ = ρ ∆ ln ρ + ρ | Im z | = − ρ divIm z + ρ | Im z | , we find ∂ t ρ = κ (cid:16) − (1 + ε √ b )divIm z −
12 (1 + ε √ b )Re h z, z i − ε √ b ) ε √ ε b (cid:17) − div (cid:0) (1 + ε √ b )Re z (cid:1) , as we wanted. (cid:3) Proof of Proposition 2.2.
We present now the proof of Proposition 2.2. In all thisparagraph, C stands for a number depending only on s and N , which possibly changes froma line to another. We will make use of the identity ε √ ∇ b = − (1 + ε √ b )Im z. (4.1)As we want to rely on the estimates already performed for the Gross-Pitaevskii equationin [2], it is convenient to write the equations for ( b, z ) as follows ( ∂ t b = κf d ( b, z ) + f s ( b, z ) ∂ t z = κg d ( b, z ) + g s ( b, z ) , AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 11 where we have introduced the dissipative part f d ( b, z ) = − ( √ ε + b )div(Im z ) −
12 ( √ ε + b )Re h z, z i − √ ε ( √ ε + b ) b,g d ( b, z ) = ∆ z + i ∇h z, z i + i √ ε ∇ b and the dispersive part f s ( b, z ) = − div (cid:0) ( √ ε + b )Re z (cid:1) ,g s ( b, z ) = i ∆ z − ∇h z, z i − √ ε ∇ b. Let k ∈ N ∗ . We compute ddt Γ k ( b, z ) = ddt Z R N (1 + ε √ b ) D k z · D k z + D k b D k b = 2 Z R N (1 + ε √ b ) D k z · D k ∂ t z + D k b D k ∂ t b + Z R N ε∂ t b √ D k z · D k z = I s + I d , where I s = 2 Z R N (1 + ε √ b ) D k z · D k g s + D k b D k f s + Z R N εf s √ D k z · D k z and κ − I d = 2 Z R N (1 + ε √ b ) D k z · D k g d + D k b D k f d + Z R N εf d √ D k z · D k z. To estimate the first term I s we invoke Proposition 1 in [2] : | I s | ≤ C (1 + ε k b k ∞ ) k ( Db, Dz ) k L ∞ (cid:16) Γ k ( b, z ) + E ε (Ψ ε ) (cid:17) , so we only need to estimate the term I d . Inserting the expressions of f d and g d we find I d = κ (2 I + 2 J + K ) , where I = Z R N (1 + ε √ b ) (cid:16) D k z · D k ∆ z + 12 D k z · iD k ∇h z, z i + √ ε D k z · iD k ∇ b (cid:17) = I + I + I ,J = Z R N − D k b D k (cid:16) ( √ ε + b )div(Im z ) (cid:17) − D k b D k (cid:16) ( √ ε + b )Re h z, z i (cid:17) − D k b D k (cid:16) √ ε ( √ ε + b ) b (cid:17) = J + J + J , and K = − Z R N (1 + ε √ b ) (cid:16) div(Im z ) + 12 Re h z, z i + √ ε b (cid:17) D k z · D k z. Step 1 : estimate for I .Integrating by parts in I , then inserting (4.1) we find I = − Z R N (1 + ε √ b ) ∇ D k z : ∇ D k z − ε √ ∇ b · ( D k z · ∇ D k z )= − Z R N (1 + ε √ b ) |∇ D k z | + Z R N (1 + ε √ b )Im z · ( D k z · ∇ D k z ) ≤ − Z R N (1 + ε √ b ) |∇ D k z | + Z R N (1 + ε √ b ) / | Im z || D k z | (1 + ε √ b ) / |∇ D k z | . Applying Young inequality to the second term in the right-hand side, we obtain I ≤ − Z R N (1 + ε √ b ) |∇ D k z | + 12 Z R N (1 + ε √ b ) | Im z | | D k z | , so finally I ≤ − Z R N (1 + ε √ b ) |∇ D k z | + C (1 + ε k b k ∞ ) k Im z k ∞ k z k H k . Step 2 : estimate for I .Expanding I thanks to Leibniz formula, we obtain I = Z R N (1 + ε √ b ) D k z · D k ( i h z, ∇ z i )= Z R N (1 + ε √ b ) D k z · i h z, ∇ D k z i + k − X j =0 C jk Z R N (1 + ε √ b ) D k z · i h D k − j z, D j ( ∇ z ) i . Applying then Young inequality to the first term in the right-hand side, we infer that I ≤ Z R N (1 + ε √ b ) |∇ D k z | + C (1 + ε k b k ∞ ) k z k ∞ k z k H k + C k − X j =0 (cid:12)(cid:12)(cid:12) Z R N (1 + ε √ b ) D k z · i h D k − j z, D j ( ∇ z ) i (cid:12)(cid:12)(cid:12) . For each 0 ≤ j ≤ k −
1, we apply first Cauchy-Schwarz, then Gagliardo-Nirenberg (see Lemma7.4 in the appendix) inequalities. This yields (cid:12)(cid:12)(cid:12) Z R N (1 + ε √ b ) D k z · i h D k − j z, D j ( ∇ z ) i (cid:12)(cid:12)(cid:12) ≤ C (1 + ε k b k ∞ ) k D k z k L k| D k − j z || D j ( ∇ z ) |k L ≤ C (1 + ε k b k ∞ ) k D k z k L k Dz k ∞ k z k H k , and we are led to I ≤ Z R N (1 + ε √ b ) |∇ D k z | + C (1 + ε k b k ∞ )( k z k ∞ + k Dz k ∞ ) k z k H k . Step 3 : estimate for I .Since D k ∇ b ∈ R N we have by definition of the complex product I = Z R N (1 + ε √ b ) √ ε D k z · iD k ∇ b = Z R N (1 + ε √ b ) √ ε D k Im z · D k ∇ b. AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 13
Inserting first (4.1) and using then Leibniz formula we get I = − ε Z R N (1 + ε √ b ) D k Im z · D k (cid:0) (1 + ε √ b )Im z (cid:1) = − ε Z R N (1 + ε √ b ) | D k Im z | − ε k X j =1 C jk Z R N (1 + ε √ b ) D k Im z · (cid:0) D j (1 + ε √ b ) D k − j Im z (cid:1) . Now, we observe that for each j ≥
1, we have D j (1 + ε √ b ) = ε √ D j b. Consequently, applying Young inequality to each term of the sum we find I ≤ − ε Z R N (1 + ε √ b ) | D k Im z | + C k X j =1 Z R N | D j b D k − j Im z | , and we finally infer from Gagliardo-Nirenberg inequality that I ≤ C (cid:0) k b k ∞ + k Im z k ∞ (cid:1) k ( b, z ) k H k . Step 4 : estimate for J .A short calculation using (4.1) yields J = − Z R N D k b D k (cid:16) ( √ ε + b )div(Im z ) (cid:17) = − Z R N D k b D k div (cid:16) ( √ ε + b )Im z (cid:17) + Z R N D k b D k ( ∇ b · Im z )= Z R N D k b D k div( ∇ b ) + Z R N D k b D k ( ∇ b · Im z ) . After integrating by parts in the first term in the right-hand side and expanding the secondterm by means of Leibniz formula we obtain J = − Z R N |∇ D k b | + Z R N D k b ( D k ∇ b ) · Im z + k X j =1 C jk Z R N D k b ( D k − j ∇ b ) · D j Im z. Next, combining Young, Cauchy-Schwarz and Gagliardo-Nirenberg inequalities we find J ≤ − Z R N |∇ D k b | + C k Im z k ∞ k b k H k + C k b k H k ( k∇ b k ∞ + k Dz k ∞ ) k ( b, z ) k H k , so that J ≤ − Z R N |∇ D k b | + C (cid:0) k Im z k ∞ + k ( ∇ b, Dz ) k ∞ (cid:1) k ( b, z ) k H k . Step 5 : estimate for J .Similarly, we compute thanks to Leibniz formula J = − Z R N D k b D k (cid:16) ( √ ε + b )Re h z, z i (cid:17) = − Z R N D k b ( √ ε + b ) D k (Re h z, z i ) + 12 k X j =1 C jk Z R N D k b D j bD k − j (Re h z, z i )= − ε √ Z R N D k b D k (Re h z, z i ) − Z R N bD k bD k (Re h z, z i )+ 12 k X j =1 C jk Z R N D k b D j bD k − j (Re h z, z i ) . Invoking Young and Cauchy-Schwarz inequalities, we obtain J ≤ ε Z R N | D k b | + C kh z, z ik H k + C (cid:0) k b k ∞ k b k H k kh z, z ik H k + k b k H k k X j =1 k D j bD k − j h z, z ik L (cid:1) , so that by virtue of Lemma 7.4, J ≤ ε Z R N | D k b | + C k ( b, z ) k ∞ k ( b, z ) k H k . Step 6 : estimate for J .We have J = − √ ε Z R N D k b D k (cid:16) b ( √ ε + b ) (cid:17) = − ε Z R N | D k b | − √ ε Z R N D k bD k ( b ) , so, thanks to Cauchy-Schwarz inequality and Lemma 7.4, J ≤ − ε Z R N | D k b | + Cε k b k ∞ k b k H k . Step 7 : estimate for K .We readily obtain | K | ≤ C (1 + ε k b k ∞ ) (cid:16) k b k ∞ ε + k Dz k ∞ + k z k ∞ (cid:17) k z k H k . Gathering the previous steps we obtain ddt Γ k ( b, z ) + κ k +1 ( b, z ) + 2 κε Γ k ( b, ≤ C (1 + ε k b k ∞ ) (cid:16) κ (cid:0) k ( b, z ) k ∞ + ε − k b k ∞ (cid:1) + k ( ∇ b, Dz ) k ∞ (cid:17) k ( b, z ) k H k , AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 15 holding for any 1 ≤ k ≤ s . Following step by step the previous computations we readily checkthat it also holds for k = 0. Finally, we have by assumption12 ≤ εb √ ≤
32 on [0 , T ] × R N , from which we infer that k ( b, z ) k H k ≤ C Γ k ( b, z ) for all 0 ≤ k ≤ s . Therefore the proof ofProposition 2.2 is complete. (cid:3) Proof of Proposition 2.3.
To show the first inequality we add the inequalities obtainedin Proposition 2.2 for k varying from 1 to s . Since 1 / ≤ εb/ √ ≤ /
2, this yields ddt k ( b, z ) k H s ≤ C ( ν ε k b k ∞ + κ k ( b, z ) k ∞ + k ( Db, Dz ) k ∞ ) k ( b, z ) k H s ≤ C (cid:0) ν ε k b k ∞ + ( κ k ( b, z ) k H s + 1) k ( b, z ) k H s (cid:1) k ( b, z ) k H s . After integrating on [0 , T ] and using Cauchy-Schwarz inequality this leads to k ( b, z )( T ) k H s ≤ k ( b, z )(0) k H s + C k ( b, z ) k L ∞ T ( H s ) (cid:16) ν ε k b k L T ( L ∞ ) k ( b, z ) k L T ( H s ) + ( κ k ( b, z ) k L ∞ T ( H s ) + 1) k ( b, z ) k L T ( H s ) (cid:17) , for all T ∈ [0 , T ]. Considering the supremum over T ∈ [0 , t ] and applying Young inequalityin the right-hand-side we find the result.Finally the second inequality in Proposition 2.3 is obtained by integrating on [0 , t ] andusing Sobolev and Cauchy-Schwarz inequalities. (cid:3) Proof of Proposition 2.4.
In this paragraph again, C refers to a constant depending only on s and N and possiblychanging from a line to another.First, we formulate System (2.3)-(2.4) with second members involving only b and z . By thesame computations as those in Paragraph 4.2 we find ∂ t b + √ ε div v + 2 ν ε ε b − κ ∆ b = f ( b, z ) ∂ t v + √ ε ∇ b − κ ∆ v − ε √ ∇ ∆ b = g ( b, z ) , (5.1)where f = ˜ f and g = ˜ g − ε √ ∇ ∆ b are defined by f ( b, z ) = ν ε (cid:18) − √ ε √ b ) | z | − √ b (cid:19) − div( b Re z ) g ( b, z ) = − κ ∇ (Re z · Im z ) + ε √ ∇ div( b Im z ) − ∇ Re h z, z i . (5.2)5.1. Some notations and preliminary results.
As in [2], we symmetrize System (5.1) byintroducing the new functions c = (1 − ε / b, d = ( − ∆) − / div v, and F = (1 − ε / f, G = ( − ∆) − / div g. We remark that, knowing d , one can retrieve v since v is a gradient. We have ∂ t c + 2 ν ε ε c − κ ∆ c + √ ε ( − ∆) / (1 − ε / d = F∂ t d − κ ∆ d − √ ε ( − ∆) / (1 − ε / c = G. (5.3)In the following, we denote by ξ ∈ R N the Fourier variable, by ˆ f the Fourier transform of f and by F − the inverse Fourier transform.In view of the definition of ( c, d ), it is useful to introduce the frequency threshold | ξ | ∼ ε − .More precisely, let us fix some R > χ denote the characteristic function on B (0 , R ).For f ∈ L ( R N ), we define the low and high frequencies parts of ff l = F − (cid:0) χ ( εξ ) ˆ f (cid:1) and f h = F − (cid:0) (1 − χ ( εξ )) ˆ f (cid:1) , so that b f l and b f h are supported in {| ξ | ≤ Rε − } and {| ξ | ≥ Rε − } respectively. Lemma 5.1.
There exists C = C ( s, N, R ) > such that the following holds for all ≤ m ≤ s and t ∈ [0 , T ] : k g ( t ) k H m ≈ k G ( t ) k H m , k f l ( t ) k H m ≈ k F l ( t ) k H m and k ( ε ∇ f ) h ( t ) k H m ≈ k F h ( t ) k H m . In addition, k v ( t ) k H m ≈ k d ( t ) k H m , k b l ( t ) k H m ≈ k c l ( t ) k H m and k ( ε ∇ b ) h ( t ) k H m ≈ k c h ( t ) k H m . Finally, k ( b, z )( t ) k H m ≈ k ( b, v ) l ( t ) k H m + k ( ε ∇ b, v ) h ( t ) k H m . Here we have set for f , f ∈ H m k f k H m ≈ k f k H m if and only if C − k f k H m ≤ k f k H m ≤ C k f k H m . Proof.
For the first two statements it suffices to consider the Fourier transforms of the func-tions and to use their support properties. The last statement is already established in [2],Lemma 1. (cid:3)
Lemma 5.1 guarantees that for 0 ≤ m ≤ s , k ( b, v )( t ) k H m + ε k b ( t ) k H m +1 ≈ k ( b, z )( t ) k H m and k ( b, z )( t ) k H m ≈ k ( c, d )( t ) k H m , (5.4)therefore we have k ( c, d )(0) k H s ≤ CM , where M is defined in Theorem 1.1.On the other side, when s − > N/
2, Sobolev embedding yields k b l ( t ) k ∞ ≤ C k b l ( t ) k H s − ≤ C k c l ( t ) k H s − and k b h ( t ) k ∞ ≤ C k b h ( t ) k H s − ≤ C k ( ε ∇ b ) h ( t ) k H s − ≤ C k c h ( t ) k H s − . Therefore it suffices to establish the first inequality of Proposition 2.4 for k ( c, d ) k L t ( H s ) andthe second inequality for k c k L t ( H s − ) .Next, we have ddt (cid:18) ˆ c ˆ d (cid:19) + M ( ξ ) (cid:18) ˆ c ˆ d (cid:19) = (cid:18) ˆ F ˆ G (cid:19) , AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 17 where M ( ξ ) = ν ε ε ε | ξ | | ξ | ν ε (2 + ε | ξ | ) / − | ξ | ν ε (2 + ε | ξ | ) / ε | ξ | . By Duhamel formula we have [ ( c, d )( t, ξ ) = e − tM ( ξ ) [ ( c, d )(0 , ξ ) + Z t e − ( t − τ ) M ( ξ ) \ ( F, G )( τ, ξ ) dτ. Our next result, which is proved in the appendix, establishes pointwise estimates for e − tM ( ξ ) . Lemma 5.2.
There exist positive numbers κ , r , c and C such that for all ( a, b ) ∈ C , wehave for < ε ≤ , κ < κ and t ≥ If | ξ | ≤ rν ε then (cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12) ≤ C exp( − ν ε ε | ξ | t ) (cid:20) exp (cid:16) − ν ε ε t (cid:17) ( | a | + | b | ) + exp (cid:18) − c | ξ | ν ε ε t (cid:19) ( ν − ε | ξ || a | + | b | ) (cid:21) . (2) If | ξ | ≥ rν ε then (cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12) ≤ C exp (cid:18) − ν ε (1 + ε | ξ | )2 ε t (cid:19) ( | a | + | b | ) . Here for A = ( a, b ) ∈ C we have set | A | = | a | + | b | . Lemma 5.2 reveals the new frequency threshold | ξ | ∼ ν ε . We may choose R > r , so that rν ε < Rε − . We are therefore led to split the frequency space into three regions R N = R ∪ R ∪ R , where • R = {| ξ | ≤ rν ε } denotes the low frequencies region, in which the semi-group is composedof a parabolic part (exp( − ( ν ε ε ) − | ξ | t )), and a damping part (exp( − ν ε ε − t )). • R = { rν ε ≤ | ξ | ≤ Rε − } denotes the intermediate frequencies region, in which the damp-ing effect exp( − ν ε ε − t ) is prevalent with respect to the parabolic contribution exp( − ν ε ε | ξ | t ). • R = {| ξ | ≥ Rε − } denotes the high frequencies region, in which the parabolic contribu-tion is strong and dominates the damping.With respect to this decomposition we introduce the small, intermediate and high frequen-cies parts of f ∈ L ( R N ) as follows f s = F − (cid:0) χ | ξ |≤ rν ε ˆ f (cid:1) , f m = F − (cid:0) χ rν ε ≤| ξ |≤ Rε − ˆ f (cid:1) and f h = F − (cid:0) χ | ξ |≥ Rε − ˆ f (cid:1) , where χ E denotes the characteristic function on the set E . Note that we have f = f s + f m + f h = f l + f h . Proof of Proposition 2.4.
We first introduce some notations. Let L ( b, z )( t ) = k (1 + εb ( t )) | z ( t ) | k H s + k b ( t ) k H s + k b ( t ) z ( t ) k H s + kh z, z i ( t ) k H s . Next, we sort the terms in the definitions of f ( b, z ) and g ( b, z ) in System (5.2) as follows. Weset f ( b, z ) = ν ε f ( b, z ) + f ( b, z )and g ( b, z ) = g ( b, z ) + εg ( b, z ) = ∇ h ( b, z ) + ε ∇ h ( b, z ) , where the subscript j = 0 , , f ( b, z ) = − √ ε √ b ) | z | − √ b f ( b, z ) = − div( b Re z )and g ( b, z ) = − κ ∇ (Re z · Im z ) − / ∇ Re h z, z i = ∇ h ( b, z ) g ( b, z ) = 1 √ ∇ div( b Im z ) = ∇ h ( b, z ) . The proof of Proposition 2.4 relies on several lemmas which we present now separately.
Lemma 5.3.
Under the assumptions of Proposition 2.4 we have for T ∈ [0 , T ] C − k ( c, d ) s k L T ( H s ) ≤ κ / max(1 , ν − ε ) M + ε k L ( b, z ) k L T + κ / k L ( b, z ) k L T . Proof.
By virtue of Lemma 5.2 we have | [ ( c, d ) s ( t, ξ ) | ≤ C ( I ( t, ξ ) + J ( t, ξ )) , where I ( t, ξ ) = e − νεε t | [ ( c, d ) s (0 , ξ ) | + Z t e − νεε ( t − τ ) | \ ( F, G ) s ( τ, ξ ) | dτ and J ( t, ξ ) = e − c | ξ | νεε t (cid:12)(cid:12) ( | ξ | ν − ε b c s (0) , b d s (0)) (cid:12)(cid:12) + Z t e − c | ξ | νεε ( t − τ ) (cid:12)(cid:12) ( | ξ | ν − ε c F s , c G s ) (cid:12)(cid:12) dτ = J L ( t, ξ ) + J NL ( t, ξ ) . We set ˇ I = F − I and ˇ J = F − J , so that k ( c, d ) s k L T ( H s ) ≤ C ( k ˇ I k L T ( H s ) + k ˇ J k L T ( H s ) ). First step : estimate for k ˇ I k L T ( H s ) .Invoking Lemma 7.3 we obtain k ˇ I k L T ( H s ) ≤ C (cid:0) ( εν − ε ) / k ( c, d ) s (0) k H s + εν − ε k ( f, g ) s k L T ( H s ) (cid:1) . Let h ∈ H s . We observe that thanks to the support properties of b h s , we have k D k h s k H s ≤ Cν kε k h s k H s , k ∈ N . Applying this inequality to the higher order derivatives f , g and g , we see that k ( f, g ) s ( t ) k H s ≤ C ( ν ε + εν ε ) L ( b, z )( t ) ≤ Cν ε L ( b, z )( t ) , and we conclude that k ˇ I k L T ( H s ) ≤ C (cid:0) ( εν − ε ) / M + ε k L ( b, z ) k L T (cid:1) . (5.5) Second step : estimate for k ˇ J k L T ( H s ) .We have k ˇ J k L T ( H s ) ≤ C ( k ˇ J L k L T ( H s ) + k ˇ J NL k L T ( H s ) ) . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 19
For the linear term we obtain k ˇ J L k L T ( H s ) ≤ (cid:13)(cid:13) (1 + | ξ | s ) e − c | ξ | νεε t ( | ξ | ν − ε | b c s (0) | + | b d s (0) | ) (cid:13)(cid:13) L T ( L ) ≤ C (cid:13)(cid:13) (1 + | ξ | s ) e − c | ξ | νεε t | ξ | ( ν − ε | b c s (0) | + | ξ | − | b d s (0) | ) (cid:13)(cid:13) L T ( L ) ≤ C max(1 , ν − ε ) (cid:13)(cid:13) (1 + | ξ | s ) e − c | ξ | νεε t | ξ | ( | b c s (0) | + | c ϕ s (0) | ) (cid:13)(cid:13) L T ( L ) , because d (0) = − − ∆) / ϕ (0). By virtue of Lemma 7.1 in the appendix, this yields k ˇ J L k L T ( H s ) ≤ C max(1 , ν − ε )( εν ε ) / (cid:0) k c s (0) k H s + k ϕ s (0) k H s (cid:1) ≤ C max(1 , ν − ε ) κ / M . On the other side, Lemma 5.1 yields k ˇ J NL k L T ( H s ) ≤ (cid:13)(cid:13)(cid:13) Z t (1 + | ξ | s ) e − c | ξ | νεε ( t − τ ) (cid:0) | ξ | ν − ε | c F s | + | c G s | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) ≤ (cid:13)(cid:13)(cid:13) Z t (1 + | ξ | ) s e − c | ξ | νεε ( t − τ ) (cid:0) | ξ | ν − ε | b f s | + | b g s | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) . Inserting the expressions f = ν ε f + f and g = ∇ h + ε ∇ h we obtain (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) (1 + | ξ | s ) (cid:0) | ξ | ν − ε | b f s | + | b g s | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) ≤ (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s ) (cid:0) ν − ε | ξ | − | b f | + ε | ξ | − | c h | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) + (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s ) (cid:0) | b f | + | c h | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) . First, invoking Lemma 7.1, we find (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s ) (cid:0) ν − ε | ξ | − | b f | + ε | ξ | − | c h | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) ≤ Cεν ε k (1 + | ξ | s )( ν − ε | ξ | − b f , ε | ξ | − c h ) k L T ( L ) ≤ Cεν ε ( ν − ε + ε ) k (1 + | ξ | s )( | [ b Re z | + | [ b Im z | ) k L T ( L ) ≤ Cεν ε ( ν − ε + ε ) k b · z k L T ( H s ) ≤ Cε k L ( b, z ) k L T . Next, we infer from Lemma 7.2 in the appendix that (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s ) (cid:0) | b f | + | c h | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) ≤ C ( εν ε ) / k (1 + | ξ | s )( | b f | + | c h | ) k L T ( L ) ≤ Cκ / k L ( b, z ) k L T . Gathering the previous steps and noticing that ( εν − ε ) / ≤ κ / max(1 , ν − ε ), we concludethe proof of the lemma. (cid:3) Lemma 5.4.
Under the assumptions of Proposition 2.4 we have for T ∈ [0 , T ] C − (cid:16) k ( c, d ) m k L T ( H s ) + k ( c, d ) h k L T ( H s ) (cid:17) ≤ ( εν − ε ) / M + ( ε + ν − ε ) k L ( b, z ) k L T . Proof.
We divide the proof into several steps.
First step : intermediate frequencies rν ε ≤ | ξ | ≤ Rε − .Another application of Lemma 5.2 yields | [ ( c, d ) m ( t, ξ ) | ≤ Ce − νε ε t | [ ( c, d ) m (0 , ξ ) | + C Z t e − νε ε ( t − τ ) | \ ( F, G ) m ( τ, ξ ) | dτ, whence, according to Lemma 7.3, k ( c, d ) m k L T ( H s ) ≤ C ( εν − ε ) / k ( c, d )(0) k H s + Cεν − ε k ( F, G ) m k L T ( H s ) . Let us set (
F, G ) m = A m + B m , where A m and B m ∈ L T ( H s × H s ), to be determined later on, are such that d A m ( t, · ) and c B m ( t, · ) are compactly supported in (cid:0) R ∪ R = {| ξ | ≤ Rε − } (cid:1) . Owing to these supportproperties we find k ( F, G ) m k L T ( H s ) ≤ kA m k L T ( H s ) + kB m k L T ( H s ) ≤ C ( ε − kA m k L T ( H s − ) + ε − kB m k L T ( H s − ) ) , so finally C − k ( c, d ) m k L T ( H s ) ≤ ( εν − ε ) / M + ν − ε (cid:0) kA m k L T ( H s − ) + ε − kB m k L T ( H s − ) (cid:1) . (5.6) Second step : high frequencies | ξ | ≥ Rε − .For the high frequencies we neglect the contribution of the damping e − νε ε t and only take thecontribution of e − ν ε ε | ξ | t into account. Exploiting again Lemma 5.2 we have | [ ( c, d ) h ( t, ξ ) | ≤ Ce − ν ε ε | ξ | t | [ ( c, d ) h (0 , ξ ) | + C Z t e − ν ε ε | ξ | ( t − τ ) | \ ( F, G ) h ( τ, ξ ) | dτ ≤ Cε | ξ | e − ν ε ε | ξ | t | [ ( c, d ) h (0 , ξ ) | + C Z t e − ν ε ε | ξ | ( t − τ ) | \ ( F, G ) h ( τ, ξ ) | dτ, where the second inequality is due to the fact that 1 ≤ Cε | ξ | on the support of [ ( c, d ) h . Byvirtue of Lemma 7.1 we obtain k ( c, d ) h k L T ( H s ) ≤ C (cid:0) ( εν − ε ) / k ( c, d ) h (0) k H s + ( ν ε ε ) − k ( F, G ) h k L T ( H s − ) (cid:1) . (5.7)As in the first step, we set ( F, G ) h = A h + B h , where A h and B h ∈ L T ( H s − × H s − ) will be set in such a way that c A h ( t, · ) and c B h ( t, · ) aresupported in the region (cid:0) R = {| ξ | ≥ Rε − } (cid:1) . Thanks to these support properties we cansave one factor ε to the detriment of one derivative : k ( F, G ) h k L T ( H s − ) ≤ kA h k L T ( H s − ) + kB h k L T ( H s − ) ≤ C ( ε kA h k L T ( H s − ) + kB h k L T ( H s − ) ) . Therefore in view of (5.7) we are led to C − k ( c, d ) h k L T ( H s ) ≤ ( εν − ε ) / M + ν − ε (cid:0) kA h k L T ( H s − ) + ε − kB h k L T ( H s − ) (cid:1) . (5.8) Third step .The last step consists in choosing suitable A and B . We recall that( F, G ) = ((1 − − ε ∆) / f, ( − ∆) / div g ) , AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 21 and f ( b, z ) = ν ε f ( b, z ) + f ( b, z ) , g ( b, z ) = g ( b, z ) + εg ( b, z ) . Now, for the intermediate frequencies we define ( A m = (cid:0) (1 − − ε ∆) / f m , ( − ∆) − / div( g ) m (cid:1) B m = (cid:0) , ε ( − ∆) − / div( g ) m (cid:1) , and for the high frequencies ( A h = (cid:0) ν ε (1 − − ε ∆) / ( f ) h , ( − ∆) − / div( g ) h (cid:1) B h = (cid:0) (1 − − ε ∆) / ( f ) h , ε ( − ∆) − / div( g ) h (cid:1) . Clearly A m + B m = ( F, G ) m and A h + B h = ( F, G ) h . Moreover, we readily check that kA m k H s − ≈ k ( f, g ) m k H s − and kA h k H s − ≈ k ( ν ε ε ∇ f , g ) h k H s − (5.9)and kB m k H s − ≈ ε k ( g ) m k H s − and kB h k H s − ≈ k ( ε ∇ f , εg ) h k H s − . (5.10)On the one hand we have k g k H s − + k g k H s − ≤ C ( k z · z k H s + k b Im z k H s ) ≤ CL ( b, z ) . (5.11)On the other hand, the support properties of d ( f ) m imply that k ( f ) m k H s − ≤ C min(1 , ν − ε ) k ( f ) m k H s , so that k f m k H s − ≤ ν ε k ( f ) m k H s − + k ( f ) m k H s − ≤ C ( k ( f ) m k H s + k ( f ) m k H s − ) , and finally k f m k H s − ≤ CL ( b, z ) . (5.12)Arguing similarly we obtain ν ε k ( ε ∇ f ) h k H s − ≤ Cν ε ε k f k H s ≤ CL ( b, z ) (5.13)and k ( ε ∇ f ) h k H s − ≤ ε k f k H s − ≤ CεL ( b, z ) . (5.14)We infer from (5.9), (5.11), (5.12) and (5.13) that kA m k H s − + kA h k H s − ≤ CL ( b, z ) . (5.15)Moreover (5.10), (5.11) and (5.14) yield kB m k H s − + kB h k H s − ≤ CεL ( b, z ) , (5.16)so that the conclusion of Lemma 5.4 finally follows from (5.6), (5.8), (5.15) and (5.16). (cid:3) Next, in order to establish the second part of Proposition 2.4 involving the norm k b k L ( L ∞ ) ,we show the following analogs of Lemmas 5.3 and 5.4 involving k c k L ( H s − ) . Lemma 5.5.
Under the assumptions of Proposition 2.4 we have for T ∈ [0 , T ] C − k c k L T ( H s − ) ≤ ( εν − ε ) / M + ε max(1 , ν − ε ) k L ( b, z ) k L T . Proof.
We closely follow the proofs of Lemmas 5.3 and 5.4, handling again the regions R , R and R separately. First step : low frequencies | ξ | ≤ rν ε .For low frequencies one may even improve the estimates given by Lemma 5.2 for the semi-group acting on c . Indeed, according to identity (7.1) stated in the proof of Lemma 5.2, weget the bound | b c s ( t, ξ ) | ≤ C ( I ( t, ξ ) + J ( t, ξ )) , where I ( t, ξ ) = e − νε ε t (cid:12)(cid:12) [ ( c, d ) s (0 , ξ ) (cid:12)(cid:12) + Z t e − νε ε ( t − τ ) | \ ( F, G )( τ, ξ ) | dτ and J ( t, ξ ) = e − c | ξ | νεε t (cid:12)(cid:12) ( | ξ | ν − ε b c s , | ξ | ν − ε b d s )(0) (cid:12)(cid:12) + Z t e − c | ξ | νεε ( t − τ ) | ( | ξ | ν − ε c F s , | ξ | ν − ε c G s ) | dτ = J L ( t, ξ ) + J NL ( t, ξ ) . Here again we set ˇ I = F − I and ˇ J = F − J . In view of the first step in the proof of Lemma5.3 (see (5.5)) we already know that k ˇ I k L T ( H s ) ≤ C (cid:0) ( εν − ε ) / M + ε k L ( b, z ) k L T (cid:1) . Next, since | ξ | ν − ε ≤ r we have k ˇ J L k L T ( H s − ) ≤ (cid:13)(cid:13) e − c | ξ | νεε t (1 + | ξ | s − ) (cid:0) | ξ | ν − ε | b c s (0) | + | ξ | ν − ε | b d s (0) | (cid:1)(cid:13)(cid:13) L T ( L ) ≤ Cν − ε (cid:13)(cid:13) e − c | ξ | νεε t | ξ | (1 + | ξ | s − ) (cid:0) | b c s (0) | + | b d s (0) | (cid:1)(cid:13)(cid:13) L T ( L ) ≤ Cν − ε ( εν ε ) / M , where the last inequality is a consequence of Lemma 7.1.On the other side we have k ˇ J NL k L T ( H s − ) ≤ C (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) (1 + | ξ | s − ) (cid:0) | ξ | ν − ε | c F s | + | ξ | ν − ε | c G s | (cid:1) dτ (cid:13)(cid:13)(cid:13) L T ( L ) ≤ Cν − ε (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s − ) | c F s | dτ (cid:13)(cid:13)(cid:13) L T ( L ) + Cν − ε (cid:13)(cid:13)(cid:13) Z t e − c | ξ | νεε ( t − τ ) | ξ | (1 + | ξ | s − ) | ξ | − | c G s | dτ (cid:13)(cid:13)(cid:13) L T ( L ) . Applying Lemma 7.1 to each term we obtain k ˇ J NL k L T ( H s − ) ≤ C ( ν ε εν − ε k F s k L T ( H s − ) + ν ε εν − ε k D − G s k L T ( H s − ) ) ≤ C ( εν − ε k f s k L T ( H s − ) + ε k D − g s k L T ( H s − ) ) ≤ Cε k L ( b, z ) k L T . We have used the support properties of f s in the last inequality above.Finally, we gather the previous inequalities to find C − k c s k L T ( H s − ) ≤ ( εν − ε ) / M + ε k L ( b, z ) k L T . (5.17) AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 23
Second step : intermediate frequencies rν ε ≤ | ξ | ≤ Rε − .In contrast with the previous step, we may here imitate the first step of the proof of Lemma5.4, estimating the H s − norm instead : k c m k L T ( H s − ) ≤ k ( c, d ) m k L T ( H s − ) ≤ C (cid:0) ( εν − ε ) / k ( c, d )(0) k H s − + εν − ε k ( F, G ) m k L T ( H s − ) (cid:1) . Recalling that (
F, G ) m = A m + B m , where d A m and c B m are compactly supported in the region {| ξ | ≤ Rε − } , we obtain k ( F, G ) m k L T ( H s − ) ≤ kA m k L T ( H s − ) + kB m k L T ( H s − ) ≤ kA m k L T ( H s − ) + Cε − kB m k L T ( H s − ) . In view of the third step of the proof of Lemma 5.4 (see (5.15) and (5.16)) we get k ( F, G ) m k H s − ≤ CL ( b, z )and we conclude that C − k c m k L T ( H s − ) ≤ ( εν − ε ) / M + εν − ε k L ( b, z ) k L T . (5.18) Third step : high frequencies | ξ | ≥ Rε − .With ( F, G ) h = A h + B h we obtain, arguing exactly as in the second step of the proof ofLemma 5.4, the analog of (5.7): k ( c, d ) h k L T ( H s − ) ≤ C (cid:0) ( εν − ε ) / M + ( εν ε ) − k ( F, G ) h k L T ( H s − ) (cid:1) ≤ C (cid:0) ( εν − ε ) / M + ν − ε k ( F, G ) h k L T ( H s − ) (cid:1) ≤ C (cid:0) ( εν − ε ) / M + ν − ε ε ( kA h k L T ( H s − ) + ε − kB h k L T ( H s − ) ) (cid:1) . Hence we infer from estimates (5.15) and (5.16) for A h and B h that C − k c h k L T ( H s − ) ≤ ( εν − ε ) / M + εν − ε k L ( b, z ) k L T . (5.19)The conclusion finally follows from estimates (5.17), (5.18) and (5.19). (cid:3) Invoking the previous results we may now complete the
Proof of Proposition 2.4 .First, Cagliardo-Nirenberg inequality yields k| z | k H s + k b k H s + k bz k H s + kh z, z ik H s ≤ C k ( b, z ) k ∞ k ( b, z ) k H s and k εb | z | k H s ≤ Cε k ( b, z ) k ∞ k ( b, z ) k H s , so that L ( b, z ) ≤ C (1 + ε k ( b, z ) k ∞ ) k ( b, z ) k ∞ k ( b, z ) k H s . By Sobolev embedding and Cauchy-Schwarz inequality we obtain k L ( b, z ) k L T ≤ C (cid:0) ε k ( b, z ) k L ∞ T ( H s ) (cid:1) k ( b, z ) k L ∞ T ( H s ) k ( b, z ) k L T ( H s ) and k L ( b, z ) k L T ≤ C (cid:0) ε k ( b, z ) k L ∞ T ( H s ) (cid:1) k ( b, z ) k L T ( H s ) . Proposition 2.4 finally follows from both estimates above together with Lemmas 5.3, 5.4 and5.5. (cid:3)
We conclude this section with a result that will be needed in the course of the next section.We omit the proof, which is a straightforward adaptation of the proof of Lemma 5.5.
Proposition 5.1.
Under the assumptions of Proposition 2.4 we have for all T ∈ [0 , T ] C − k c k L T ( H s ) ≤ ( εν − ε ) / M + ν − ε k ( b, z ) k L ∞ T ( H s ) k ( b, z ) k L T ( H s ) (1 + ε k ( b, z ) k L ∞ T ( H s ) ) . Proofs of Theorems 1.1 and 1.3.
Proof of Theorem 1.1.
This paragraph is devoted to the proof of Theorem 1.1. LetΨ ∈ W + H s +1 such thatΨ = ρ exp( iϕ ) = (cid:0) ε √ a (cid:1) / exp( iϕ ) , where ( a , ϕ ) satisfies the assumptions of Theorem 1.1. Let Ψ ∈ W + C ([0 , T ∗ ) , H s +1 ) denotethe corresponding solution to (C ε ) provided by Theorem 3.1.With c ( s, N ) denoting a constant corresponding to the Sobolev embedding H s ( R N ) ⊂ L ∞ ( R N ), we first assume that the constant K ( s, N ) in Theorem 1.1 satisfies K ( s, N ) > √ c ( s, N ) . (6.1)Hence k| Ψ | − k ∞ = ε √ k a k ∞ < , so that the assumptions of Corollary 3.1 are satisfied. Let ( b, v ) be the solution given byCorollary 3.1 on [0 , T ), with T ≤ T ∗ maximal.We introduce the following control function H ( t ) = k ( b, z ) k L ∞ t ( H s ) + k ( b, z ) k L t ( H s ) κ / max(1 , ν − ε ) + k b k L t ( L ∞ ) ( εν − ε ) / ,H = H (0) . (6.2)Note that, according to (5.4) we have H ≤ C ( s, N ) M and k ( b, v )( t ) k H s + ε k b ( t ) k H s +1 ≤ C ( s, N ) H ( t ) , where the constant C ( s, N ) depends only on s and N . We recall that M is defined inTheorem 1.1. Increasing possibly the number K ( s, N ) introduced in Theorem 1.1, we mayassume that C ( s, N ) < K ( s, N ).We define the stopping time T ε = sup { t ∈ [0 , T ) such that H ( t ) < C ( s, N ) M } , where C ( s, N ) denotes a constant (to be specified later) satisfying C ( s, N ) < C ( s, N ) < K ( s, N ) . (6.3)We remark that T ε > t H ( t ).We next choose κ ( s, N ) in such a way that κ ( s, N ) C ( s, N ) < K ( s, N ) √ c ( s, N ) . (6.4)By assumption on M , this implies that for κ ≤ κ ( s, N ) C ( s, N ) M < C ( s, N ) ν ε K ( s, N ) ≤ √ c ( s, N ) ε . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 25
In particular, since k ( b, z )( t ) k H s ≤ H ( t ), it follows that k| Ψ | ( t ) − k ∞ < , ∀ t ∈ [0 , T ε ) . (6.5)Our next purpose is to show that T ε = T = T ∗ = + ∞ .First, mollifying possibly ( a , u ) we may asume that ( b, z ) ∈ C ([0 , T ) , H s +1 ). By (6.5),Propositions 2.3 and 2.4 hold on [0 , T ε ), so that C ( s, N ) − H ≤ H + H (cid:16)(cid:0) κH (cid:1) κ max(1 , ν − ε ) + (cid:0) εH (cid:1) ( κ max(1 , ν − ε ) + ε + ν − ε ) (cid:17) ≤ H + H (cid:0) ε, κ ) H (cid:1) ( κ max(1 , ν − ε ) + ε + ν − ε ) . Observing that κ max(1 , ν − ε ) + ε + ν − ε ≤ κ + ν − ε ) , we find C ( s, N ) − H ≤ H + max( κ, ν − ε ) H (cid:0) ε, κ ) H (cid:1) . Here C ( s, N ) is a constant depending only on s and N , which can be assumed to be largerthan max( C ( s, N ) , t ∈ [0 , T ε ] we have according to (6.3) and byassumption on M max( ε, κ ) H ≤ max( ε, κ ) C ( s, N ) M ≤ , so that H ≤ C ( s, N ) (cid:16) M + max( κ, ν − ε ) H (cid:17) . (6.6)At this stage we may choose the constants C ( s, N ) and K ( s, N ) as follows: C ( s, N ) = 4 C ( s, N ) and K ( s, N ) > C ( s, N ) max( √ c ( s, N ) , , so that all conditions (6.1), (6.3) and (6.4) are met.We now show that T ε = T : otherwise T ε is finite. Hence, considering (6.6) at time T ε weobtain 4 C ( s, N ) M ≤ C ( s, N )( M + 16 max( κ, ν − ε ) C ( s, N ) M ) , whence 1 ≤ C ( s, N ) max( κ, ν − ε ) M ≤ C ( s, N ) K ( s, N ) . By definition of K ( s, N ), this leads to a contradiction, therefore T ε = T .Now, since (6.5) holds on [0 , T ), Corollary 3.1 and a standard continuation argument implythat T = T ∗ . Invoking again (6.5) we easily show that k∇ Ψ( t ) k H s ≤ C (cid:0) k ( b, v )( t ) k H s +1 × H s (cid:1) , ∀ t ∈ [0 , T ∗ )for a constant C . In view of the previous estimates we obtainlim sup t → T ∗ k∇ Ψ( t ) k H s ≤ lim sup t → T ∗ C (1 + H ( t ) ) < ∞ . We finally conclude that T ∗ = + ∞ thanks to Theorem 3.1. (cid:3) Proof of Theorem 1.3.
We present here the proof of Theorem 1.3. Here again, C always stands for a constant depending only on s and N . We define ( b ℓ , v ℓ )( t, x ) =( a ℓ , u ℓ )( ε − t, x ), where ( a ℓ , u ℓ ) is the solution to the linear equation (1.6) with initial datum( b , v ) = ( a , u ). Introducing ( b , v ) = ( b − b ℓ , v − v ℓ ), we have ∂ t b + √ ε div v + 2 ν ε ε b − κ ∆ b = f ( b, z ) ∂ t v + √ ε ∇ b − κ ∆ v = g ( b, z ) + ε √ ∇ ∆ b. The proof of Theorem 1.3 relies on energy estimates, since the method used in Section 5 is notconvenient to establish uniform in time estimates. For 0 ≤ k ≤ s we compute by integrationby parts12 ddt k ( D k b , D k v )( t ) k L = Z R N D k b D k ∂ t b + D k v · D k ∂ t v = − ν ε ε Z R N | D k b | − κ Z R N |∇ D k b | − κ Z R N |∇ D k v | + Z R N D k b D k f ( b, z ) + Z R N D k v · D k g ( b, z ) + ε √ Z R N D k v · D k ∇ ∆ b. We recall the decompositions f = ν ε f + f and g = g + εg = ∇ h + ε ∇ h , where the f i , g i , h i , i = 0 , ,
2, which have been defined in Paragraph 5.2, are i -order derivatives ofquadratic functions in ( b, z ). We obtain12 ddt k ( D k b , D k v )( t ) k L ≤ I + J + K, where I = − ν ε ε Z R N | D k b | + ν ε Z R N D k b D k f ( b, z ) J = Z R N D k b D k f ( b, z ) + Z R N D k v · D k g ( b, z ) ,K = − κ Z R N |∇ D k v | + ε Z R N D k v · D k g ( b, z ) + ε √ Z R N D k v · D k ∇ ∆ b. Estimates for I and J . By virtue of Lemma 7.4 and by Sobolev embedding we find I ≤ − ν ε ε Z R N | D k b | + Cεν ε Z R N | D k f | ≤ Cκ k f k H k ≤ Cκ k ( b, z ) k H s . Next, Cauchy-Schwarz inequality yields J ≤ k ( D k b , D k v ) k L k ( f , g ) k H k ≤ C k ( D k b , D k v ) k L k ( b, z ) k H k +1 . Estimate for K . We perform an integration by parts in the last two integrals and insert the fact that g = ∇ h to obtain K = − κ Z R N |∇ D k v | − ε Z R N div D k v D k h − ε √ Z R N div D k v D k ∆ b ≤ − κ Z R N |∇ D k v | + C ε κ Z R N | D k h | + Cκ Z R N | ε ∆ D k b | . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 27
First, by virtue of Cagliardo and Sobolev inequalities we have k D k h k L ≤ C k ( b, z ) k ∞ k ( b, z ) k H k +1 ≤ C k ( b, z ) k H s k ( b, z ) k H k +1 . Therefore: • If 0 ≤ k ≤ s − K ≤ Cκ − ε (cid:0) k ( b, z ) k H s + k b k H s (cid:1) . • If k = s − k ε ∆ D k b k L ≤ C k c k H s , where c = (1 − ε ∆ / / b is definedin the beginning of Section 5. So we find K ≤ Cκ − (cid:0) ε k ( b, z ) k H s + k c k H s (cid:1) . • If k = s , similar arguments using that k ε ∆ D k b k L ≤ C k c k H s +1 ≤ C k ( b, z ) k H s +1 (see(5.4)) yield K ≤ Cκ − k ( b, z ) k H s +1 (cid:0) ε k ( b, z ) k H s (cid:1) . Integrating the previous estimates for I , J and K on [0 , t ] we find: • If 0 ≤ k ≤ s − k ( D k b , D k v )( t ) k L ≤ C Z t k ( D k b , D k v ) k L k ( b, z ) k H s dτ + C Z t (cid:16) ( κ + κ − ε ) k ( b, z ) k H s + κ − ε k ( b, z ) k H s (cid:17) dτ. Appyling Young inequality to the first term in the right-hand side we infer that C − k ( D k b , D k v ) k L ∞ t ( L ) ≤ k ( b, z ) k L t ( H s ) +( κ + κ − ε ) k ( b, z ) k L ∞ t ( H s ) k ( b, z ) k L t ( H s ) + κ − ε k ( b, z ) k L t ( H s ) . (6.7) • Similarly, if k = s − C − k ( D k b , D k v ) k L ∞ t ( L ) ≤ k ( b, z ) k L t ( H s ) +( κ + κ − ε ) k ( b, z ) k L ∞ t ( H s ) k ( b, z ) k L t ( H s ) + κ − k c k L t ( H s ) . (6.8) • If k = s then C − k ( D k b , D k v ) k L ∞ t ( L ) ≤ k ( b, z ) k L t ( H s +1 ) + κ k ( b, z ) k L ∞ t ( H s ) k ( b, z ) k L t ( H s ) + κ − (cid:0) ε k ( b, z ) k L ∞ t ( H s ) (cid:1) k ( b, z ) k L t ( H s +1 ) . (6.9) Proof of the uniform in time comparison estimates in Theorem 1.3.
We control each term in the right-hand sides in (6.7), (6.8) and (6.9) by means of the variousestimates established in the previous sections. We recall that the control function H ( t ),which is defined in (6.2), satisfies H ( t ) ≤ CM . This controls the quantities k ( b, z ) k L t ( H s ) and k ( b, z ) k L ∞ t ( H s ) in terms of M . We use Proposition 5.1 to estimate k c k L t ( H s ) . Finally, tocontrol k ( b, z ) k L t ( H s +1 ) we rely on the second inequality in Proposition 2.3. Straightforwardcomputations then lead to the uniform comparison estimates in Theorem 1.3. Proof of the time dependent comparison estimates in Theorem 1.3.
We go back to the previous energy estimates. • If 0 ≤ k ≤ s − C − k ( D k b , D k v ) k L ∞ t ( L ) ≤ t k ( b, z ) k L ∞ t ( H s ) k ( b, z ) k L t ( H s ) + t ( κ + κ − ε ) k ( b, z ) k L ∞ t ( H s ) + tκ − ε k ( b, z ) k L ∞ t ( H s ) . • If k = s − C − k ( D k b , D k v ) k L ∞ t ( L ) ≤ t k ( b, z ) k L ∞ t ( H s ) k ( b, z ) k L t ( H s ) + t ( κ + κ − ε ) k ( b, z ) k L ∞ t ( H s ) + tκ − k ( b, z ) k L ∞ t ( H s ) . Using that H ( t ) ≤ CM , the assumptions on M as well as the fact that ( a ε , u ε )( t ) =( b ε , v ε )( εt ) we are led to the desired estimates. We omit the details. (cid:3) Appendix.
In this appendix we gather some helpful results.7.1.
Some parabolic estimates and useful tools.
The following result is an immediateconsequence of maximal regularity for the heat operator e t ∆ . We refer to [8] for further details. Lemma 7.1.
There exists
C > such that for all λ > , a ∈ L ( R N ) , a = a ( s ) ∈ L ( R + × R N ) and T > k e λt ∆ a k L T ( ˙ H ) ≤ C √ λ k a k L and (cid:13)(cid:13)(cid:13)(cid:13) ∆ Z t e λ ( t − s )∆ a ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) L T ( L ) ≤ Cλ k a k L T ( L ) . We also have the following
Lemma 7.2.
There exists
C > such that for all λ > and H ∈ L ( R + × R N ) (cid:13)(cid:13)(cid:13) Z t e λ ( t − s )∆ H ( s ) ds (cid:13)(cid:13)(cid:13) L T ( ˙ H ) ≤ C √ λ Z T k H ( t ) k L dt. Proof.
We may assume that H is smooth, compactly supported, and that the function u ( t ) = R t e λ ( t − s )∆ H ( s ) ds is the smooth solution to ∂ t u − λ ∆ u = H and u (0) = 0 . We infer that 12 ddt k u ( t ) k L = Z R N uH − λ Z R N |∇ u | , so that λ k∇ u k L T ( L ) ≤ C Z T Z R N | u || H | dt dx ≤ C sup t ∈ [0 ,T ] k u ( t ) k L k H k L T ( L ) . But u (0) = 0, therefore we also have k u ( t ) k L ≤ C R t R | uH | . This yieldssup t ∈ [0 ,T ] k u ( t ) k L ≤ C k H k L T ( L ) and the conclusion follows. (cid:3) Lemma 7.3.
There exists
C > such that for all λ > , a ∈ L , a ∈ L ( R + × R N ) and T > k e − λt a k L T ≤ C √ λ k a k L and (cid:13)(cid:13)(cid:13) Z t e − λ ( t − s ) a ( s, · ) ds (cid:13)(cid:13)(cid:13) L T ( L ) ≤ Cλ k a k L T ( L ) . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 29
Proof.
We only establish the second estimate. We set ˜ a ( s ) = a ( s ) for s ∈ [0 , T ] and ˜ a = 0 for s / ∈ [0 , T ], so that (cid:13)(cid:13)(cid:13) Z t e − λ ( t − s ) a ( s ) ds (cid:13)(cid:13)(cid:13) L T ( L ) ≤ (cid:13)(cid:13)(cid:13) Z T e − λ ( t − s ) k ˜ a ( s ) k L ( R N ) ds (cid:13)(cid:13)(cid:13) L T = k e − λ · ∗ k ˜ a ( · ) k L ( R N ) k L . By Young inequality for the convolution, we then have (cid:13)(cid:13)(cid:13) Z t e − λ ( t − s ) a ( s ) ds (cid:13)(cid:13)(cid:13) L T ( L ) ≤ C k e − λ · k L k ˜ a k L ( R + ,L ) . We conclude by definition of ˜ a . (cid:3) We conclude this paragraph with the following result, which is a consequence of Gagliardo-Nirenberg inequality.
Lemma 7.4 (see [2], Lemma 3) . Let k ∈ N and j ∈ { , . . . , k } . There exists a constant C ( k, N ) such that k D j uD k − j v k L ≤ C ( k, N ) (cid:16) k u k ∞ k D k v k L + k v k ∞ k D k u k L (cid:17) and k uv k H k ≤ C ( k, N ) ( k u k ∞ k v k H k + k v k ∞ k u k H k ) . Proof of Lemma 5.2.
In all the following C denotes a numerical constant. In order tosimplify the notations we introduce the quantities ω = ε | ξ | and µ = 1 ν ε | ξ |√ ω, and we express M as follows M = ν ε ε (cid:18) ω µ − µ ω (cid:19) . First we compute the eigenvalues λ and λ of M . Setting∆ = 1 − µ , we have λ = ν ε ε ( ω + 1 − C √ ∆) and λ = ν ε ε ( ω + 1 + C √ ∆) , where C √ ∆ is √ ∆ if ∆ ≥ i √− ∆ if ∆ <
0. Hence M = P − DP , where D = diag( λ , λ )and P − = 1 µ − α (cid:18) − µ αα − µ (cid:19) , P = (cid:18) − µ − α − α − µ (cid:19) , with α = 1 + C √ ∆ . Finally for all ( a, b ) ∈ C we have e − tM (cid:18) ab (cid:19) = P − e − tD P (cid:18) ab (cid:19) = 1 µ − α (cid:18) ( µ a + αµb ) e − λ t − ( α a + αµb ) e − λ t ( αµa + µ b ) e − λ t − ( αµa + α b ) e − λ t (cid:19) = e − νεε (1+ ω ) t µ − α ( µ a + αµb ) e t νεε C √ ∆ − ( α a + αµb ) e − t νεε C √ ∆ ( αµa + µ b ) e − t νεε C √ ∆ − ( αµa + α b ) e t νεε C √ ∆ ! , or equivalently e − tM (cid:18) ab (cid:19) = e − νεε (1+ ω ) t " e − t νεε C √ ∆ (cid:18) ab (cid:19) + e t νεε C √ ∆ − e − t νεε C √ ∆ µ − α (cid:18) αµb + µ a − αµa − α b (cid:19) . (7.1) First case | ξ | ≥ ν ε / µ ≥ /
4, hence ∆ ≤ /
4. We need to examine the following subcases. • ≤ ∆ ≤ / . It follows that C √ ∆ = √ ∆ and µ − α = − √ ∆), so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp( t ν ε ε √ ∆) − exp( − t ν ε ε √ ∆) µ − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sinh (cid:16) t ν ε ε √ ∆ (cid:17) √ ∆ ≤ C sinh (cid:18) ν ε t ε (cid:19) , where the second inequality is due to the fact that x sinh( x ) /x is an increasing functionon R + . We infer that (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ε t (cid:17) exp (cid:16) − ν ε ωε t (cid:17) (cid:0) | a | + | b | (cid:1) . (7.2) • − ≤ ∆ < . Then C √ ∆ = i √− ∆ and µ − α = − i √− ∆), therefore | µ − α | = 2 p ∆ − ∆ ≥ √− ∆ . It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp( it ν ε ε √− ∆) − exp( − it ν ε ε √− ∆) µ − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12) sin (cid:0) t ν ε ε √− ∆ (cid:1)(cid:12)(cid:12) √− ∆ ≤ C ν ε tε , where in the last inequality we have inserted that | sin x | ≤ x for all x ≥
0. Since | µ | ≤ C and | α | ≤ C this yields (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ε (1 + ω ) t (cid:17) (cid:16) ν ε ε t (cid:17) (cid:0) | a | + | b | (cid:1) , so finally (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ε (1 + ω ) t (cid:17) (cid:0) | a | + | b | (cid:1) . (7.3) • ∆ ≤ − . We have | µ − α | = 2 p ∆ − ∆ ≥ | ∆ | ≥ Cµ , while | α | = √ − ∆ = µ . Hence we find (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ε (1 + ω ) t (cid:17) (cid:0) | a | + | b | (cid:1) . (7.4) Second case | ξ | ≤ ν ε / µ ≤ κ / /
8, therefore 1 / ≤ ∆ ≤ κ < κ = p / . Moreover C − ≤ | µ − α | ≤ C, α ≤ C, µ ≤ C and µ ≤ C | ξ | ν ε . In addition, ν ε ε ( − √ ∆) = − ν ε ε − ∆1 + √ ∆ = − ν ε ε µ √ ∆ ≤ − C ν ε ε µ . Therefore in view of (7.1) (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ε (1 + ω ) t (cid:17) (cid:0) | a | + | b | (cid:1) + C exp (cid:16) − ν ε ωε t (cid:17) exp (cid:18) − Cν ε µ ε t (cid:19) (cid:18) | ξ | ν ε | a | + | b | (cid:19) . AMPED WAVE DYNAMICS FOR A GL EQUATION WITH LOW DISSIPATION 31
Now, since C | ξ | ν ε ≥ µ = | ξ | ν ε (2 + ω ) ≥ | ξ | ν ε we obtain (cid:12)(cid:12)(cid:12) e − tM ( ξ ) ( a, b ) (cid:12)(cid:12)(cid:12) ≤ C exp (cid:16) − ν ε ωε t (cid:17) (cid:18) exp (cid:16) − ν ε ε t (cid:17) + exp (cid:18) − C | ξ | ν ε ε t (cid:19)(cid:19) (cid:18) | ξ | ν ε | a | + | b | (cid:19) . (7.5)Gathering estimates (7.2) to (7.5) and setting r = p / (cid:3) Acknowlegments.
I warmly thank Didier Smets for his help. This work was partly sup-ported by the grant JC05-51279 of the Agence Nationale de la Recherche.
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