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Mathematical Modelling and Numerical Analysis Willbesetbythepublisher ModélisationMathématiqueetAnalyseNumérique 7 0 0 INFLUENCE OF BOTTOM TOPOGRAPHY ON LONG WATER WAVES 2 n a Florent Chazel1 J 8 Abstract. Wefocushereonthewaterwavesproblemforunevenbottomsinthelong-waveregime,on ] P anunboundedtwoorthree-dimensionaldomain. Inordertoderiveasymptoticmodelsforthisproblem, weconsidertwodifferentregimesofbottomtopography,oneforsmallvariationsinamplitude,andone A for strong variations. Starting from theZakharov formulation of this problem, we rigorously compute . h the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global t strategyintroducedbyBona,ColinandLannesin[6], newsymetricasymptoticmodelsarederivedfor a eachregimeofbottomtopography. Solutionsofthesesystemsareprovedtogivegood approximations m of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as [ well as for spatially periodic ones. 1 v Résumé. Nous nous intéressons ici au problème d’Euler surface libre pour des fonds non plats en 7 régime d’ondes longues, sur un domaine non borné à deux ou trois dimensions. Afin de construire 2 des modèles asymptotiques pour ce problème, nous considèrons deux régimes topographiques sur le 2 fond du domaine, l’un pour de petites variations en amplitude, et l’autre pour de fortes variations. A 1 partirdelaformulationdeZakhzarov,nouscontruisonsrigoureusementledéveloppementasymptotique 0 del’opérateur deDirichlet-Neumannrelatif au problème. En suivant lastratégie globale proposée par 7 Bona,ColinetLannesdans[6],nousobtenonsensuitedenouveauxmodèlesasymptotiquessymétriques 0 pourchaque régime de variation topographique du fond. Nousprouvons alors queles solutions deces / h systèmes fournissent de bonnes approximations aux solutions deséquations d’Euler surface libre. Ces t résultats sont valables aussi bien pour des solutions évanescentes à l’infini que pour des solutions a m spatialement périodiques. : 1991 Mathematics Subject Classification. 76B15, 35L55,35C20, 35Q35. v i X . r a Introduction Generalities This paper deals with the water waves problem for uneven bottoms which consists in describing the motion of the free surface and the evolution of the velocity field of a layer of fluid, under the following assumptions : the fluid is ideal, incompressible, irrotationnal,and under the only influence of gravity. Earlier works have set a good theoretical background for this problem : its well-posedness has been discussed Keywords and phrases: Water waves, uneven bottoms, bottom topography, long-wave approximation, asymptotic expansion, hyperbolicsystems,Dirichlet-Neumannoperator 1 LaboratoiredeMathématiques AppliquéesdeBordeaux,UniversitéBordeaux1,351CoursdelaLibération,F-33405Talence cedex;e-mail: [email protected] c EDPSciences,SMAI1999 (cid:13) 2 TITLEWILLBESETBYTHEPUBLISHER among others by Nalimov ( [18], 1974), Yoshihara ( [29], 1982), Craig ( [9], 1985), Wu ( [27], 1997, [28], 1999) and Lannes ( [16], 2005). Nevertheless, the solutions of these equations are very difficult to describe, because of the complexity of these equations. Atthispoint,aclassicalmethodistochooseanasymptoticregime,inwhichwelookforapproximate models and hence for approximate solutions. We consider in this paper the so-called long-wave regime, where the ratio of the typical amplitude of the waves over the mean depth and the ratio of the square of the mean depth over the square of the typical wave-length are both neglictible in front of 1 and of the same order. In2002,Bona,ChenandSautconstructedin[5]alargeclassofsystemsforthis regimeandperformedaformal study in the two-dimensional case. A significant step forward has been made in 2005 by Bona, Colin and Lannes in [6]; they rigorously justified the systems of Bona, Chen and Saut, and derived a new specific class of symmetric systems. Solutions of these systems areprovedto tend to solutions ofthe water wavesproblemona longtimescale,astheamplitudebecomessmallandthewavelengthlarge. Thankstotheirsymmetricstructure, computing solutions ofsuchsystemsis significantly easierthancomputing directly solutions ofthe water waves problem. Another significantworkin this field is the one of Lannes and Saut( [17], 2006)onweakly transverse Boussinesq systems. However, all these results only hold for flat bottoms. The case of uneven bottoms has been less investigated ; some of the significant references are Peregrine ( [23], 1967), Madsen et al. ( [19], 1991), Nwogu ( [22], 1993), and Chen ( [8], 2004). Peregrine was the first one to formulate the classical Boussinesq equations for waves in shallow water with variable depth on a three-dimensionnal domain. Following this work, Madsen et al. and Nwogu derived new Boussinesq-like systems for uneven bottoms with improved linear dispersion properties. Recently, Chenperformeda formalstudy ofthe waterwavesproblemfor unevenbottoms with smallvariations in amplitude, in 1D of surface, and derived a class of asymptotic models inspired by the work of Bona, Chen and Saut. To our knowledge, the only rigorously justified result on the uneven bottoms case is the work of Iguchi ( [12], 2004), who provided a rigorous approximation via a system of KdV-like equations, in the case of a slowly varying bottom. The main idea of our paper is to reconsider the water waves problem for uneven bottoms in the angle shown by Bona, Colin and Lannes. Moreover, our goal is to consider two different types of bottoms : bottoms with small variations in amplitude, and bottoms with strong variations in amplitude. To this end, we introduce a newparametertocharacterizetheshapeofthebottom. Intheend,newasymptoticmodelsarederived,studied and rigorously justified under the assumption that long time solutions to the water waves equations exist. Presentation and formulation of the problem In this paper, we work indifferently in two or three dimensions. Let us denote by X Rd the transverse ∈ variable, d being equal to 1 or 2. In the two-dimensional case, d=1 and X =x corresponds to the coordinate alongtheprimarydirectionofpropagationwhilstinthethree-dimensionalcase,d=2andX =(x,y)represents thehorizontalvariables. Werestrictourstudytothecasewherethefreesurfaceandthebottomcanbedescribed by the graph of two functions (t,X) η(t,X) and X b(X) defined respectively over the surface z = 0 and → → the mean depth z = h both at the steady state, t corresponding to the time variable. The time-dependant 0 − domain Ω of the fluid is thus taken of the form : t Ω = (X,z), X Rd, h +b(X) z η(t,X) . t 0 { ∈ − ≤ ≤ } In order to avoid some special physical cases such as the presence of islands or beaches, we set a condition of minimal water depth : there exists a strictly positive constant h such that min η(t,X)+h b(X) h , (t,X) R R2 . (0.1) 0 min − ≥ ∈ × For the sake of simplicity, we assume here that b and all its derivatives are bounded. 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1 ( η,1)T denotestheoutwardnormalvectortothesurfaceandn = 1 ( b, 1)T + √1+ η2 −∇ − √1+ b2 ∇ − |∇ | |∇ | denotes the outward normal vector to the bottom. The first equation corresponds to the Euler equation for a perfect fluid under the influence of gravity (which is characterized by the term ge where e denotes the z z − base vector along the vertical component). The second and third one characterize the incompressibility and irrotationnalityof the fluid. The fourth andlast ones deal with the boundary conditions atthe surface andthe bottom. These are given by the usual assumption that they are both bounding surfaces, i.e. surfaces across whichno fluidparticlesaretransported. As far asthe pressureP is concerned,we assumethatit is constantat the surface by neglicting the surface tension. Up to a renormalization, we can assume that it is equal to zero at the surface. In this paper, we use the Bernoulli formulation of the water-wavesequations. The conditions of incompressibility and irrotationnality ensure the existence of a potential flow φ such that V = φ. From now on, we X,z separate the transverse variable X Rd and the vertical variable z R : the operat∇ors and ∆ act only on the transverse variable X Rd so t∈hat we have V = φ+∂2φ. The∈use of the potential∇flow φ instead of the ∈ ∇ z velocity V leads to the following formulation of (0.2) : 4 TITLEWILLBESETBYTHEPUBLISHER ∂ φ+ 1 φ2+ ∂ φ2 +gz = P inΩ , t 0 , t 2 |∇ | | z | − t ≥  (cid:2) ∆φ+∂2φ=(cid:3)0 inΩ , t 0 ,  ∂tη− 1+|∇η|2 ∂zn+φ|z=η(t,X) =0 fortt≥0,≥X ∈Rd , (0.3)  p∂n−φ|z=−h0+b(X) =0 fort≥0, X ∈Rd , where we used the notations ∂n− =n−· ∇∂z and ∂n+ =n+· ∇∂z . (cid:18) (cid:19) (cid:18) (cid:19) Separating the variables X and z in the boundary conditions and taking the trace of (0.3) on the free surface thus leads to the system : ∆φ+∂2φ=0 inΩ , t 0 , z t ≥  ∂ φ+ 1 φ2+ ∂ φ2 +gη =0 atz =η(t,X), X Rd, t 0 ,  t ∂t2η(cid:2)+|∇∇η|·∇φ| −z ∂|zφ(cid:3) =0 atz =η(t,X), X ∈∈Rd, t≥≥0 , (0.4) b φ ∂ φ=0 atz = h +b(X), X Rd, t 0 .  ∇ ·∇ − z − 0 ∈ ≥ Wenowperformanon-dimensionalisationoftheseequationsusingthefollowingparameters: λisthetypical wavelength, a the typical amplitude of the waves, h the mean depth of the fluid, b the typical amplitude 0 0 of the bottom, t = λ a typical period of time (√gh corresponding to sound velocity in the fluid) and 0 √gh0 0 φ = λa√gh . Introducing the following parameters : 0 h0 0 a b aλ2 0 ǫ= ; β = ; S = , h h h3 0 0 0 and taking the Stokes number S to be equal to one, one gets for the non-dimensionnalized version of (0.4) : ε∆φ+∂2φ=0 1+βb z εη, X Rd, t 0 , z − ≤ ≤ ∈ ≥  ∂ φ+ 1 ε φ2+ ∂ φ2 +gη=0 atz =εη, X Rd, t 0 ,  t ∂tη2+(cid:2) ε|∇∇η|·∇φ|−z 1ε|∂z(cid:3)φ=0 atz =εη, X ∈∈Rd, t≥≥0 , (0.5) ∂ φ εβ b φ=0 atz = 1+βb, X Rd, t 0 .  z − ∇ ·∇ − ∈ ≥ The final step consists in recovering the Zakharov formulation by reducing the previous system (0.5) to a system expressed at the free surface. To this end, we introduce the trace of the velocity potential φ at the free surface, namely ψ : ψ(t,X)=φ(t,X,εη(t,X)) , TITLEWILLBESETBYTHEPUBLISHER 5 and the operator Z (εη,βb) which maps ψ to ∂ φ . This operator is defined for any f (C1 W1, )(Rd) ε z z=εη ∞ | ∈ ∩ by : H23(Rd) H12(Rd) −→  f ∂zu with u solution of :  7−→ |z=εη Zε(εη,βb)f : ε∆u+∂z2u=0, −1+βb≤z ≤εη ,  . (0.6)  ∂ u εβ b u=0, z = 1+βb,   z   − ∇ ·∇ −   u(X,εη)=f, X Rd .   ∈    Using this operator and computing the derivatives of ψ in terms of ψ and η, the final formulation (S ) of the 0 water waves problem reads : ∂ ψ ε∂ ηZ (εη,βb)ψ+ 1 ε ψ ε ηZ (εη,βb)ψ 2+ Z (εη,βb)ψ 2 +η =0 , t − t ε 2 |∇ − ∇ ε | | ε | (S ) (0.7) 0  h i  ∂tη+ε∇η·[∇ψ−ε∇ηZε(εη,βb)ψ]= 1εZε(εη,βb)ψ .  Organization of the paper The aim of this paper is to derive and study two different asymptotic regimes based each on a specific assumption on the parameter β which characterizes the topography of the bottom. The first assumption deals with the case β =O(ε) which corresponds to the physical case of a bottom with small variations in amplitude. The secondone dealswith the more complexcaseβ =O(1) whichcorrespondsto the physicalcaseofabottom with high variations in amplitude. The following part will be devoted to the asymptotic expansion of the operator Z (εη,βb) in the two regimes ε mentionned above. To this end, a general method is introduced and rigorously proved which aims at deriving asymptotic expansions of Dirichlet-Neumann operatorsfor a large classof elliptic problems. This result is then applied in each regime, wherein a formal expansion is performed and an asymptotic Boussinesq-like model of (0.7) is derived. The second and third part are both devoted to the derivation of new classes of equivalent systems, following the strategy developped in [6]. In the end, completely symmetric systems are obtained for each bottom topography regime : convergence results are proved showing that solutions of these symmetric asymptotic systems tend to associated solutions of the water waves problem. 1. Asymptotic expansion of the operator Zε(εη,βb) This section is devoted to the asymptotic expansion of the operator Z (εη,βb) defined in the previous section ε as ε tends to zero, in both regimes β = O(ε) and β = O(1). To this end, we first enounce some general results on elliptic equations on a strip : the final proposition gives a general rigourously justified method for determining an approximation of Dirichlet-Neumann operators. This result is then applied to the case of the operator Z (εη,βb) and two asymptotic models with bottom effects are derived. ε 1.1. Elliptic equations on a strip In this part, we aim at studying a general elliptic equation on a domain Ω given by : Ω= (X,z) Rd+1/X Rd, h +B(X)<z <η(X) , 0 { ∈ ∈ − } where the functions B and η satisfy the following condition : 6 TITLEWILLBESETBYTHEPUBLISHER h >0, X Rd , η(X) B(X)+h h . (1.1) min 0 min ∃ ∀ ∈ − ≥ Let us consider the following general elliptic boundary value problem set on the domain Ω : .P u=0 inΩ , (1.2) X,z X,z −∇ ∇ u =f and ∂ u =0 , (1.3) n |z=η(X) |z=−h0+B(X) where P is a diagonal(d+1) (d+1)matrix whosecoefficients (p ) are constantandstrictly positive. i 1 i d+1 × ≤ ≤ StraightforwardlyP is coercive. We denote by ∂ u the outwardconormalderivative associatedto n |z=−h0+B(X) P of u at the lower boundary z = h +B(X) , namely : 0 { − } ∂ u = n P u , n X,z |z=−h0+B(X) − − · ∇ |z=−h0+B(X) wheren denotestheoutwardnormalvectortothelowerboundaryofΩ. Forthesakeofsimplicity,thenotation − ∂ will always denote the outward conormal derivative associated to the elliptic problem under consideration. n Remark 1.1. When no confusion can be made, we denote by . X ∇ ∇ As in [6,16,21]we transformthe boundary value problem (1.2)(1.3) into a new boundary problem defined over the flat band = (X,z) Rd+1/X Rd, 1<z <0 . S { ∈ ∈ − } Let S be the following diffeomorphism mapping to Ω : S Ω S : S −→ . (1.4) (X,z) s(X,z)=(η(X) B(X)+h )z+η(X) 0 (cid:18) 7−→ − (cid:19) Remark 1.2. As shownin [16],a more complex ”regularizing”diffeomorphism must be used instead of ( ) to S obtain a shard dependence on η in terms of regularity, but since the trivial diffeomorphism ( ) suffices for our S present purpose, we use it for the sake of simplicity. Clearly, if v is defined over Ω then v = v S is defined over . As a consequence, we can set an equivalent ◦ S problem to (1.2)(1.3) on the flat band using the following proposition (see [15] for a proof) : S Proposition 1.3. u is solution of (1.2)(1.3) if and only if u=u S is solution of the boundary value problem ◦ .P u=0 in , (1.5) X,z X,z −∇ ∇ S u =f and ∂ u =0 , (1.6) n |z=0 |z=−1 where P(X,z) is given by 1 P(X,z)= MT P M , η+h B 0 − (η+h B)I (z+1) η+z B with M(X,z)= 0− d×d − ∇ ∇ . 0 1 (cid:18) (cid:19) TITLEWILLBESETBYTHEPUBLISHER 7 Consequently, let us consider boundary value problems belonging to the class (1.5)(1.6). From now on, all references to the problem set on will be labelled with an underscore. S On the class (1.5)(1.6) of problems set on the flat band , we have the following classical existence theorem : S assuming that P and all its derivatives are bounded on , if f Hk+23(Rd) then there exists a unique solution S ∈ u Hk+2( ) to (1.5)(1.6). The proof is very classical and we omit it here. ∈ S As previously seen, we need to consider the following operator Z(η,B) which maps the value of u at the upper bound to the value of ∂ u : z z=η | H23(Rd) H12(Rd) −→ Z(η,B): .  f ∂zuz=η with u solution of (1.2)(1.3)  7−→ |   εI 0 Remark1.4. TheoperatorZ definedin(0.6)correspondstotheoperatorZ inthecasewhereP = d ε 0 1 (cid:18) (cid:19) in (1.2)(1.3). ToconstructanapproximationofthisoperatorZ(η,B),weneedthe followinglemmawhichgivesacoercitivity result taking into account the anisotropy of (1.2)(1.3). Lemma 1.5. Let η W1, (Rd) and B W1, (Rd). Then for all V Rd+1 : ∞ ∞ ∈ ∈ ∈ (V , P V) c ( η , B ) √P V 2 , 0 W1,∞ W1,∞ ≥ || || || || | | where c is a strictly positive function given by 0 p d+1 min c0(x,y)= (dh+mi1n)2 min1,h (x+1 h +y),1≤(xi≤+d yp)2i  . min 0     Proof. Using Proposition 1.3 , we can write, with δ(X)=η(X)+h B(X) : 0 − 1 (V , PV) = V , MT P M V δ (cid:16)1 (cid:17) = M V , P M V δ (cid:16)1 (cid:17) = √P M V , √P M V δ (cid:16) 1 2 (cid:17) = (√P V) √δ M (cid:12) (cid:12) (cid:12) (cid:12) where = √P M(√P) 1. Thanks to the cond(cid:12)ition (1.1), we d(cid:12)educe the invertibility of M and hence the − invertibMility of . This yields the following norm inequality for all U Rd+1 : M ∈ 1 U (d+1) √δ 1 U , − | |≤ M √δ M (cid:12) (cid:12)∞ (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 8 TITLEWILLBESETBYTHEPUBLISHER with 1I 1 √Pd((z+1) η z B) −1 = δ d×d δ√pd+1 ∇ − ∇ . M 0 1 ! where |A|∞ =1 is,jupd+1|ai,j|L∞(Rd) and Pd is the d×d diagonal matrix whose coefficients are (pi)1≤i≤d. ≤ ≤ If we apply the previous inequality to our problem, one gets : 1 2 (V , P V) √P V . ≥ 2 (d+1)2 √δ 1 − (cid:12) (cid:12) M (cid:12) (cid:12) (cid:12) (cid:12)∞ (cid:12) (cid:12) (cid:12) (cid:12) Thanks to the expression of 1 given above, we ob(cid:12)tain the fo(cid:12)llowing inequality : − M (V , P V) c ( η , B ) √P V 2 , 0 W1,∞ W1,∞ ≥ || || || || | | where c as in the statement of the Lemma 1.5. (cid:3) 0 Let us introduce the space Hk,0( ) : S 1 0 2 Hk,0( )= v L2( ), v := v(,z) 2 dz <+ . S { ∈ S || ||Hk,0 | · |Hk(Rd) ∞} (cid:18)Z−1 (cid:19) The result of this subsection consists in the following theorem which aims at giving a rigourous method for deriving an asymptotic development of Z(η,B). Of course, P, and thus P, as well as the boundaries η and B, can depend on ε in the following theorem. In such cases, the proof can be easily adapted just by remembering that 0<ε<1. Theorem 1.6. Let p N ,k N , η Wk+2, (Rd) and B Wk+2, (Rd). Let 0 < ε < 1 and u be such ∗ ∗ ∞ ∞ app ∈ ∈ ∈ ∈ that P u =εpRε in , (1.7) X,z X,z app −∇ · ∇ S u =f , ∂ u =εprε , (1.8) app|z=0 n app|z=−1 where (Rε) and (rε) are bounded independently of ε respectively in Hk+1,0( ) and Hk+1(Rd). 0<ε<1 0<ε<1 Assuming that h is independent of ε and that the coefficients (p ) of P are suSch that ( pi ) are min i 1≤i≤d+1 pd+1 1≤i≤d bounded by a constant γ independent of ε, we have 1 εp Z(η,B)f (∂ u ) C ( Rε + rε ) , (cid:12) − η+h0−B z app |z=0(cid:12)Hk+12 ≤ √pd+1 k+2 || ||Hk+1,0 | |Hk+1 (cid:12) (cid:12) (cid:12) (cid:12) where C =(cid:12) C(η , B ) and C is a(cid:12)non decreasing function of its arguments, independent of the k+2 Wk+2,∞ Wk+2,∞ | | | | coefficients (p ) . i 1 i d+1 ≤≤ TITLEWILLBESETBYTHEPUBLISHER 9 Proof. In this proof, we often use the notation C =C(η , B ,h ,h ,k,d,γ) where C is an unde- k Wk,∞ Wk,∞ 0 min | | | | fined non decreasing function of its arguments. The notation C can thus refer to different constants, but of k the same kind. A simple calculus shows that Z(η,B) can be expressedin terms of the solution u of (1.5)(1.6) via the following relation : 1 Z(η,B)f = ∂ u . z η+h0 B |z=0 − Using this fact, we can write 1 1 Z(η,B)f ∂ u = ∂ (u u ) . − η+h0 B z app|z=0 η+h0 B z − app |z=0 − − Introducing ϕ:=u u we use a trace theorem (see Metivier [20] p.23-27) to get app − 1 Z(η,B)f (∂ u ) C ( ∂ ϕ + ∂2ϕ ) . (1.9) | − η+h0 B z app |z=0|Hk+12 ≤ k+1 || z ||Hk+1,0 || z ||Hk,0 − It is clear that the proof relies on finding an adequate control of ∂ ϕ and ∂2ϕ . The rest of this || z ||Hk+1,0 || z ||Hk,0 proof will hence be devoted to the estimate of both terms. 1. Let us begin with the estimate of ∂ ϕ . To deal correctly with this problem, we introduce the z Hk+1,0 || || following norm . defined by : ||||H˙1 ϕ := √P ϕ . || ||H˙1 || ∇X,z ||L2(S) First remark that for all α Nd such that α k, ∂αϕ solves : ∈ | |≤ P ∂αϕ=εp∂αRε+ [∂α,P] ϕ , X,z X,z X,z X,z −∇ · ∇ ∇ · ∇ (1.10) ( ∂αϕ z=0 =0, ∂n(∂αϕ) z= 1+∂[∂nα,P]ϕ z= 1 =εp∂αrε . | | − | − In order to get an adequate control of the norm ∂ ϕ , we prove the following estimate by induction on z Hk+1,0 || || α k: | |≤ εp α Nd / α k, ∂αϕ C ( Rε + rε ) . (1.11) ∀ ∈ | |≤ || ||H˙1 ≤ √pd+1 k+1 || ||Hk,0 | |Hk The proof of (1.11) is hence divided into two parts : initialization of the induction and heredity. Initialization : α =0. • | | Taking α=0, multiplying (1.10) by ϕ and integrating by parts leads to : (P ϕ, ϕ) + ∂ ϕ ϕ ∂ ϕ ϕ =(εpRε, ϕ) . ∇X,z ∇X,z L2(S) ZRd n |z=0 |z=0 −ZRd n |z=−1 |z=−1 L2(S) The boundary term at the free surface vanish because of the condition ϕ = 0 and using the condition at z=0 the bottom leads to : | (P ϕ, ϕ) =(εpRε, ϕ) +εp rεϕ . ∇X,z ∇X,z L2(S) L2(S) ZRd |z=−1 Finally, using Cauchy-Schwartzinequality, one gets : (P ∇X,zϕ, ∇X,zϕ)L2(S) ≤ εp||Rε||L2(S)||ϕ||L2(S)+εp|rε|L2(Rd)|ϕ|z=−1|L2(Rd) . (1.12) 10 TITLEWILLBESETBYTHEPUBLISHER Recalling that ϕ = 0 and that the band is bounded in the vertical direction, one can use Poincaré |z=0 S inequality so that ||ϕ||L2(S) ≤||∂zϕ||L2(S) and |ϕ|z=−1|L2(Rd) ≤||∂zϕ||L2(S). Therefore, (1.12) yields εp εp (P ∇X,zϕ, ∇X,zϕ)L2(S) ≤ √pd+1 ||Rε||L2(S)||ϕ||H˙1 + √pd+1 |rε|L2(Rd)||ϕ||H˙1 . (1.13) Using Lemma 1.5 to bound (P ϕ, ϕ) from below, one finally gets : ∇X,z ∇X,z L2( ) S εp c ( η , B ) ϕ 2 ( Rε + rε ) ϕ . 0 | |W1,∞ | |W1,∞ || ||H˙1 ≤ √pd+1 || ||H0,0 | |H0 || ||H˙1 Sincec ( η , B )dependsonlyonh ,dandγ throughthequantitymin pd+1 (byLemma1.5), 0 | |W1,∞ | |W1,∞ min 1≤i≤d pi and since the function c is a decreasing function of its arguments (again by Lemma 1.5), we get the following 0 desired estimate : εp ϕ C ( Rε + rε ) , || ||H˙1 ≤ √pd+1 1 || ||H0,0 | |H0 which ends the initialization of the induction. Heredity : for m N fixed such that m k, we suppose that (1.11) is verified for all α Nd such that ∗ • ∈ ≤ ∈ α m 1. L| e|t≤α −Nd such that α =m. Multiplying (1.10) by ∂αϕ and integrating by parts on leads to : ∈ | | S (P ∂αϕ, ∂αϕ) + ∂αϕ ∂ ∂αϕ ∂αϕ ∂ ∂αϕ =(εp∂αRε, ∂αϕ) ∇X,z ∇X,z L2(S) ZRd |z=0 n |z=0−ZRd |z=−1 n |z=−1 L2(S) ([∂α,P] ϕ, ∂αϕ) ∂αϕ ∂[∂α,P]ϕ + ∂αϕ ∂[∂α,P]ϕ . − ∇X,z ∇X,z L2(S)−ZRd |z=0 n |z=0 ZRd |z=−1 n |z=−1 The boundary terms at z =0 vanish because of the condition ∂αϕ = 0, and using the second boundary z=0 condition ∂ (∂αϕ) +∂[∂α,P]ϕ =εp∂αrε, one gets : | n z= 1 n z= 1 | − | − (P ∂αϕ, ∂αϕ) =(εp∂αRε, ∂αϕ) + εp ∂αϕ ∂αrε ([∂α,P] ϕ, ∂αϕ) , ∇X,z ∇X,z L2 L2 ZRd |z=−1 − ∇X,z ∇X,z L2 and with Cauchy-Schwartz : (P ∇X,z∂αϕ, ∇X,z∂αϕ)L2(S) ≤ εp||∂αRε||L2(S)||∂αϕ||L2(S)+εp|∂αrε|L2(Rd)|∂αϕ|z=−1|L2(Rd) + ([∂α,P] ϕ, ∂αϕ) . ∇X,z ∇X,z L2( ) S (cid:12) (cid:12) (cid:12) (cid:12) By using the same method and arguments as(cid:12)in the initialization, the following(cid:12) inequality arises : εp c ( η , B ) ∂αϕ 2 ( Rε + rε ) ∂αϕ + ([∂α,P] ϕ, ∂ ϕ) . 0 | |W1,∞ | |W1,∞ || ||H˙1 ≤ √pd+1 || ||Hk,0 | |Hk || ||H˙1 ∇X,z ∇X,z α L2(S) (cid:12) (cid:12) (cid:12) (1.1(cid:12)4) (cid:12) (cid:12) Let us now focus on the second term of the left hand side of (1.14). In order to get an adequate controlof this term, we have to write explicitly the commutator [∂α,P] : [∂α,P] ϕ= C(α , α )∂α′P ∂α′′ϕ , X,z ′ ′′ X,z ∇ | | | | ∇ α′+Xα′′=α α′=0 6