Cauchy theory for the water waves system in an analytic framework
aa r X i v : . [ m a t h . A P ] J u l CAUCHY THEORY FOR THE WATER WAVES SYSTEM IN ANANALYTIC FRAMEWORK.
THOMAS ALAZARD, NICOLAS BURQ AND CLAUDE ZUILY
Abstract.
In this paper we consider the Cauchy problem for gravity water waves, in adomain with a flat bottom and in arbitrary space dimension. We prove that if the data areof size ε in a space of analytic functions which have a holomorphic extension in a strip ofsize σ , then the solution exists up to a time of size C/ε in a space of analytic functions havingat time t a holomorphic extension in a strip of size σ − C ′ εt . Introduction.
The water-wave problem consists in describing, by means of the Euler equations, the dynam-ics of the free surface of a fluid. There are many different equations associated with thisproblem. Indeed, there are many different factors that dictate the dynamics of water-waves:the equations may be incompressible or not, irrotationnal or not, the fluid may have a fixed ormoving bottom, and the restoring forces may be determined by gravitation or surface tension.The study of these equations has received a lot of attention during the last decades and thereare now many cases in which the mathematical analysis is well developed. In particular, thereare many recent results concerning the well-posedness of the water-waves equations in Sobolevspaces in large time, including global existence results (see [50, 51, 18, 5, 26, 23, 49, 53] forthe equations without surface tension).In addition to the analysis of the Cauchy problem, another line of research is the mathematicaljustification of the derivation of approximate equations describing water-waves dynamics inasymptotic regimes. The most famous examples are the equations introduced by Boussinesqand Korteweg-de Vries (see [45, 35, 44] and references there-in). Kano and Nishida [29, 28]gave, in the two dimensional case, the first justification of the Friedrichs expansion for thewater-waves equations in terms of the shallowness parameter (by definition this is the ratio ofthe mean depth to the wavelength). In order to guarantee the existence of the solution for thefull equations, they used an abstract Cauchy-Kowalevski theorem in a scale of Banach spaces,so that analyticity of the initial data is required (see also Kano [27] and Kano-Nishida [30]).These results have been extended to include initial data belonging to usual Sobolev spaces, byCraig [15], Iguchi [24, 25], Bona, Lannes and Saut [13] or Alvarez-Samaniego and Lannes [9](see [35] for more references).The study of various nonlinear partial differential equations in spaces of analytic functionshas also received a great attention. We can mention the well-posedness results in analyticspaces by Kato and Masuda [31] which apply to many equations in fluid dynamics, the studyof the Rayleigh-Taylor instability by Sulem-Sulem [47], the study of the Cauchy problem or the semi-linear one dimensional Schr¨odinger equations by Bona-Grujic and Kalisch [12],Selberg-D.O.da Silva [46], the work on the KdV equation by Hayashi [20], Tesfahun [48]and on the periodic BBM equation by Himonas-Petronilho [21], the work by Kucavica-Vicol[32] on the Euler equation, the work on quasilinear wave equations and other quasilinearsystems by Alinhac and M´etivier [8] and Kuksin-Nadirashvili [33], the work by Matsuyamaand Ruzhansky [36] on the Kirchhoff equation, Gancedo-Granero-Belinch´on-Scrobogna [17]for the Muskat problem and the one of Pierre [43] for the MHD equations. We should alsomention the recent works by Mouhot-Villani [37], Bedrossian-Masmoudi-Mouhot [11] and byGrenier-Nguyen-Rodnianski [19] on the Landau damping for analytic and Gevrey data.Inspired by the pioneering works of Kano-Nishida, our goal is to revisit the analysis of thewater-problem with analytic data, using tools and methods that we developed previously tostudy the Cauchy problem with rough initial data. Our main result in this direction statesthat the solutions remain analytic for large time intervals.Let us now state our problem more precisely. We are mainly interested by the study ofthe Cauchy problem for the gravity wave system, in any space dimension. There are manypossible formulation for this problem. Here we use the classical Eulerian formulation and workwith the so-called Craig-Sulem-Zaharov formulation, following [16, 52]. In this formulation,there are two unknowns: ( i ) the free surface elevation η and ( ii ) the trace ψ of the velocitypotential on the free surface. These two unknowns depend on the time variable t and thehorizontal space variable x . Motivated by possible applications to control theory ([1, 54]), weassume below that x belongs to the d dimensional torus T d = ( R / π Z ) d , which means thatthe solutions are 2 π -periodic in each variable x j , 1 ≤ j ≤ d .We consider initial data in spaces of functions having a holomorphic extension to a fixed stripin the complex plane. Furthermore, we assume that the fluid domain has a flat bottom andconsider a source term on the bottom which belongs merely to a classical Sobolev space. Thisproblem can be written as follows. Given functions η , ψ on T d , and b on R × T d , solve thesystem,(1.1) ∂ t η − G ( η )( ψ, b ) = 0 ,∂ t ψ + gη + 12 |∇ x ψ | − (cid:0) ∇ x η · ∇ x ψ + G ( η )( ψ, b ) (cid:1) |∇ x η | = 0 ,η | t =0 = η , ψ | t =0 = ψ . Here G ( η ) denotes the Dirichlet-Neuman operator, which is is defined as follows. Given h > t , introduce the fluid domainΩ( t ) = { ( x, y ) ∈ T d × R : − h < y < η ( t, x ) } . Then, define the potential φ = φ ( t, x, y ) as the unique solution of the problem,(1.2) ∆ φ = 0 in Ω( t ) , φ | y = η ( t,x ) = ψ ( t, x ) , ∂ y φ | y = − h = b ( t, x ) . Then the Dirichlet-Neumann operator is defined by, G ( η )( ψ, b )( t, x ) = p |∇ x η | ∂ n φ | y = η ( t,x ) = ∂ y φ − ∇ x η · ∇ x φ y = η ( t,x ) . We refer to [3] for the proof that from solutions of (1.1) one may define solutions of theoriginal Euler equations. s in [28], we shall work in the spaces defined as follows. Let d ≥ T d = ( R / π Z ) d .Given σ ≥ s ≥
0, we define, H σ, s ( T d ) = n u ∈ L ( T d ) : k u k H σ, s := X ξ ∈ Z d e σ | ξ | h ξ i s | b u ( ξ ) | < + ∞ o with b u ( ξ ) = Z T d e − ix · ξ u ( x ) dx, h ξ i = (cid:0) | ξ | (cid:1) / . Several properties of these spaces are gathered in Appendix 8.Roughly speaking, the main result of this paper asserts that if the norms of the data η , ψ in such spaces and the norm of b in some Sobolev space are of size ε >
0, then our systemhas a unique solution in these spaces up to the time c ∗ /ε for some c ∗ >
0. It is classical sincethe work of Kato and Masuda [31] that, for solutions with analytic initial data, the width ofthe strip of analyticity might decrease with time. The main novelty here is that we show thatfor small data of size ε , the decrease is at most linear in ε . To prove this result, we cannotrely on an abstract Cauchy-Kowalevski theorem (as the ones introduced by Nirenberg [38],Ovsjannikov [40], Nishida [39] or Baouendi-Goulaouic [10]; used by Ovsjannikov [41, 42] andCastro-C´ordoba-Fefferman-Gancedo-G´omez-Serrano [14] to study the Cauchy problem for thewater-waves equations). A key difference between our work and previous ones is that we shalluse energy estimates on weighted Sobolev spaces, using the methods introduced in [7, 2, 4] tostudy the water-waves equations. To achieve these estimates we begin by a precise analysisof the Dirichlet-Neuman operator in the spaces of analytic functions. This requires a carefulstudy of elliptic equations with variable coefficients, which is of independent interest.1.1. The spaces of analytic functions and their characterizations.
It is well knownthat functions in H σ, s ( T d ) can be expressed as the traces on the real of functions which areholomorphic in a strip of the form S σ = { z ∈ C d : Re z ∈ T d , | Im z | < σ } . More precisely,for U ∈ H ol ( S σ ) and | y | < σ we shall denote by U y the function from T d to C defined by x U ( x + iy ) (here y ∈ R d and | y | denotes its euclidean norm). Then, for any u ∈ H σ, s ( T d ),there exists U ∈ H ol ( S σ ) such that U = u andsup | y | <σ k U y k H s x ( T d ) ≤ C k u k H σ, s . In Appendix 8, we prove a result of independent interest which clarifies the reciprocal.
Theorem 1.1.
Let σ > and s ≥ .1. Let U ∈ H ol ( S σ ) be such that M := sup | y | <σ k U y k H s x ( T d ) < + ∞ and set u = U .(i) If d = 1 , then u belongs to H σ, s ( T d ) and k u k H σ, s ≤ M . (ii) If d ≥ , then u belongs to H δ, s ( T d ) for any δ < σ and there exists a constant C δ > such that k u k H δ, s ≤ C δ M .2. Let U ∈ H ol ( S σ ) be such that, M := sup | y | <σ k U y k H s ′ x ( T d ) < + ∞ with s ′ > s + d − . Then the function u = U belongs to H σ, s ( T d ) and there exists a constant C > such that k u k H σ, s ≤ CM . emark 1.2. (i) In the case (1ii), in general we do not have u ∈ H σ, s ( T d ).(ii) If U is radial it is enough to assume in (2) that sup | y | <σ k U y k H s + d − x ( T d ) is finite.(iii) All the properties of these spaces needed in this paper are gathered in the Appendix.(iv) The same results hold with T d replaced by R d .1.2. Local in time well-posedness.
Consider the Cauchy-problem (1.1). Our first resultstates that, for data η , ψ of size ε , there exists a unique solution in the space of analyticfunctions on a time interval of size 1. Definition 1.3.
Given a real number s and time dependent index σ = σ ( t ) ≥ , we denote by C (cid:0) [0 , T ] , H σ, s (cid:1) the subspace of C (cid:0) [0 , T ] , H s (cid:1) which consists of those functions f such that F ∈ C (cid:0) [0 , T ] , H s (cid:1) where F ( t, · ) = e σ ( t ) | D x | f ( t, · ) . Proposition 1.4.
Let d ≥ and s > d . There exist positive constants ε , K, M anda positive time T ≤ ( λh ) /K such that, for all ε ≤ ε , for all < λ < , all ( η , ψ ) ∈H λh, s × H λh, s , all b ∈ L ∞ ( R , H s − ( T d )) ∩ L ( R , H s − ( T d )) such that, k b k L ∞ ( R ,H s − ) ∩ L ( R ,H s − ) + k η k H λh, s + k ψ k H λh, s ≤ ε, the Cauchy problem (1.1) has a unique solution (1.3) ( η, ψ ) ∈ C (cid:0) [0 , T ] , H σ, s × H σ, s (cid:1) ∩ L (cid:0) (0 , T ) , H σ, s + × H σ, s + (cid:1) with σ ( t ) = λh − Kt, such that, sup t ∈ [0 ,T ] k η ( t ) k H σ, s + sup t ∈ [0 ,T ] k ψ ( t ) k H σ, s + K Z T (cid:0) k η ( t ) k H σ, s + 12 + k ψ ( t ) k H σ, s + 12 (cid:1) dt ≤ M ε.
Remark 1.5.
This result complements the analysis by Kano and Nishida [28] and Kano [27]in that we allow a non-zero and non-analytic source term b .1.3. Well-posedness on large time intervals.
Our main result improves Proposition 1.4by showing that the solution exists and remains analytic on a large time interval whose sizeis proportional to the inverse of the size of the initial data. To state this result, we need tointroduce two auxiliary functions. Following [4], we set B = G ( η )( ψ, b ) + ∇ x η · ∇ x ψ |∇ x η | , V = ∇ x ψ − B ∇ x η. They are the traces on the free surface of the eulerian velocity field (see § N s ( b ) = k b k L ∞ ( R ,H s + 12 ) + k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) . Theorem 1.6.
Let d ≥ and consider real-numbers s > d and s ′ ∈ [ s − , s ) . There existpositive constants ε ∗ , K ∗ , c ∗ such that, for all ε ≤ ε ∗ and all ≤ λ < , if (1.4) N s ( b ) + k η k H λh, s + 12 + k a ( D x ) ψ k H λh, s + k V k H λh, s + k B k H λh, s ≤ ε, then the Cauchy problem (1.1) has a unique solution on the time interval [0 , c ∗ ε ] such that, (1.5) ( η, ψ, V, B ) ∈ C (cid:16)h , c ∗ ε i , H σ, s ′ + × H σ, s ′ + × H σ, s ′ × H σ, s ′ (cid:17) , with σ ( t ) = λh − K ∗ εt . emark 1.7. • One can assume without loss of generality that λh > K ∗ c ∗ , so that σ ( t ) > t in [0 , c ∗ /ε ]. • A loss in the radius of analyticity of size ε is optimal. • With a little extra work one can prove that the above result holds with s ′ = s .1.4. Organization of the paper.
In the next three Sections, (see § § §
4) we proveauxillary elliptic regularity results and apply them to study several different properties of theDirichlet-Neumann operator G ( η ) when η belongs to some analytic space. Then in Section § B, V ) introduced above. Proposition 1.4 isproved in Section § §
7. In the appendix, we gatherseveral results concerning analytic spaces, including the proof of Theorem 1.1.2.
Elliptic regularity.
All functions considered here will be real valued. We fix two real-numbers s , h and a function η = η ( x ) such that s > d , h > , η ∈ H h, s +1 ( T d ) , inf x ∈ R d η ( x ) > − h. Set Ω = { ( x, y ) : x ∈ T d , − h < y < η ( x ) } , Σ = { ( x, y ) : x ∈ T d , y = η ( x ) } , Γ = { ( x, y ) : x ∈ T d , y = − h } . We denote by n the unit normal to Σ and by ∂ n the normal derivative: n = 1 p |∇ x η | (cid:18) −∇ η (cid:19) , ∂ n = 1 p |∇ x η | ( ∂ y − ∇ x η · ∇ x ) . Given two functions ψ = ψ ( x ) and b = b ( x ), we consider the following elliptic problem:(2.1) ∆ x,y u = 0 in Ω , u | y = η = ψ, ∂ y u | y = − h = b. where ∆ x,y = ∂ y + ∆ x . Hereafter, given a function f = f ( x, y ), we use f | y = η as a shortnotation for the function x f ( x, η ( x )).The goal of this section is to obtain elliptic regularity results for the solutions of (2.1) in thespaces of analytic functions.2.1. Preliminaries.
For h > I h = ( − h, . Straightening the free surface.
We begin by making a change of variables to reduce theproblem to a fixed domain of the form e Ω = T d × I h = { ( x, z ) : x ∈ T d , − h < z < } . This change of variables will take ∆ x,y to a strictly elliptic operator and the normal derivative ∂ n to a vector field which is transverse to the boundary { z = 0 } . e consider a simple change of variables of the form ( x, z ) ( x, ρ ( x, z )). The simplestchange of variables reads ( x, z ) (cid:0) x, z + hh η ( x ) + z (cid:1) . For technical reasons, we will consideranother choice and introduce a smoothing change of variables (following Lannes [34]). Thismeans that the function ρ is given by(2.2) ρ ( x, z ) := 1 h ( z + h )( e z | D x | η )( x ) + z, x ∈ T d , − h ≤ z ≤ , where e z | D x | is the Fourier multiplier with symbol e z | ξ | . Since z ≤
0, this is a smoothingoperator, bounded from H µ ( T d ) to H z,µ ( T d ) for any real number µ . Notice that, ρ ( x,
0) = η ( x ) and ρ ( x, − h ) = − h .Since ∂ z ρ ( · , z ) − h e z | D x | η + h ( z + h ) e z | D x | | D x | η , for s > d/
2, the Sobolev embeddingimplies that, for all z ∈ I h , we have(2.3) k ∂ z ρ ( · , z ) − k L ∞ ( T d ) . k ∂ z ρ ( · , z ) − k H s ( T d ) . k η k H s ( T d ) . We refer the reader to Lemma 8.9 in the Appendix for the proof of more general estimates.Therefore, if k η k H s ≤ ε with ε small enough, the map ( x, z ) ( x, ρ ( x, z )) is a diffeo-morphism from e Ω to Ω. By this change of variables, the derivatives ∂ y and ∇ x become,respectively, Λ = 1 ∂ z ρ ∂ z , Λ = ∇ x − ∇ x ρ∂ z ρ ∂ z . More precisely, set(2.4) e u ( x, z ) = u ( x, ρ ( x, z )) , then e u solves(2.5) (Λ + Λ ) e u = 0 , in e Ω , e u | z =0 = ψ, ( ∂ z e u ) | z = − h = ( ∂ z ρ | z = − h ) b. Using the chain rule, one can expand Λ + Λ as follows:(2.6) Λ + Λ = 1 ∂ z ρ (cid:16) α∂ z + β ∆ x + γ · ∇ x ∂ z − δ∂ z (cid:17) where ,α = 1 + |∇ x ρ | ∂ z ρ , β = ∂ z ρ, γ = − ∇ x ρ,δ = 1 + |∇ x ρ | ∂ z ρ ∂ z ρ + ∂ z ρ ∆ x ρ − ∇ x ρ · ∇ x ∂ z ρ. It will be useful to observe that Λ + Λ is a perturbation of ∆ x,z = ∂ z + ∆ x , which can bewritten in divergence form. More precisely, by a direct computation, one can verify that(2.7) ( ∂ z ρ )(Λ + Λ ) e u = ∂ z (cid:16) |∇ x ρ | ∂ z ρ ∂ z e u − ∇ x ρ · ∇ x e u (cid:17) + div x (cid:0) ∂ z ρ ∇ x e u − ∂ z e u ∇ x ρ (cid:1) . Consequently, it follows from (2.5) that(2.8) ∆ x,z e u + R e u = 0 in e Ω , where(2.9) R e u = ∂ z (cid:16) |∇ x ρ | − ∂ z ρ∂ z ρ ∂ z e u − ∇ x ρ · ∇ x e u (cid:17) + div x (cid:16) ( ∂ z ρ − ∇ x e u − ∇ x ρ ∂ z e u (cid:17) . .1.2. The lifting of the trace.
Another standard approach consists in further transformingthe problem by simplifying the Dirichlet boundary condition on z = 0. To do so, given afunction ψ = ψ ( x, z ) satisfying ψ ( x,
0) = ψ ( x ), we shall set v = e u − ψ , solution to(2.10) (∆ x,z + R ) v = − (∆ x,z + R ) ψ in e Ω = T d × I h ,v | z =0 = 0 , ( ∂ z v ) | z = − h = ( ∂ z ρ | z = − h ) b. Parallel to the choice of the coordinate ρ in the above paragraph, a convenient choice for ψ is to consider the solution of an elliptic problem, to gain some extra regularity inside thedomain e Ω. Namely, we determine ψ by solving the problem,(2.11) ( ∂ z + ∆ x ) ψ = 0 in e Ω , ψ | z =0 = ψ, ∂ z ψ | z = − h = 0 . Notice that this problem can be explicitly solved using the Fourier transform in x . Moreprecisely we have, ψ ( x, z ) = (2 π ) − d P ξ ∈ Z d e ix · ξ b ψ ( ξ, z ) , where,(2.12) b ψ ( ξ, z ) = e z | ξ | e − h | ξ | b ψ ( ξ ) + e − h | ξ |− z | ξ | e − h | ξ | b ψ ( ξ ) . Now, we set(2.13) G (0) ψ = ∂ z ψ | z =0 . This is the Dirichlet-Neumann operator associated to the problem (2.11). By using (2.12),we find(2.14) G (0) = | D x | tanh( h | D x | ) . To shorten the notations we shall set in the sequel,(2.15) a ( D x ) = G (0) . By using the previous notation, we have the following result (proved in the appendix, seeLemma 8.14).
Lemma 2.1.
For all µ ∈ R , there exists C > such that for all σ ≥ and all ψ such that a ( D x ) ψ ∈ H σ,µ , there holds k∇ x,z ψ k L ( I h , H σ,µ ) ≤ C k a ( D x ) ψ k H σ,µ , k ∂ z ψ k L ( I h , H σ,µ − ) ≤ C k a ( D x ) ψ k H σ,µ , k∇ x,z ψ k L ∞ ( I h , H σ,µ − ) ≤ C k a ( D x ) ψ k H σ,µ . Remark 2.2.
There exists
C > σ ≥ µ ∈ R we have,(2.16) k| D x | ψ k H σ,µ − + k∇ x ψ k H σ,µ − ≤ C k a ( D x ) ψ k H σ,µ . This follows from the inequality | ξ | ≤ C h ξ i tanh ( h | ξ | ) for all ξ ∈ R d . .2. Elliptic regularity in analytic spaces.
In this paragraph, we specify the spaces inwhich we shall work to study the elliptic regularity theory. Recall the notation I h = ( − h, ∂ z + ∆ x ) w = 0 in T d × I h , w | z =0 = ψ, ∂ z w | z = − h = θ. Then, by using a Fourier calculation analogous to (2.12), one verifies that if ψ ∈ H h,µ ( T d )and θ ∈ H µ ( T d ), then e ( h + z ) | D x | w ∈ C ([ − h, , H µ ( T d )) , which is equivalent to e z | D x | w ∈ C ([ − h, , H h,µ ( T d )) . Our aim is to obtain a similar result for solutions to the general problem with variablecoefficients. However for the latter problem we will loose on the radius of analyticity. Namely,we will replace e z | D x | (resp. H h,µ ( T d )) by e λz | D x | (resp. H λh,µ ( T d )) for some λ ∈ [0 , Definition 2.3.
Let λ ∈ [0 , . For µ ∈ R , we introduce the spaces, (2.17) E λ,µ = { u : e λz | D x | u ∈ C ([ − h, , H λh,µ ( T d )) } ,F λ,µ = { u : e λz | D x | u ∈ L ( I h , H λh,µ ( T d )) } , X λ,µ = E λ,µ ∩ F λ,µ + . Remark 2.4.
Lemma 8.2 shows that ∇ x,z u ∈ F σ,µ + and D αx,z u ∈ F σ,µ − for | α | = 2 implythat ∇ x,z u ∈ E σ,µ . We are now in position to state our first two results concerning elliptic regularity in analyticSobolev spaces.
Proposition 2.5.
Consider real numbers λ , s , µ such that, ≤ λ < , s > d , ≤ µ ≤ s − . There exists ε > and C > such that for all ≤ λ ≤ λ , all η ∈ H λh, s ( T d ) satisfying k η k H λh, s ≤ ε , all F ∈ F λ,µ − , all θ ∈ H µ − ( T d ) and all w ∈ L ( I h , H µ +1 ( T d )) solution ofthe problem, (2.18) (∆ x,z + R ) w = F in T d × I h , w | z =0 = 0 , ( ∂ z w ) | z = − h = θ, the function w belongs to F λ,µ and satisfies (2.19) k∇ x,z w k F λ,µ ≤ C (cid:16) k F k F λ,µ − + k θ k H µ − ( T d ) (cid:17) . Remark 2.6.
For our purposes the estimate (2.19) is interesting for λ close to 1.Before proving this result, we pause to show how to deduce a variant of Proposition 2.5 witha non-vanishing trace on z = 0, assuming that the index µ is equal to s − orollary 2.7. Consider two real numbers λ , s such that, ≤ λ < , s > d . There exists ε > and C > such that, for all ≤ λ ≤ λ , all η ∈ H λh, s ( T d ) satisfying k η k H λh, s ≤ ε , all ψ ∈ H λh, s ( T d ) , all F ∈ F λ, s − , all θ ∈ H s − ( T d ) and all e w ∈ L ( I h , H s ( T d )) solution of the problem, (∆ x,z + R ) e w = F in T d × I h , e w | z =0 = ψ, ( ∂ z e w ) | z = − h = θ, the function ∇ x,z e w belongs to F λ, s − and satisfies (2.20) k∇ x,z e w k F λ, s − ≤ C (cid:16) k F k F λ, s − + (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − + k θ k H s − ( T d ) (cid:17) . Proof.
Since by (2.11) we have, ∆ x,z ψ = 0 , the function w = e w − ψ satisfies(∆ x,z + R ) w = F − Rψ in T d × I h , w | z =0 = 0 , ( ∂ z w ) | z = − h = θ. Consequently, (2.20) follows from the estimate (2.19) given by Proposition 2.5, applied with µ = s −
1, together with the estimate (8.26) for the remainder Rψ . (cid:3) Proof of Proposition 2.5.
As we have seen in the previous paragraph, one can used the Fouriertransform to study the analytic regularity of the solutions to the linearized problem∆ x,z w = F in T d × I h , w | z =0 = 0 , ( ∂ z w ) | z = − h = θ. However, since the operator R is a differential operator with variable coefficients, to study theregularity of the solution to (2.18), we must proceed differently. We will use the multipliermethod. More precisely, our strategy consists in conjugating the operator ∆ x,z + R by theweight e λ ( h + z ) | D x | . The trick here is that, when λ <
1, we obtain another coercive operatorand then the desired estimate (2.19) will follow from an energy estimate. The proof thusconsists in estimating the function e λ ( h + z ) | D x | w . To rigorously justify the computations, weshall truncate the symbol e λ ( h + z ) | ξ | , using the following lemma. Lemma 2.8.
Given ε > , λ ∈ [0 , and z ∈ I h , define the symbol q ε ( z, · ) : R d → R by q ε ( z, ξ ) = λ (cid:16) h | ξ | ε | ξ | + z | ξ | (cid:17) . Then, for all ξ, ζ ∈ R d we have, (2.21) q ε ( z, ξ ) ≤ λ ( h + z ) | ξ | , q ε ( − h, ξ ) = − λhε | ξ | ε | ξ | ≤ ,q ε ( z, ξ ) − q ε ( z, ζ ) ≤ λh | ξ − ζ | . Proof.
The two first claims are obvious. Then set h εξ i = 1 + ε | ξ | . We have, q ε ( z, ξ ) − q ε ( z, ζ ) = λ ( | ξ | − | ζ | ) (cid:18) h h εξ ih εζ i + z (cid:19) . If | ξ | − | ζ | ≥ h h εξ ih εζ i ≤ h , z ≤
0, together with the inequality | ξ | − | ζ | ≤| ξ − ζ | . If | ξ | − | ζ | ≤ h h εξ ih εζ i ≥ ≤ − z ≤ h. (cid:3) ow we fix s , µ satisfying s > d/ ≤ µ ≤ s and setΛ ε ( z ) = e q ε ( z,D x ) h D x i µ , h D x i µ = ( I − ∆ x ) µ/ , where q ε is defined in Lemma 2.8. Given a function f = f ( x, z ) defined for x ∈ T d and z ∈ I h ,we define Λ ε f as usual by (Λ ε f )( · , z ) = Λ ε ( z ) f ( · , z ). Then we set,(2.22) w ε = Λ ε w. Notice that this definition is meaningful since the symbol e q ε ( x,ξ ) is bounded and w belongsto L ( I h , H ( T d )). Our goal is to estimate the H ( e Ω) norm of w ε uniformly in ε , this willimply the wanted result by means of Fatou’s lemma.To form an equation on w ε , we notice that, for any function f = f ( x, z ),(2.23) ( ∂ z − λ | D x | ) e q ε ( z,D x ) f = e q ε ( z,D x ) ∂ z f, ∇ x (cid:0) e q ε ( z,D x ) f (cid:1) = e q ε ( z,D x ) ∇ x f. Therefore, setting(2.24) P λ = ( ∂ z − λ | D x | ) + ∆ x , we obtain P λ w ε = Λ ε ( ∂ z + ∆ x ) w . Since v solves (2.18), we conclude that w ε is solution ofthe problem,(2.25) P λ w ε = Λ ε ( − Rw + F ) ,w ε | z =0 = 0 , ( ∂ z w ε − λ | D x | w ε ) | z = − h = e q ε ( − h,D x ) h D x i µ θ =: θ ε . Now, the rest of the proof is divided in three steps: • Firstly, we will prove that the operator P λ is elliptic. • The second step is elementary. We check that the contributions of the Cauchy data F and θ are estimated by the right-hand side of (2.19). • In the third step, we prove a commutator estimate in analytic spaces and use it todeduce that the contribution of the source term e q ε ( z,D x ) Rv can be absorbed by theelliptic regularity, under a smallness assumption on the coefficients in the operator R . Step 1: The conjugated operator.
We begin by studying the operator P λ introducedin (2.24). We will see that it is an elliptic operator and prove some elementary ellipticestimates.Recall that, by notation, e Ω = T d × I h . We denote by H ( e Ω) the subspace of H ( e Ω) whichconsists of those functions whose trace on z = 0 vanishes, equipped with the H ( e Ω)-norm.Poincar´e’s inequality applies in this setting and there is a positive constant C e Ω such that(2.26) k u k L ( e Ω) ≤ C e Ω k∇ x,z u k L ( e Ω) , ∀ u ∈ H ( e Ω) . Now, consider the bilinear form, a ( u, v ) = (cid:0) ∂ z u, ∂ z v (cid:1) L ( e Ω) + (1 − λ ) (cid:0) ∇ x u, ∇ x v (cid:1) L ( e Ω) + λ (cid:0) ∂ z u, | D x | v (cid:1) L ( e Ω) − λ (cid:0) | D x | u, ∂ z v (cid:1) L ( e Ω) . his is a continuous bilinear form on H ( e Ω) × H ( e Ω). Moreover if u ∈ H ( e Ω) ∩ H ( e Ω) we canmake the following computations. (cid:0) ∂ z u, ∂ z v (cid:1) L ( e Ω) = − (cid:0) ∂ z u, v (cid:1) L ( e Ω) − Z T d ( ∂ z u ) v | z = − h dx, (cid:0) ∇ x u, ∇ x v (cid:1) L ( e Ω) = − (cid:0) ∆ x u, v ) L ( e Ω) , (cid:0) ∂ z u, | D x | v (cid:1) L ( e Ω) = (cid:0) | D x | ∂ z u, v (cid:1) L ( e Ω) , − (cid:0) | D x | u, ∂ z v (cid:1) L ( e Ω) = (cid:0) | D x | ∂ z u, v (cid:1) L ( e Ω) + Z T d ( | D x | u ) v | z = − h dx. It follows that(2.27) a ( u, v ) = (cid:0) − P λ u, v (cid:1) L ( e Ω) − Z T d (cid:2) ( ∂ z u − λ | D x | u ) v (cid:3) | z = − h dx. On the other hand, using the assumption λ <
1, and remembering that we are consideringreal-valued functions, we have,(2.28) a ( u, u ) = k ∂ z u k L ( e Ω) + (1 − λ ) k∇ x u k L ( e Ω) ≥ C (1 − λ ) k u k H ( e Ω) , where we used the classical Poincar´e inequality (2.26). Here C > λ. With the notations in (2.25) consider the linear form on H ( e Ω) , (2.29) L ( f ) = − (cid:10) Λ ε ( − Rw + F ) , f (cid:11) − (( θ ε , f | z = − h )) , where (cid:10) · , · (cid:11) denotes the duality between L z (cid:0) I h , H − ( T d ) (cid:1) and L z (cid:0) I h , H ( T d ) (cid:1) and (( · , · ))denotes the duality between H − ( T d ) and H ( T d ). We deduce from (2.27) that w ε is solutionof the problem,(2.30) a ( w ε , f ) = L ( f ) , ∀ f ∈ H ( e Ω) . Recall from (2.9) that Rw is given by,(2.31) Rv = ∂ z F + div x F , where(2.32) F = 1 + |∇ x ρ | − ∂ z ρ∂ z ρ ∂ z w − ∇ x ρ · ∇ x w, F = ( ∂ z ρ − ∇ x w − ∂ z w ∇ x ρ. Parallel to the computations above we immediately verify that,Λ ε ∂ z F = ( ∂ z − λ | D x | ) (cid:0) Λ ε F (cid:1) , Λ ε div x F = div x (cid:0) Λ ε F (cid:1) , so, Λ ε Rw = ( ∂ z − λ | D x | ) (cid:0) Λ ε F (cid:1) + div x (cid:0) Λ ε F (cid:1) . Integrating by parts with respect to z or x , we find that, (cid:10) Λ ε Rw, f (cid:11) = (cid:10) Λ ε F , ( ∂ z − λ | D x | ) f (cid:11) − (( e q ε ( − h,D x ) h D x i µ F | z = − h , f | z = − h ))+ (cid:10) Λ ε F , ∇ x f (cid:11) . The absolute value of the first and last term in the right-hand side above are estimated bymeans of the Cauchy-Schwarz inequality by, (cid:16)(cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) + (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) (cid:17) k∇ x,z f k L ( e Ω) . o estimate the second term, we use the fact that q ε ( − h, ξ ) ≤ (cid:12)(cid:12)(cid:12) (( e q ε ( − h,D x ) h D x i µ F | z = − h , f | z = − h )) (cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) e q ε ( − h,D x ) h D x i µ F | z = − h (cid:13)(cid:13)(cid:13) H − k f | z = − h k H ( T d ) . kh D x i µ F | z = − h k H − k f k H ( e Ω) . kh D x i µ F | z = − h k H − k∇ x,z f k L ( e Ω) , where we used the Poincar´e’s inequality. Similarly we have, | (( θ ε , f | z = − h )) | . k θ ε k H − k∇ x,z f k L ( e Ω) . We conclude that,(2.33) | L ( f ) | ≤ (cid:0) A + B + C + D (cid:1) k∇ x,z f k L ( e Ω) where A = (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( I h ,H − ( T d )) , B = k θ ε k H − ,C = (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) + (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) ,D = (cid:13)(cid:13) h D x i µ F | z = − h (cid:13)(cid:13) H − . By combining (2.30) and (2.33) (applied with f = w ε ) together with the coercive inequal-ity (2.28), we conclude that,(2.34) k∇ x,z w ε k L z ( I h ,L ) ≤ C λ ( A + B + C + D ) , where C λ = C − λ . Step 2: Estimates of A and B . Directly from Lemma 2.8 and (2.25) we have that,(2.35) k θ ε k H − ( T d ) ≤ k θ k H µ − ( T d ) . Similarly, using the property q ε ( z, ξ ) ≤ λ ( h + z ) | ξ | , we obtain, k Λ ε F k L z ( I h ,H − ( T d )) ≤ k e λz | D x | h D x i µ F k L z ( I h , H λh, − ) ≤ k F k F λ,µ − . Then the terms A and B in (2.34) are estimated by the right-hand side of the wanted in-equality (2.19). Step 3: estimate of C . We shall use the following lemma. Lemma 2.9.
Consider two real numbers ν > d , ≤ µ ≤ ν. Given two functions r, u : T d × I h → R set, U ( · , z ) = e q ε ( z,D x ) h D x i µ ( r ( · , z ) u ( · , z )) . Then there exists C = C ( d, ν, µ, h ) > such that, k U k L z ( I h ,L ( T d )) ≤ C k r k L ∞ z ( I h , H λh,ν ) k e q ε ( z,D x ) h D x i µ u k L z ( I h ,L ( T d )) . roof. By Lemma 2.8 we have, e q ε ( z,ξ ) = e q ε ( z,ξ ) − q ε ( z,ζ ) e q ε ( z,ζ ) ≤ e λh | ξ − ζ | e q ε ( z,ζ ) . So writing, b U ( ξ, z ) = 1(2 π ) d Z h ξ i µ h ξ − ζ i ν h ζ i µ e q ε ( z,ξ ) − q ε ( z,ζ ) h ξ − ζ i ν b r ( ξ − ζ, z ) e q ε ( z,ζ ) h ζ i µ b u ( ζ, z ) dζ, we find that (cid:12)(cid:12) b U ( ξ, z ) (cid:12)(cid:12) ≤ π ) d Z F ( ξ, ζ ) f ( ξ − ζ, z ) f ( ζ, z ) dζ where F ( ξ, ζ ) = h ξ i µ h ξ − ζ i − ν h ζ i − µ , and f ( θ, z ) = e λh | θ | h θ i ν | b r ( θ, z ) | , f ( ζ, z ) = e q ε ( z,ζ ) h ζ i µ | b u ( ζ, z ) | . Now, we are exactly in the setting introduced by H¨ormander to study the continuity of theproduct in Sobolev spaces (see Theorem 8.4 in the appendix and take s = µ , s = ν and s = µ ). We infer that, for any fixed z ∈ I h , k U ( · , z ) k L ≤ C k f ( · , z ) k L k f ( · , z ) k L . Using the Plancherel’s identity and then integrating in z , we obtain the desired result. (cid:3) We are now in position to estimate (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) + (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) . Remembering that, F = 1 + |∇ x ρ | − ∂ z ρ∂ z ρ ∂ z w − ∇ x ρ · ∇ x w, F = ( ∂ z ρ − ∇ x w − ∂ z w ∇ x ρ, we see that it is sufficient to estimate the L ( e Ω)-norm of a term of the form Λ ε ( αβ ) wherethe factors of the product are, α ∈ ( |∇ x ρ | − ∂ z ρ∂ z ρ , ∇ x ρ, ∂ z ρ − ) , β ∈ {∇ x w, ∂ z w } . Since s > d/ ≤ µ ≤ s − ν = s − k Λ ε ( αβ ) k L ( e Ω) . k α k L ∞ ( I h , H λh, s − ) k Λ ε β k L ( e Ω) . Now we claim that there exists
C > k Λ ε β k L ( e Ω) ≤ C k∇ x,z w ε k L ( e Ω) , (2.36) k α k L ∞ ( I h , H λh, s − ) ≤ Cε provided k η k H λh, s ≤ ε. (2.37)The proof of (2.36) is straightforward; indeedΛ ε ∂ x j w = ∂ x j Λ ε w = ∂ x j w ε , Λ ε ∂ z w = ( ∂ z − λ | D x | )Λ ε w = ( ∂ z − λ | D x | ) w ε . The second estimate follows from Lemma 8.11. Now, by using (2.37) and (2.36), we obtainthat, if k η k H λh, s ≤ ε we have, (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) + (cid:13)(cid:13) Λ ε F (cid:13)(cid:13) L ( e Ω) . ε k∇ x,z w ε k L ( e Ω) . tep 4: estimate of D . It remains only to estimate D = (cid:13)(cid:13) h D x i µ F | z = − h (cid:13)(cid:13) H − , where recall that F = 1 + |∇ x ρ | − ∂ z ρ∂ z ρ ∂ z w − ∇ x ρ · ∇ x w. By definition of ρ (see (8.13)) we have, ∂ z ρ | z = − h = 1 + 1 h e − h | D x | η ( x ) , ∇ x ρ | z = − h = 0 . Since ∂ z v | z = − h = θ we obtain, F | z = − h = − e − h | D x | ηh + e − h | D x | η θ. Then D ≤ C k θ k H µ − , where C depends only on h . Step 5: end of the proof.
By plugging the previous estimates in (2.34), we conclude that k∇ x,z w ε k L z ( I h ,L ) ≤ C λ (cid:0) k θ k H µ − ( T d ) + k F k F λ,µ − (cid:1) + C ′ λ ε (cid:13)(cid:13) ∇ x,z w ε (cid:13)(cid:13) L z ( I h ,L ) . Now, taking ε such that C ′ λ ε <
1, we obtain the uniform estimate,sup ε ∈ (0 ,ε ] k∇ x,z w ε k L z ( I h ,L ) ≤ C ′′ λ (cid:0) k θ k H µ − ( T d ) + k F k F λ,µ − (cid:1) . It follows from Fatou’s lemma that k∇ x,z w k L z ( I h ,L ) ≤ C ′′ λ (cid:0) k θ k H µ − ( T d ) + k F k F λ,µ − (cid:1) . This completes the proof of Proposition 2.5. (cid:3)
Sharp elliptic estimates.
We consider again the problem,(2.38) (∆ x,z + R ) e w = F in T d × I h , e w | z =0 = ψ, ∂ z e w | z = − h = θ, and our purpose is to refine the result proved in Corollary 2.7.Let us recall some elliptic estimates proved in [7, 4] for this problem:(2.39) k∇ x,z e w k L ( I h ; H s ) ≤ F (cid:16) k η k H s + 12 (cid:17) (cid:16) k F k L z ( I h ; H s − ) + k ψ k H s + 12 + k θ k H s − (cid:17) , for some non-decreasing function F : R + → R + . Here we will prove an analogue of thisestimate in the analytic setting. As in the previous paragraph, we will make a smallnessassumption on η . But we will only assume that the H λh, s -norm of η is small, and not the H λh, s + -norm. Also, for later purposes, we will prove a sharp estimate which is tame in thesense that the H λh, s + -norm of η will multiply only the H λh, s -norm of ψ . Theorem 2.10.
Consider two real numbers λ , s such that ≤ λ < , s > d . here exists ε > and a constant C > such that for all ≤ λ ≤ λ , all η ∈ H λh, s + ( T d ) satisfying k η k H λh, s ≤ ε , all ψ ∈ H λh, s + ( T d ) , all θ ∈ H s − ( T d ) and all solution e w of problem (2.38) , if we set, I = k∇ x,z e w k F λ, s , I = k ∂ z e w k F λ, s − , I = k∇ x,z e w k E λ, s − , then (2.40) X k =1 I k ≤ C (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s + k θ k H s − (cid:17) + C k η k H λh, s + 12 (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s − + k θ k H s − (cid:17) , where recall that a ( D x ) = G (0) as introduced in (2.15) . Remark 2.11.
Compared to the statement of Corollary 2.7 we notice that, under the samesmallness assumption on η , we are considering smoother data ψ , F and θ as well as smoothersolutions. Indeed, now θ and F belongs to H s − / ( T d ) and F λ, s − respectively, while weconsidered before the case where they belong to H µ − / ( T d ) and F λ,µ − for some µ ≤ s − ∇ x e w .The rest of this section is devoted to the proof of this result. Proof of Theorem 2.10.
Step 1. Notice that by interpolation, (see Lemma 8.8, iv ) with µ = s − ) we have,(2.41) I ≤ C ( I + I ) . Therefore, it is enough to estimate I and I .Step 2. We now prove that,(2.42) I ≤ C (cid:0) I + k F k F λ, s − + k η k H λh, s + 12 (cid:0) k F k F λ, s − + (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − + k θ k H s − ( T d ) (cid:1)(cid:1) . Since (∆ x,z + R ) w = F , we have ∂ z w = F − ∆ x w − Rw . Now, we use the technical esti-mate (8.17) in the appendix to handle the contribution of Rw . It follows that, k ∂ z e w k F λ, s − ≤ k F k F λ, s − + k∇ x e w k F λ, s − + Cε (cid:0) k∇ x,z e w k F λ, s − + k ∂ z e w k F λ, s − (cid:1) . Taking ε such that Cε ≤ we deduce that,(2.43) k ∂ z e w k F λ, s − ≤ C (cid:0) k F k F λ, s − + k∇ x,z e w k F λ, s − (cid:1) . By the same way, using (8.19) we get, k ∂ z e w k F λ, s − ≤ k F k F λ, s − + k∇ x e w k F λ, s + Cε (cid:0) k∇ x e w k F λ, s + (cid:13)(cid:13) ∂ z e w (cid:13)(cid:13) F λ, s − )+ C (cid:0) k ∂ z e w k F λ, s − + k η k H λh, s + 12 ( k ∂ z e w k F λ, s − + k ∂ z e w k F λ, s − (cid:1) . As before taking ε so small that Cε ≤ , we deduce that, k ∂ z e w k F λ, s − ≤ C (cid:0) k F k F λ, s − + k∇ x,z e w k F λ, s + k η k H λh, s + 12 ( k ∂ z e w k F λ, s − + k ∂ z e w k F λ, s − (cid:1) . ow, since s − ≤ s − s − ≤ s − k ∂ z e w k F λ, s − ≤ C (cid:0) k F k F λ, s − + k∇ x,z e w k F λ, s + k η k H λh, s + 12 ( k F k F λ, s − + k∇ x,z e w k F λ, s − ) (cid:1) . Now by Corollary 2.7 we have,(2.45) k∇ x,z e w k F λ, s − ≤ C (cid:16) k F k F λ, s − + (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − + k θ k H s − ( T d ) (cid:17) . Since k∇ x,z e w k F λ, s = I by definition, we obtain the desired estimate (2.42) by plugging theprevious inequality in (2.44).Step 3. We are left with the estimate of I . We shall prove that,(2.46) I ≤ C (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s + k θ k H s − (cid:17) + C k η k H λh, s + 12 (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s − + k θ k H s − (cid:17) + C k η k H λh, s ( I + I ) . This will conclude the proof of desired estimate (2.40); by taking an appropriate linear com-bination of (2.41), (2.42) and (2.46) and then taking ε ≥ k η k H λh, s small enough to absorb thecontribution of I + I in the right-hand side of (2.46).Notice that(2.47) I = (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s ≤ (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − + (cid:13)(cid:13) ∇ x,z ∇ x e w (cid:13)(cid:13) F λ, s − . We will prove that these two terms are bounded by the right-hand side of (2.46).The first term k∇ x,z e w k F λ, s − has been estimated in Corollary 2.7. By (2.20) we have, k∇ x,z e w k F λ, s − ≤ C (cid:16) k F k F λ, s − + (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − + k θ k H s − ( T d ) (cid:17) , which immediately implies that it is bounded by the right-hand side of (2.46).Hence, it remains only to estimate (cid:13)(cid:13) ∇ x,z ∇ x e w (cid:13)(cid:13) F λ, s − . We will estimate the F λ, s − -norm of ∇ x,z ∂ j e w for 1 ≤ j ≤ d . Notice that ∂ j e w satisfies(2.48) (∆ x,z + R ) ∂ j e w = − [ ∂ j , R ] w + ∂ j F in T d × I h , e w | z =0 = ∂ j ψ, ∂ z e w | z = − h = ∂ j θ, where [ ∂ j , R ] e w = ∂ j ( R e w ) − R∂ j e w . It follows from Corollary 2.7 that,(2.49) k∇ x,z ∂ j e w k F λ, s − ≤ C (cid:16)(cid:13)(cid:13) [ ∂ j , R ] e w (cid:13)(cid:13) F λ, s − + k ∂ j F k F λ, s − + (cid:13)(cid:13) a ( D x ) ∂ j ψ (cid:13)(cid:13) H λh, s − + k ∂ j θ k H s − (cid:17) . The key point is to estimate the commutator [ ∂ j , R ] w . We claim that(2.50) (cid:13)(cid:13) [ ∂ j , R ] e w (cid:13)(cid:13) F λ, s − ≤ C k η k H λh, s (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − + C k η k H λh, s + 12 (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − . et us assume this claim and conclude the proof. By combining (2.49), (2.50) and (2.45), wehave(2.51) k∇ x,z ∂ j e w k F λ, s − ≤ C (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s + k θ k H s − (cid:17) + C k η k H λh, s + 12 (cid:16) k F k F λ, s − + k a ( D x ) ψ k H λh, s − + k θ k H s − (cid:17) . + C k η k H λh, s (cid:13)(cid:13) ∇ x,z w (cid:13)(cid:13) F λ, s − . By summing all these estimates for 1 ≤ j ≤ d , this will give the wanted estimate for (cid:13)(cid:13) ∇ x,z ∇ x w (cid:13)(cid:13) F λ, s − , which will conclude the proof of the theorem as explained after (2.46).We now have to prove the claim (2.50). Recall that,(2.52) R = a∂ z + b ∆ x + c · ∇ x ∂ z − d∂ z , where ,a = 1 + |∇ x ρ | ∂ z ρ − , b = ∂ z ρ − , c = − ∇ x ρ,d = 1 + |∇ x ρ | ∂ z ρ ∂ z ρ + ∂ z ρ ∆ x ρ − ∇ x ρ · ∇ x ∂ z ρ. We have, [ ∂ j , R ] w = ( ∂ j a ) ∂ z w + ( ∂ j b )∆ x w + ( ∂ j c ) · ∇ x ∂ z w − ( ∂ j d ) ∂ z w. Then we use statement i ) in Lemma 8.8 (applied with s = s = s = s − > d/
2) to writethat, (cid:13)(cid:13) ( ∂ j a ) ∂ z e w (cid:13)(cid:13) F λ, s − . k ∂ j a k E λ, s − (cid:13)(cid:13) ∂ z e w (cid:13)(cid:13) F λ, s − , (cid:13)(cid:13) ( ∂ j b )∆ x e w (cid:13)(cid:13) F λ, s − . k ∂ j b k E λ, s − (cid:13)(cid:13) ∆ x e w (cid:13)(cid:13) F λ, s − , (cid:13)(cid:13) ( ∂ j c ) · ∇ x ∂ z e w (cid:13)(cid:13) F λ, s − . k ∂ j c k E λ, s − (cid:13)(cid:13) ∇ x ∂ z e w (cid:13)(cid:13) F λ, s − . By lemma 8.11 we have, k a k E λ, s − + k b k E λ, s − + k c k E λ, s − . k η k H λh, s . Therefore (cid:13)(cid:13) ( ∂ j a ) ∂ z e w (cid:13)(cid:13) F λ, s − + (cid:13)(cid:13) ( ∂ j b )∆ x e w (cid:13)(cid:13) F λ, s − + (cid:13)(cid:13) ( ∂ j d ) ∂ z e w (cid:13)(cid:13) F λ, s − . k η k H λh, s (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − . It remains to estimate the term ( ∂ j d ) ∂ z e w . To do so, again, we begin by applying the productrule given by ii ) in Lemma 8.8 (applied with s replaced by s − > d/
2) to write that (cid:13)(cid:13) ( ∂ j d ) ∂ z e w (cid:13)(cid:13) F λ, s − . k ∂ j d k F λ, s − (cid:13)(cid:13) ∂ z e w (cid:13)(cid:13) E λ, s − . Then, by Lemma 8.11 we have, k ∂ j d k F λ, s − ≤ k d k F λ, s − . k η k H λh, s + 12 . By combining the previous estimates, we see that, to complete the proof of the claim (2.50),it remains to estimate (cid:13)(cid:13) ∂ z e w (cid:13)(cid:13) E λ, s − and (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − in terms of (cid:13)(cid:13) ∇ x,z e w (cid:13)(cid:13) F λ, s − . Since wewill need to prove a similar result later on, we pause here to prove a general result. Lemma 2.12.
Consider two real numbers s > d , λ ∈ [0 , . here exist ε > and a constant C > such that for all η ∈ H λh, s ( T d ) satisfying k η k H λh, s ≤ ε ,if v satisfies ∆ x,z v + Rv = f , then (2.53) k∇ x,z v k E λ, s − + (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − ≤ C k∇ x,z v k F λ, s − + C k f k F λ, s − . Proof.
By interpolation (see statement iv ) in Lemma 8.8), we have (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) E λ, s − . (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − + (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − . Therefore, it is sufficient to prove that (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − is estimated by the right-hand side of(2.53). To do so, we repeat the arguments used in Step 2. Namely, we write ∂ z v = − ∆ x v − Rv + f , to infer that (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − ≤ k∇ x v k F λ, s − + k Rv k F λ, s − + k f k F λ, s − . Then we use the estimate (8.17) to estimate the contribution of Rv , which implies that, (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − ≤ k∇ x v k F λ, s − + Cε (cid:0) k∇ x,z v k F λ, s − + (cid:13)(cid:13) ∂ z v (cid:13)(cid:13) F λ, s − (cid:1) + k f k F λ, s − . Then we conclude the proof by taking ε so small that Cε ≤ / (cid:3) By applying the previous lemma to ( v, f ) = ( e w, F ), we complete the proof of the claim (2.50),which in turn concludes the proof of the theorem. (cid:3) We consider eventually the problem,(2.54) (∆ x,z + R ) e u = 0 in T d × I h , e u | z =0 = ψ, ∂ z e u | z = − h = ( ∂ z ρ | z = − h ) b. Corollary 2.13.
Consider two real numbers λ , s such that ≤ λ < , s > d . There exists ε > and a constant C > such that for all ≤ λ ≤ λ , all η ∈ H λh, s + ( T d ) satisfying k η k H λh, s ≤ ε , all ψ ∈ H λh, s + ( T d ) , all b ∈ H s − ( T d ) and all solution e u of problem (2.54) , if we set, I = k∇ x,z e u k F λ, s , I = k ∂ z e u k F λ, s − , I = k∇ x,z e u k E λ, s − then, (2.55) X k =1 I k ≤ C (cid:16) k a ( D x ) ψ k H λh, s + k b k H s − + k η k H λh, s + 12 (cid:16) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:17)(cid:17) , where recall that a ( D x ) = G (0) as introduced in (2.15) .Proof. This follows from Theorem 2.10 and the fact that, since ∂ z ρ | z = − h = 1 + h e − h | D x | η with k η k H λh, s ≤
1, we have, for µ = s − , µ = s − , k ( ∂ z ρ | z = − h ) b k H µ ≤ C k b k H µ . (cid:3) . The Dirichlet Neumann operator.
Given functions ψ , b we consider the problem,(3.1) ∆ x,y u = 0 in Ω , u | y = η = ψ, ∂ y u | y = − h = b. We set,(3.2) G ( η )( ψ, b ) = p |∇ x η | ∂ n u | y = η = (cid:0) ∂ y u − ∇ x η · ∇ x u (cid:1) | y = η . This is the Dirichlet-Neumann operator associated to problem (3.1). Notice that using thenotations in (2.13) we have G (0) ψ = G (0)( ψ,
0) = a ( D x ) ψ = | D x | tanh( h | D x | ) ψ. We have then the following result.
Theorem 3.1.
Consider a real number s > d/ . For all ≤ λ < , there exist ε > and C > such that,for all ≤ λ ≤ λ , for all η ∈ H λh, s + ( T d ) satisfying k η k H λh, s ≤ ε , all ψ such that a ( D x ) ψ ∈ H λh, s ( T d ) , all b ∈ H s − ( T d ) we have, (3.3) k G ( η )( ψ, b ) k H λh, s − ≤ C (cid:0) k a ( D x ) ψ k H λh, s + k b k H s − + k η k H λh, s + 12 (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1)(cid:1) . Remark 3.2. (1) We insist on the fact that the constants ε and C in the above Theoremdepend on λ but not on λ as soon as 0 ≤ λ ≤ λ . (2) Assume s > δ + d , b ∈ H s − ( T d ) and η j ∈ H λh, s + ( T d ) with k η j k H λh, s ≤ ε, a ( D x ) ψ j ∈ H λh, s ( T d ) ( j = 1 , . Then we may apply the above Theorem with s ′ = s − δ and we obtain an estimate of k G ( η )( ψ, b ) k H λh, s − − δ by the right hand side of (3.3) where s is replaced by s − δ. Proof.
We use the notations introduced in § x, z ) ∈ T d × ( − h, G ( η )( ψ, b ) = (cid:16) |∇ x ρ | ∂ z ρ ∂ z e u − ∇ x ρ · ∇ x e u (cid:17) z =0 , where e u = u ( x, ρ ( x, z )) satisfies the following elliptic boundary value problem:(∆ x,z + R ) e u = 0 , e u | z =0 = ψ, ( ∂ z e u ) | z = − h = ( ∂ z ρ | z = − h ) b. Introduce the function, U = 1 + |∇ x ρ | ∂ z ρ ∂ z e u − ∇ x ρ · ∇ x e u, and set I h = [ − h, G ( η )( ψ, b ) = U | z =0 , by definition of the spaces E λ,µ we have, k G ( η )( ψ, b ) k H λh, s − ≤ k U k E λ, s − . Now, we use an interpolation argument (see statement iv ) in Lemma 8.8), to infer that(3.5) k G ( η )( ψ, b ) k H λh, s − ≤ k U k E λ, s − ≤ C k ∂ z U k F λ, s − + C k U k F λ, s . he rest of the proof consists in estimating U and ∂ z U in terms of ∇ x,z e u , so that the wantedestimate (3.3) will be a consequence of Proposition 2.7 and Theorem 2.10. Lemma 3.3.
For all s > d/ and all ≤ λ < , there exist ε > and C > such that,for all η ∈ H λh, s + satisfying k η k H λh, s ≤ ε , there holds, (3.6) k U k F λ, s ≤ C ( k∇ x,z e u k F λ, s + k η k H λh, s + 12 k∇ x,z e u k F λ, s − ) , and (3.7) k ∂ z U k F λ, s − ≤ C ( k∇ x,z e u k F λ, s + k η k H λh, s + 12 k∇ x,z e u k F λ, s − ) . Proof.
Write U under the form U = (1 + a ) ∂ z e u − ∇ x ρ · ∇ x e u , where a is as defined in (2.52).Then, the tame product estimate (8.12) (applied with s = s −
2) implies that k U k F λ, s . (1 + k a k E λ, s − ) k ∂ z e u k F λ, s + k∇ x ρ k E λ, s − k∇ x e u k F λ, s + k a k F λ, s k ∂ z e u k E λ, s − + k∇ x ρ k F λ, s k∇ x e u k E λ, s − . The contribution of ρ is estimated by means of Lemma 8.9 and the one of a is estimatedby Lemma 8.11 (which implies that a belongs to the space E ). Consequently, provided k η k H λh, s ≤ ε ≤
1, we have, k a k E λ, s − + k∇ x ρ k E λ, s − ≤ C k η k H λh, s ≤ C, k a k F λ, s + k∇ x ρ k F λ, s ≤ C k η k H λh, s + 12 . By combining the previous estimates, we obtain that(3.8) k U k F λ, s ≤ C k∇ x,z e u k F λ, s + k η k H λh, s + 12 k∇ x,z e u k E λ, s − , Now, since ∆ x,z e u + R e u = 0, it follows from Lemma 2.12 that(3.9) k∇ x,z e u k E λ, s − ≤ C k∇ x,z e u k F λ, s − . By plugging this bound in (3.8), we conclude the proof of the estimate (3.6).We now estimate ∂ z U . To do so, we exploit the fact that the equation ∆ x,z e u + R e u = 0 canbe written in divergence form, as we have seen in (2.7). More precisely, we have,(3.10) ∂ z (cid:16) |∇ x ρ | ∂ z ρ ∂ z e u − ∇ x ρ · ∇ x e u (cid:17) + div x (cid:0) ∂ z ρ ∇ x e u − ∂ z e u ∇ x ρ (cid:1) = 0 . This immediately implies that k ∂ z U k F λ, s − ≤ k ∂ z ρ ∇ x e u − ∂ z e u ∇ x ρ k F λ, s . Now, as above, we apply the tame product estimate (8.12) to infer that k ∂ z U k F λ, s − . k∇ x,z ρ k E λ, s − k∇ x,z e u k F λ, s + k∇ x,z ρ k F λ, s k∇ x,z e u k E λ, s − . Then we use Lemma 8.9 to obtain(3.11) k ∂ z U k F λ, s − ≤ C k∇ x,z e u k F λ, s + k η k H λh, s + 12 k∇ x,z e u k E λ, s − , and hence the desired estimate (3.7) follows from (3.9). (cid:3) In view of (3.5) and the previous lemma, the estimate (3.3) follows directly from Corollary 2.7and Theorem 2.10. (cid:3) n the following result we shall prove that, in a certain sense, the Dirichlet-Neuman operatoris Lipschitz with respect to ( ψ, η ). Let us introduce some notations. If ( θ , θ ) is a givencouple of functions and t ∈ R we shall set, θ = θ − θ , k ( θ , θ ) k H λh,t = X j =1 k θ j k H λh,t , and we shall use these notations if θ = η or a ( D x ) ψ. Moreover we shall set,(3.12) G j = G ( η j )( ψ j , b ) , j = 1 , ,λ = k ( η , η ) k H λh, s + 12 (cid:0) k ( a ( D x ) ψ , a ( D x ) ψ ) k H λh, s − + k b k H s − (cid:1) + k ( a ( D x ) ψ , a ( D x ) ψ ) k H λh, s + k b k H s − ,λ = k ( a ( D x ) ψ , a ( D x ) ψ ) k H λh, s − + k b k H s − ,λ = k ( η , η ) k H λh, s + 12 . Then we have the following result.
Theorem 3.4.
For all s > d/ and all ≤ λ < , there exist C > and ε > , suchthat for all ≤ λ ≤ λ , for all η j ∈ H λh, s + ( T d ) satisfying k η j k H λh, s ≤ ε, all ψ j such that a ( D x ) ψ j ∈ H λh, s ( T d ) , j = 1 , and all b ∈ H s − ( T d ) we have, kG − G k H λh, s − ≤ C (cid:16) λ k η − η k H λh, s + λ k η − η k H λh, s + 12 + λ k a ( D x ) ( ψ − ψ ) k H λh, s − + k a ( D x ) ( ψ − ψ ) k H λh, s (cid:17) . Remark 3.5. (1) The constants ε and C in the above Theorem depend on λ but noton λ for 0 ≤ λ ≤ λ . (2) Assume s > δ + d , δ > b ∈ H s − ( T d ) and, η j ∈ H λh, s + ( T d ) with k η j k H λh, s ≤ ε, a ( D x ) ψ j ∈ H λh, s ( T d ) ( j = 1 , . Then we may apply the above Theorem to s ′ = s − δ and we get an estimate of theterm, kG − G k H λh, s − − δ by the right hand side where s is replaced by s − δ. Proof.
Introduce for j = 1 , , the functions U j = |∇ x ρ j | ∂ z ρ j ∂ z e u j − ∇ x ρ j · ∇ x e u j where,(3.13) (∆ x,z + R j ) e u j = 0 , e u j | z =0 = ψ j , ( ∂ z e u j ) | z = − h = ( ∂ z ρ j | z = − h ) b. Then by definition we have G − G = ( U − U ) | z =0 . As in (3.5) we have, kG − G k H λh, s − ≤ k U − U k E λ, s − ≤ C ( k U − U k F λ, s + k ∂ z ( U − U ) k F λ, s − ) . Now according to (3.10) our equation on e u j reads, ∂ z U j + div x V j = 0 with,(3.14) U j = 1 + |∇ x ρ j | ∂ z ρ j ∂ z e u j − ∇ x ρ j · ∇ x e u j , V j = ∂ z ρ j ∇ x e u j − ∂ z e u j ∇ x ρ j , It follows that,(3.15) kG − G k H λh, s − ≤ C ( k U − U k F λ, s + k V − V k F λ, s ) . ecall that,(3.16) ∂ z ρ j = 1 + q j , where q j = 1 h e z | D x | η j + 1 h ( z + h ) e z | D x | | D x | η j ,∂ z ρ j | z = − h = 1 + 1 h e − h | D x | η j , ∇ x ρ j = 1 h ( z + h ) e z | D x | ∇ x η j , |∇ x ρ j | ∂ z ρ j = 1 + A j , A j = − f ( q j ) − |∇ x ρ j | f ( q j ) + |∇ x ρ j | , where f ( q j ) = q j q j . With this notations and using (3.14) we can write,(3.17) e u = e u − e u ,U − U = ( A − A ) ∂ z e u − ( ∇ x ρ − ∇ x ρ ) · ∇ x e u + (1 + A ) ∂ z e u − ∇ x ρ · ∇ x e u,V − V = ( ∂ z ρ − ∂ z ρ ) ∇ x e u − ( ∇ x ρ − ∇ x ρ ) ∂ z e u + ∂ z ρ ∇ x e u − ∇ x ρ ∂ z e u. The first two terms in the right hand side of (3.17) are of the form( p − p ) ∇ x,z e u with p ∈ F , by Lemma 8.13, where F has been defined in Definition 8.12.They are estimated as follows. Using Lemma 8.8 iii ) with s = s − , we can write, k ( p − p ) ∇ x,z e u k F λh, s . k p − p k E λ, s − k∇ x,z e u k F λ, s + k p − p k F λ, s k∇ x,z e u k E λ, s − , therefore, k ( p − p ) ∇ x,z e u k F λh, s . k p − p k L ∞ ( I h , H λh, s − ) k∇ x,z e u k F λ, s + k p − p k L ( I h , H λh, s ) k∇ x,z e u k E λh, s − . Using the definition of F , (3.13), Corollary 2.13 , Lemma 2.12, Proposition 2.7 and thenotation in (3.12) we obtain,(3.18) k ( p − p ) ∇ x,z e u k F λh, s ≤ λ k η − η k H λh, s + λ k η − η k H λh, s + 12 . The last two terms in the right hand side of (3.17) are of the form, q ∇ x,z ( e u − e u ) = q ∇ x,z e u, with q = 1 or q ∈ E , where E has been defined in Definition 8.10.They are estimated as follows. By Lemma 8.8 with with s = s − we can write, k q ∇ x,z e u k F λ, s . k q k L ∞ ( I h , H λh, s − ) k∇ x,z e u k F λ, s + k q k L ( I h , H λh, s ) k∇ x,z e u k E λ, s − . Since q ∈ E we deduce from (8.14) that,(3.19) k q ∇ x,z e u k F λ, s . k∇ x,z e u k F λh, s + k η k H s + 12 k∇ x,z e u k E λh, s − . e see therefore that we have to estimate ∇ x,z e u. For that, according to (3.13), we notice that e u = e u − e u is solution of the problem,(∆ x,z + R ) e u = ( R − R ) e u , e u | z =0 = ψ − ψ , ∂ z e u | z = − h = 1 h e − h | D x | ( η − η ) b. Notice that for every µ > d we have,(3.20) k h e − h | D x | ( η − η ) b k H µ ≤ C k η − η k H λh,µ k b k H µ . Therefore using Theorem 2.10 we can write,(3.21) k∇ x,z e u k F λ, s . k ( R − R ) e u k F λ, s − + k a ( D x ) ( ψ − ψ ) k H λh, s + k η − η k H λh, s − k b k H s − + C k η k H λh, s + 12 k ( R − R ) e u k F λ, s − + k a ( D x ) ( ψ − ψ ) k H λh, s − + k η − η k H λh, s − k b k H s − . Moreover using Lemma 2.12 we can write, k∇ x,z e u k E λ, s − . k∇ x,z e u k F λ, s − + k ( R − R ) e u k F λ, s − . Then we can use Corollary 2.7 and we obtain,(3.22) k∇ x,z e u k E λ, s − . k ( R − R ) e u k F λ, s − + k η − η k H λh, s − k b k H s − . Using (3.19),(3.21),(3.22) we obtain, k q ∇ x,z e u k F λ, s . k ( R − R ) e u k F λ, s − + k a ( D x ) ( ψ − ψ ) k H λh, s + k η − η k H λh, s − k b k H s − + k η k H λh, s + 12 (cid:16) k ( R − R ) e u k F λ, s − + k a ( D x ) ( ψ − ψ ) k H λh, s − + k η − η k H λh, s − k b k H s − (cid:17) . Using the definition of λ j , j = 1 , , k q ∇ x,z e u k F λ, s . k ( R − R ) e u k F λ, s − + k η k H λh, s + 12 k ( R − R ) e u k F λ, s − + λ k η − η k H λh, s + λ k a ( D x ) ( ψ − ψ ) k H λh, s − + k a ( D x ) ( ψ − ψ ) k H λh, s To complete the proof we are led to estimate ( R − R ) e u . According to (2.52) we have,(3.24) ( R − R ) = ( a − a ) ∂ z + ( b − b )∆ x + ( c − c ) · ∇ x ∂ z − ( d − d ) ∂ z a j = 1 + |∇ x ρ j | ∂ z ρ − , b j = ∂ z ρ j − , c j = − ∇ x ρ j ,d j = 1 + |∇ x ρ j | ∂ z ρ j ∂ z ρ j + ∂ z ρ j ∆ x ρ j − ∇ x ρ j · ∇ x ∂ z ρ j . Estimate of k ( R − R ) e u k F λ, s − . The first three terms in ( R − R ) e u are estimated in the same manner. Since a, b, c ∈ F (see Lemma 8.13) for r ∈ { a, b, c } using Lemma 8.8, ii ) we can write, k ( r − r ) ∇ x,z e u k F λ, s − ≤ C k r − r k L ∞ ( I h , H λh, s − ) k∇ x,z e u k F λ, s − , ≤ C k η − η k H λh, s k∇ x,z e u k F λ, s − . sing Corollary 2.13 we have, k ( r − r ) ∇ x,z e u k F λ, s − ≤ k η − η k H λh, s (cid:16) k a ( D x ) ψ k H λh, s + k b k H s − + k η k H λh, s + 12 (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1)(cid:17) . With the notations in (3.12) one can deduce eventually that,(3.25) k ( r − r ) ∇ x,z e u k F λ, s − ≤ λ k η − η k H λh, s , where r = a or b or c .We consider now the term k ( d − d ) ∂ z e u k F λ, s − . Using Lemma 8.8 iii ) with s = s − k ( d − d ) ∂ z e u k F λ, s − . k d − d k L ∞ ( I h , H λh, s − ) k ∂ z e u k F λ, s − + k d − d k L ( I h , H λh, s − ) k ∂ z e u k E λ, s − . By Lemma 8.13 we have d ∈ F . Therefore, k d − d k L ∞ ( I h , H λ, s − ) . k η − η k H λh, s , k d − d k L ( I h , H λ, s − ) . k η − η k H λh, s + 12 . By Corollary 2.7 we have,(3.26) k ∂ z e u k F λ, s − . (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − + k b k H s − ( T d ) . By Lemma 2.12 and Corollary 2.7 we have, k ∂ z e u k E λ, s − . k∇ x,z e u k F λ, s − . k a ( D x ) ψ k H λ, s − + k b k H s − . Using the notation in (3.12) we obtain,(3.27) k ( d − d ) ∂ z e u k F λ, s − ≤ λ k η − η k H λh, s + λ k η − η k H λh, s + 12 . It follows from (3.25) and (3.27) that,(3.28) k ( R − R ) e u k F λ, s − . λ k η − η k H λh, s + λ k η − η k H λh, s + 12 . Estimate of k ( R − R ) e u k F λ, s − . We use the same notations as above. Since s − > d we can write, k ( r − r ) ∇ x,z e u k F λ, s − . k r − r k L ∞ ( I h , H λh, s − ) k∇ x,z e u k F λ, s − . Since r ∈ F we have, k r − r k L ∞ ( I h , H λh, s − ) . k η − η k H λh, s − . Moreover from Lemma 2.12and Corollary 2.7 we have, k∇ x,z e u k F λ, s − . k a ( D x ) ψ k H λh, s − + k b k H s − . It follows that, k ( r − r ) ∇ x,z e u k F λ, s − . k η − η k H λh, s − (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1) . Now, k ( d − d ) ∂ z e u k F λ, s − . k d − d k L ∞ ( I h , H λh, s − ) k ∂ z e u k F λ, s − . ince d ∈ F we have, k d − d k L ∞ ( I h , H λh, s − ) . k η − η k H λh, s . Using (3.26) we obtain, k ( d − d ) ∂ z e u k F λ, s − . k η − η k H λh, s (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1) . Therefore,(3.29) k ( R − R ) e u k F λ, s − . k η − η k H λh, s (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1) . It follows from (3.23), (3.28), (3.29), that,(3.30) k q ∇ x,z e u k F λ, s . λ k η − η k H λh, s + λ k η − η k H λh, s + 12 + λ k a ( D x )( ψ − ψ ) k H λh, s − + k a ( D x )( ψ − ψ ) k H λh, s . Then Theorem 3.4 follows from (3.15), (3.17), (3.18), (3.30). (cid:3)
The following result is an immediate consequence of the above theorem.
Corollary 3.6.
Let ≤ λ < and s > d . There exists ε > and C > such that forall ≤ λ ≤ λ , all η ∈ H λh, s + , ψ ∈ H λh, s + , satisfying k η k H λh, s ≤ ε we have, k G ( η )( ψ, − G (0)( ψ, k H λh, s − ≤ C (cid:0) k η k H σ, s + 12 k a ( D x ) ψ k H λh, s − + k η k H σ, s k a ( D x ) ψ k H λh, s (cid:1) Remark 3.7.
Assume s > δ + d , δ > η ∈ H λh, s + ( T d ) with k η k H λh, s ≤ ε,a ( D x ) ψ ∈ H λh, s ( T d ) , b ∈ H s − ( T d ) . Then we may apply the above Theorem to s ′ = s − δ and we obtain an estimate of k G ( η )( ψ, − G (0)( ψ, k H λh, s − − δ by the right hand side where s is replaced by s − δ. Another Dirichlet-Neumann operator.
Let h > η ∈ W , ∞ ( T d ) be such that k η k L ∞ ( T d ) ≤ ε ≪ h . We set O = { ( x, y ) : x ∈ T d , − η ( x ) − h < y < } , and we consider the Dirichlet problem,(4.1) ∆ x,y v = 0 in O , v | y =0 = b, v | y = − η ( x ) − h = B. Proposition 4.1.
For all s ∈ R there exist C > and F : R + → R + non decreasing suchthat for all solution of the problem (4.1) we have, (cid:13)(cid:13)(cid:13) ∂v∂y y =0 (cid:13)(cid:13)(cid:13) H s ( T d ) ≤ C k b k H s +1 ( T d ) + F (cid:0) k η k W , ∞ ( T d ) ) (cid:0) k b k H ( T d ) + k B k H ( T d ) (cid:1) . Proof.
Let χ ∈ C ∞ ( R ) be such that χ ( y ) = 1 if − h ≤ y ≤ χ ( y ) = 0 if y ≤ − h . We set w = χ ( y ) v . Then w is solution of the problem,∆ x,y w = 2 χ ′ ( y ) ∂ y v + χ ′′ ( y ) v := F for − h < y < , w | y =0 = b, w | y ≤− h = 0 . e can solve explicitetely this problem. Indeed taking a Fourier transform with respect to x we are lead to solve the two following problems.( ∂ y + | ξ | ) w = b F , w | y ≤− h = 0 , ( ∂ y − | ξ | ) b w = w , b w | y =0 = b b. For − h ≤ y ≤ b w ( y, ξ ) = e y | ξ | b b − Z y Z s − h e ( y + σ − s ) | ξ | b F ( σ, ξ ) d s dσ. It follows that, ∂ b v∂y | y =0 = ∂ b w∂y | y =0 = | ξ | b b + Z − h e σ | ξ | ( χ ′ ( σ ) ∂ y b v ( σ, ξ ) + χ ′′ ( σ ) b v ( σ, ξ )) dσ. On the support of a derivative of χ we have σ ≤ − h . Multiplying both members by h ξ i s andusing the fact that h ξ i s e − h | ξ | is bounded on T d we obtain easily the estimate, (cid:13)(cid:13) ∂v∂y | y =0 (cid:13)(cid:13) H s ( T d ) ≤ C (cid:0) k b k H s +1 ( T d ) + k v k H (( − h , × T d ) (cid:1) . Since the problem (4.1) is variational we see easily that, k v k H (( − h , × T d ) ≤ F (cid:0) k η k W , ∞ ( T d ) ) (cid:0) k b k H ( T d ) + k B k H ( T d ) (cid:1) . This completes the proof. (cid:3)
Corollary 4.2.
Let φ be the solution of the problem (3.1) that is, ∆ x,y φ = 0 in − h < y < η ( x ) , φ | y = η ( x ) = ψ, ∂ y φ | y = − h = b. Let φ h = φ ( x, − h ) , B = ∂ y φ ( x, η ( x )) and fix s > d . Then, for all µ ∈ R there exists C > such that, k ∆ x φ h k H µ ( T d ) ≤ C k b k H µ +1 ( T d ) + F (cid:0) k η k H s ( T d ) (cid:1)(cid:0) k b k H ( T d ) + k B k H ( T d ) (cid:1) . Proof.
Set u = ∂ y φ . It satisfies,∆ x,y u = 0 , u | y = η ( x ) = B, u | y = − h = b. On the other hand, ∂ y u | y = − h = ∂ y φ | y = − h = − ∆ x φ | y = − h = − ∆ x φ h . Set v ( x, y ) = u ( x, − y − h ). Then v is a solution of the problem,∆ x,y v = 0 for − η ( x ) − h < y < , v | y =0 = b, v | y = − η ( x ) − h = B and ∂ y v | y =0 = ∂ y u | y = − h = − ∆ x φ h . Corollary 4.2 follows from Proposition 4.1 and from the fact that H s +1 ( T d ) is embedded in W , ∞ ( T d ). (cid:3) . The equations
Recall that the formulation of Craig-Sulem-Zakharov, where η and ψ are the unknowns:(5.1) ∂ t η = G ( η )( ψ, b ) ,∂ψ = gη + 12 |∇ x ψ | − ( G ( η )( ψ, b ) + ∇ x η · ∇ x ψ ) |∇ x η | ) . Our aim is to reformulate the above equations in terms of the new unknowns ( ζ, B, V ) whichare defined as follows:(5.2) ζ = ∇ x η,B = ( ∂ y φ ) | y = η = G ( η )( ψ, b ) + ∇ x η · ∇ x ψ |∇ x η | ,V = ( ∇ x φ ) | y = η = ∇ x ψ − B ∇ x η. Such a formulation was introduced in our previous paper [4], where we explained how to useit to prove energy estimates. We shall also use it in this paper to prove energy estimatesallowing to propagate analyticity in large time.
Proposition 5.1.
The unknown
V, B, ζ satisfy the equations, ( ∂ t + V · ∇ x ) B = a − g, (5.3) ( ∂ t + V · ∇ x ) V + a ζ = 0 , (5.4) ( ∂ t + V · ∇ x ) ζ = G ( η )( V, ∇ x b ) − ζ (div V ) , (5.5) where a is the Taylor coefficient, defined as a = ∂P∂y y = η . Proof.
We follow the analysis in [4]. The main novelty is that we obtain a cancellation whichallows to handle easily the source term b .Firstly, we have,(5.6) ∂ t η = G ( η ) ψ = ( ∂ y φ − ∇ x η · ∇ x φ ) | y = η = B − V · ∇ x η. Now for any smooth function f ( t, x, y ) we have,( ∂ t + V · ∇ x )[ f ( t, x, η ( t, x ))] = [ ∂ t f + ∇ x φ · ∇ x f + ( ∂ y f )( ∂ t η + V · ∇ x η )]( t, x, η ( t, x )) . It follows from the definition of B and from (5.6) that,(5.7) ( ∂ t + V · ∇ x )[ f ( t, x, η ( t, x ))] = [ ∂ t f + ∇ x,y φ · ∇ x,y f ]( t, x, η ( t, x )) . Applying this equality to f = ∂ y φ we obtain,( ∂ t + V · ∇ x ) B = ( ∂ t ∂ y φ + ∇ x,y φ · ∇ x,y ∂ y φ ]( t, x, η ( t, x )) . On the other hand we have, − P = ∂ t φ + |∇ x,y φ | + gy. Let us differentiate this equalitywith respect to y and take its trace trace on Σ . We get, a = − ∂ y P | y = η = [ ∂ t ∂ y φ + ∇ x,y φ · ∇ x,y ∂ y φ ]( t, x, η ( t, x )) + g, which proves (5.3). pplying (5.7) to f = ∂ x k φ we obtain,( ∂ t + V · ∇ x ) V k = [ ∂ t ∂ x k φ + ∇ x,y φ · ∇ x,y ∂ x k φ ]( t, x, η ( t, x )) . Let us differentiate the equation giving P with respect to x k and take its trace on Σ . Weobtain, − ∂ x k P ( t, x, η ( t, x ) = [ ∂ t ∂ x k φ + ∇ x,y φ · ∇ x,y ∂ x k φ ]( t, x, η ( t, x )) , therefore, ( ∂ t + V · ∇ x ) V k = − ∂ x k P ( t, x, η ( t, x ) . Now differentiating the equality P ( t, x, η ( t, x )) = 0 with repsct to x k we get,[ ∂ x k P + ( ∂ x k η )( ∂ y P ]( t, x, η ( t, x )) = 0which can be written, ∂ x k P ( t, x, η ( t, x )) − a ζ k = 0and proves (5.4).Let us show (5.5). Differentiating (5.6) with respect to x i , i = 1 , . . . , d and setting ∂ i = ∂ x i we get, ( ∂ t + V · ∇ x ) ζ i = ∂ i B − d X k =1 ( ∂ i V k )( ∂ k η ) . Using the definitions of
V, B we obtain, ∂ i B − d X k =1 ( ∂ i V k )( ∂ k η ) = [ ∂ i ∂ y φ + ( ∂ i η )( ∂ y φ )] | y = η − d X k =1 ( ∂ k η )[ ∂ i ∂ k φ + ( ∂ i η )( ∂ k ∂ y φ )] | y = η = [ ∂ y ∂ i φ − ∇ x η · ∇ x ∂ i φ ] + ( ∂ i η )[ ∂ y φ − ∇ x η · ∇ x ∂ y φ ] | y = η . It follows that,( ∂ t + V · ∇ x ) ζ i = [( ∂ y − ∇ x η · ∇ x ) ∂ i φ ] | y = η + ( ∂ i η )[( ∂ y − ∇ x η · ∇ x ) ∂ y φ | y = η , The function ∂ i φ is a solution of the problem,∆ x,y ∂ i φ = 0 , ∂ i φ | y = η = V i , ∂ y ∂ i φ | y = − h = ∂ i b. Therefore, ( ∂ y − ∇ x η · ∇ x ) ∂ i φ ] | y = η = G ( η )( V i , ∂ i b ) . Similarly the function ∂ y φ is a solution of the problem,∆ x,y ∂ y φ = 0 , ∂ y φ | y = η = B, ∂ y φ | y = − h = − ∆ x ( φ | y = − h )so that, ( ∂ y − ∇ x η · ∇ x ) ∂ y φ ] | y = η = G ( η )( B, ∆ x ( φ | y = − h )) . It follows that,(5.8) ( ∂ t + V · ∇ x ) ζ = G ( η )( V, ∇ x b ) + ζG ( η )( B, − ∆ x ( φ | y = − h )) . On the other hand,div V = d X j =1 ( ∂ j φ + ∂ j η∂ j ∂ y φ ) | y = η = (∆ x φ + ∇ x η · ∇ x ∂ y φ ) | y = η = ( − ∂ y φ + ∇ x η · ∇ x ∂ y φ ) | y = η , = − ( ∂ y − ∇ x η · ∇ x ) ∂ y φ | y = η . ince ∂ y φ is a solution of the problem,∆ x,y ( ∂ y φ ) = 0 , ∂ y φ | y = η = B, ∂ y ( ∂ y φ ) | y = − h = − ∆ x ( φ | y = − h )we get, G ( η )( B, − ∆ x ( φ | y = − h )) = ( ∂ y − ∇ x η · ∇ x ) ∂ y φ | y = η . It follows that,(5.9) G ( η )( B, − ∆ x ( φ | y = − h )) = − div V, which, using (5.8) proves (5.5). (cid:3) Now, G ( η )( V, ∇ x b ) = G (0)( V,
0) + G ( η )( V, − G (0)( V,
0) + G ( η )(0 , ∇ x b ) . It follows from (5.4) and (5.5) that, for i = 1 , . . . , d∂ t ζ i − G (0)( V i ,
0) = G ( η )( V i , − G (0)( V i ,
0) + G ( η )(0 , ∂ i b ) − ( V · ∇ x ) ζ i − ζ i (div V ) ,∂ t V i + gζ i = − ( V · ∇ x ) V i − ( a − g ) ζ i . Now,(5.10) G (0)( V,
0) = | D x | tanh( h | D x | ) V =: a ( D x ) V, where tanh denotes the hyperbolic tangent. We deduce that, ( ζ, V ) satisfy the system(5.11) ∂ t ζ − a ( D x ) V = F ,∂ t V + gζ = F ,F = G ( η )( V, − G (0)( V,
0) + G ( η )(0 , ∇ x b ) − ( V · ∇ x ) ζ − (div V ) ζ,F = − ( V · ∇ x ) V − ( a − g ) ζ. Let us set, u = √ gζ + ia ( D x ) V. It follows from the above equations that(5.12) ∂ t u + i (cid:0) g a ( D x ) (cid:1) u = √ gF + ia ( D x ) F . Existence of a solution on a time interval of size 1.
In this section, we prove Proposition 1.4.We will obtain the solution as the limit of an iterative scheme. To avoid confusion of notations,we denote the initial data by ( u , v ). Namely, we consider the Cauchy problem for the water-wave system (1.1) with initial data ( η, ψ ) | t =0 = ( u , v ) . Let u , v ∈ H λh, s . We consider the sequence ( η ν , ψ ν ) ν ∈ N defined by,(6.1) η = u , ψ = v , and for ν ≥ ,∂ t η ν +1 = G ( η ν )( ψ ν , b ) , η ν +1 | t =0 = u ,∂ t ψ ν +1 = − gη ν − |∇ x ψ ν | + ( G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν ) |∇ x η ν | ) , ψ ν +1 | t =0 = v . e set for s > d + 2,(6.2) m ν ( t ) = k η ν ( t ) k H σ ( t ) , s + k ψ ν ( t ) k H σ ( t ) , s + 2 K Z t (cid:0) k η ν ( τ ) k H σ ( τ ) , s + 12 + k ψ ν ( τ ) k H σ ( τ ) , s + 12 (cid:1) dτ. Proposition 6.1.
Let
T > . Assume that b ∈ L ∞ ( R , H s − ( T d )) ∩ L ( R , H s − ( T d )) . Thereexist positive constants
M, K, ε such that for all ε ≤ ε (cid:0) k b k L ( R ,H s − ) + k u k H λh, s + k v k H λh, s ≤ ε (cid:1) = ⇒ (cid:0) m ν ( t ) ≤ M ε , ∀ t ≤ T, ∀ ν ≥ (cid:1) . Proof.
First of all we shall take ε and M such that, M ε ≤ ε , where ε is defined in Theorem3.1 and M ε ≤ . Now, m ( t ) ≤ ε . Indeed we have, k u k H σ ( t ) , s + k v k H σ ( t ) , s ≤ ε and,2 K Z t k u k H σ ( τ ) , s + 12 dτ = X Z d h ξ i s e λh h ξ i | b u ( ξ ) | (cid:16) Z t K h ξ i e − Kτ h ξ i dτ (cid:17) ≤ k u k H λh, s , similarly for v .So we take M ≥ . Assume now that m j ( t ) ≤ M ε , ≤ j ≤ ν . Set, η ν = e − σ ( t ) h ξ i e η ν , ψ ν = e − σ ( t ) h ξ i e ψ ν . Notice that since σ ( t ) = λh − Kt ≤ λh and | ξ | ≤ h ξ i ≤ | ξ | we have, σ ( t ) | ξ | ≤ σ ( t ) h ξ i ≤ λh + σ ( t ) | ξ | . It follows that, k f ( t ) k H σ ( t ) ,α ≤ k e f ( t ) k H α ≤ e λh k f ( t ) k H σ ( t ) ,α . Remark 6.2.
We can write σ ( t ) = λ ( t ) h with λ ( t ) = λ − Kth . Since λ ( t ) ≤ λ we may use theestimates in section 2. and 3 with constants depending only on the fixed parameter λ. The system satisfied by ( e η ν , e ψ ν ) is then, ∂ t e η ν +1 + K h D x i e η ν +1 = e σ ( t ) h D x i G ( η ν )( ψ ν , b ) := F ν ∂ t e ψ ν +1 + K h D x i e ψ ν +1 = e σ ( t ) h D x i h − gη ν − |∇ x ψ ν | + ( G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν ) |∇ x η ν | ) i =: G ν . We have, ddt (cid:2) k e η ν +1 ( t ) k H s + k e ψ ν +1 ( t ) k H s (cid:3) = 2 (cid:16) ∂ t e η ν +1 ( t ) , e η ν +1 ( t ) (cid:17) H s + 2 (cid:16) ∂ t e ψ ν +1 ( t ) , e ψ ν +1 ( t ) (cid:17) H s . Using the above equations we get, k e η ν +1 ( t ) k H s + k e ψ ν +1 ( t ) k H s + 2 K Z t (cid:0) k e η ν +1 ( τ ) k H s + 12 + k e ψ ν +1 ( τ ) k H s + 12 (cid:1) dτ = k u k H λh, s + k v k H λh, s + 2 Z t (cid:16) F ν ( τ ) , e η ν +1 ( τ ) (cid:17) H s dτ + 2 Z t (cid:16) G ν ( τ ) , e ψ ν +1 ( τ ) (cid:17) H s dτ. et, A ( τ ) = (cid:12)(cid:12)(cid:12)(cid:16) F ν ( τ ) , e η ν +1 ( τ ) (cid:17) H s (cid:12)(cid:12)(cid:12) , B ( τ ) = (cid:12)(cid:12)(cid:12)(cid:16) G ν ( τ ) , e η ν +1 ( τ ) (cid:17) H s (cid:12)(cid:12)(cid:12) . We deduce from the hypotheses that,(6.3) m ν +1 ( t ) ≤ ε + Z t A ( τ ) dτ + Z t B ( τ ) dτ. We have, A ( τ ) ≤ CK k G ( η ν )( ψ ν , b )( τ ) k H σ ( τ ) , s − + 2 K k e η ν +1 ( τ ) k H s + 12 . For all fixed τ (which is skipped) Theorem 3.1 shows that, k G ( η ν )( ψ ν , b ) k H σ, s − ≤ C (cid:0) k e ψ ν k H s + 12 + k b k H s − + k e η ν k H s + 12 (cid:0) k e ψ ν k H s + k b k H s − (cid:1)(cid:1) . Since k e ψ ν ( τ ) k H s ≤ M ε ≤ C > s such that,(6.4) k G ( η ν )( ψ ν , b ) k H σ ( τ ) , s − ≤ C (cid:0) k e ψ ν k H s + 12 + k b k H s − + k e η ν k H s + 12 (1 + k b k H s − ) (cid:1) . Therefore, since k b k L ∞ ( R ,H s − ) ≤ ε ≤ A ( τ ) ≤ K (cid:0) k e ψ ν ( τ ) k H s + 12 + k b ( τ ) k H s − + k e η ν ( τ ) k H s + 12 (cid:1) + 2 K k e η ν +1 ( τ ) k H s + 12 . so, Z t A ( τ ) dτ ≤ CK Z t (cid:0) k e ψ ν ( τ ) k H s + 12 + k e η ν ( τ ) k H s + 12 + k b ( τ ) k H s − (cid:1) dτ + 2 K Z t k e η ν +1 ( τ ) k H s + 12 dτ. It follows that,(6.5) Z t A ( τ ) dτ ≤ CK M ε + CK k b k L ( R ,H s − ) + 120 m ν +1 ( t ) . Now,(6.6) B ( τ ) ≤ B ( τ ) + B ( τ ) + B ( τ ) ,B ( τ ) = g (cid:12)(cid:12)(cid:12)(cid:16)e η ν ( τ ) , e η ν +1 ( τ ) (cid:17) H s (cid:12)(cid:12)(cid:12) ,B ( τ ) = 12 (cid:12)(cid:12)(cid:12)(cid:16) e σ ( τ ) h D x i |∇ x ψ ν ( τ ) | , e η ν +1 ( τ ) (cid:17) H s (cid:12)(cid:12)(cid:12) B ( τ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) e σ ( τ ) h D x i ( G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν ) |∇ x η ν | ) , e η ν +1 ( τ ) (cid:17) H s (cid:12)(cid:12)(cid:12)(cid:12) . Using the Cauchy-Schwarz inequality and the induction hypothesis we obtain,(6.7) Z t B ( τ ) dτ ≤ CK M ε + 120 m ν +1 ( t ) . On the other hand, using Proposition 8.3 iii ) with s = s −
1, we obtain, B ( τ ) . k e ψ ν ( τ ) k H s k e ψ ν ( τ ) k H s + 12 k e η ν +1 ( τ ) k H s + 12 . M ε k e ψ ν ( τ ) k H s + 12 k e η ν +1 ( τ ) k H s + 12 . t follows that, Z t B ( τ ) dτ . M ε kk e ψ ν k L ((0 ,t ) ,H s + 12 ) k e η ν +1 k L ((0 ,t ) ,H s + 12 ) . Therefore, by the Cauchy-Schwarz inequality and the induction,(6.8) Z t B ( τ ) dτ ≤ CK ( M ε ) + 120 m ν +1 ( t ) . To estimate the term B set N ν = G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν and U ν = N ν |∇ x η ν | . Then wehave, B ( τ ) . k e σ ( τ ) | D x | U ν k H s − k e η ν +1 ( τ ) k H s + 12 , from which we deduce,(6.9) Z t B ( τ ) dτ ≤ CK Z t k U ν ( τ ) k H σ ( τ ) , s − dτ + 120 m ν +1 ( t ) . Using Proposition 8.3 with s = s − τ , k U ν k H σ, s − . k N ν k H σ, s − k N ν k H σ, s − + k|∇ x η ν | U ν k H σ, s − . k N ν k H σ, s − k N ν k H σ, s − + k e η ν k H s k U ν k H σ, s − + k e η ν k H s k e η ν k H s + 12 k U ν k H σ, s − . Since k e η ν ( τ ) k H s ≤ M ε taking
M, ε such that C ( M ε ) ≤ we can absorb the second termof the right hand side by the left hand side and deduce that, k U ν k H σ, s − . k N ν k H σ, s − k N ν k H σ, s − + M ε k e η ν k H s + 12 k U ν k H σ, s − . Similarly, since s − > d we have, k U ν k H σ, s − . k N ν k H σ, s − + k e η ν k H s k U ν k H σ, s − . Using again the fact that k e η ν ( τ ) k H s ≤ M ε we can absorb the second term in the right handside by the left one an deduce that, k U ν k H σ, s − . k N ν k H σ, s − . So we obtain the inequality, k U ν k H σ, s − . k N ν k H σ, s − k N ν k H σ, s − + M ε k e η ν k H s + 12 k N ν k H σ, s − . By Theorem 3.1, Remark 3.2 and the induction we have, k N ν k H σ, s − . k e ψ ν k H s + k b k H s − + k e η ν k H s k e ψ ν k H s . we have k e η ν ( τ ) k H s ≤ M ε ≤ , k e ψ ν ( τ ) k H s ≤ M ε ≤ b ∈ L ∞ ( R , H s − ) so that,(6.10) k N ν k H σ, s − . M ε + k b k H s − = O (1) . It follows that,(6.11) k U ν k H σ, s − . k N ν k H σ, s − + k e η ν k H s + 12 . It remains to estimate the term k N ν k H σ, s − . We have, k∇ x ψ ν · ∇ x η ν k H σ, s − . k e ψ ν k H s k e η ν k H s + 12 + k e ψ ν k H s + 12 k e η ν k H s , . M ε (cid:0) k e η ν k H s + 12 + k e ψ ν k H s + 12 (cid:1) . sing (6.4) we get, k N ν k H σ, s − . k e ψ ν k H s + 12 + k b k H s − + k e η ν k H s + 12 k b k H s − + M ε (cid:0) k e η ν k H s + 12 + k e ψ ν k H s + 12 (cid:1) , so,(6.12) k N ν k H σ, s − . k e ψ ν k H s + 12 + k e η ν k H s + 12 + k b k H − . Using (6.11), (6.10), (6.12) we get, k U ν k H σ, s − . k e ψ ν k H s + 12 + k e η ν k H s + 12 + k b k L ∞ ( R ,H s − ) . We deduce from (6.9) and the induction that,(6.13) Z t B ( τ ) dτ ≤ CK M ε + 120 m ν +1 ( t ) . Using (6.3), (6.5), (6.7), (6.8), (6.13) and taking K large enough, ε small enough we obtain, m ν +1 ( t ) ≤ M ε , which ends the proof of Proposition 6.1. (cid:3) Notation 6.3.
We set I = [0 , T ] and, U ν +1 = e η ν +1 − e η ν , V ν +1 = e ψ ν +1 − e ψ ν ,M ν +1 ( t ) = k U ν +1 ( t ) k H s + k V ν +1 ( t ) k H s + 2 K Z t (cid:16) k U ν +1 ( τ ) k H s + 12 + k V ν +1 ( τ ) k H s + 12 (cid:17) dτ. M ν +1 = k U ν +1 k L ∞ ( I,H s ) + k V ν +1 k L ∞ ( I,H s ) + 2 K Z T (cid:16) k U ν +1 ( τ ) k H s + 12 dτ + k V ν +1 ( τ ) k H s + 12 (cid:17) dτ. Proposition 6.4.
Assume b ∈ L ( R , H s − ) ∩ L ∞ ( R , H s − ) . There exists C > such thatfor all ν ≥ , M ν +1 ≤ CK (cid:0) K + k b k L ( R ,H s − ) (cid:1) M ν . Proof.
Since U ν +1 | t =0 = V ν +1 | t =0 = 0 we obtain as in the first part, M ν +1 ( t ) = 2 Z t (cid:16) ( F ν − F ν − )( τ ) , U ν +1 ( τ ) (cid:17) H s dτ + 2 Z t (cid:16) ( G ν − G ν − )( τ ) , V ν +1 ( τ ) (cid:17) H s dτ where,(6.14) F ν = e σ ( t ) | D x | G ( η ν )( ψ ν , b ) G ν = e σ ( t ) | D x | h − gη ν − |∇ x ψ ν | + (cid:0) G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν ) |∇ x η ν | (cid:1) i . Using the inequality | ( f, g ) H s | ≤ K k f k H s − + K k g k H s + 12 and the definition of M ν +1 we get,(6.15) M ν +1 ( t ) ≤ A ( t ) + B ( t ) where ,A ( t ) = CK Z t k ( F ν − F ν − )( τ ) k H s − dτ, B ( t ) = CK Z t k ( G ν − G ν − )( τ ) k H s − dτ. Estimate of the term A ( t ). e use Theorem 3.4 and Proposition 6.1. Keeping the notations in the Theorem we canwrite,(6.16) k F ν − F ν − k H s − . λ k e η ν − e η ν − k H s + λ k e η ν − e η ν − k H s + 12 + λ k e ψ ν − e ψ ν − k H s + k e ψ ν − e ψ ν − k H s + 12 , with, λ . (cid:16) ν X µ = ν − k e η µ k H s + 12 (cid:17)(cid:16) ν X µ = ν − k e ψ µ k H s + k b k H s − (cid:17) + ν X µ = ν − k e ψ µ k H s + 12 + k b k H s − ,λ . ν X µ = ν − k e ψ µ k H s − + k b k H s − , λ . ν X µ = ν − k e η µ k H s + 12 . Since b ∈ L ∞ ( R , H s − ) and using (6.2) together with Proposition 6.1 we deduce, λ . ν X µ = ν − (cid:0) k e η µ k H s + 12 + k e ψ µ k H s + 12 (cid:1) + k b k H s − , λ = O (1) , λ . ν X µ = ν − k e η µ k H s + 12 . It follows easily from (6.16) that,(6.17) k F ν − F ν − k H s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 + k b k H s − (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 , from which we deduce that,(6.18) A ( t ) . K (cid:16) K + k b k L ( R ,H s − ) (cid:17) M ν . Let us look to the term B ( t ). We can write,(6.19) B ( t ) ≤ C ( B ( t ) + B ( t ) + B ( t )) B ( t ) = 1 K Z t k e η ν ( τ ) − e η ν − k H s − ( τ ) dτ,B ( t ) = 1 K Z t k|∇ x e ψ ν ( t ) | − |∇ x e ψ ν − ( t ) | k H s − dτ,B ( t ) = 1 K Z t (cid:13)(cid:13)(cid:13)(cid:13) N ν ( τ )(1 + |∇ x η ν ( τ ) | ) − N ν − ( τ )(1 + |∇ x η ν − ( τ ) | ) (cid:13)(cid:13)(cid:13)(cid:13) H σ ( τ ) , s − dτ,N ν = G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν . First of all we have,(6.20) B ( t ) . K M ν . Now set I j = k ( ∂ j e ψ ν ( t )) − ( ∂ j e ψ ν − ( t )) k H s − we can write, I j . k ∂ j e ψ ν ( t ) − ∂ j e ψ ν − ( t ) k H s − k ∂ j e ψ ν ( t ) + ∂ j e ψ ν − ( t ) k H s − + k ∂ j e ψ ν ( t ) − ∂ j e ψ ν − ( t ) k H s − k ∂ j e ψ ν ( t ) + ∂ j e ψ ν − ( t ) k H s − . herefore, I j . k e ψ ν ( t ) − e ψ ν − ( t ) k H s + 12 (cid:0) k e ψ ν ( t ) k H s + k e ψ ν − ( t ) k H s (cid:1) + k e ψ ν ( t ) − e ψ ν − ( t ) k H s (cid:0) k e ψ ν ( t ) k H s + 12 + k e ψ ν − ( t ) k H s + 12 (cid:1) . It follows from Proposition 6.1 that,(6.21) B ( t ) . ( M ε ) K M ν . To estimate B set,(6.22) H ν = N ν (1 + |∇ x η ν | ) − N ν − (1 + |∇ x η ν − | ) , f ( t ) = t t . Then we can write,(6.23) H ν = (1) − (2) − (3) (1) = N ν − N ν − , (2) = N ν (cid:0) f ( |∇ x η ν | ) − f ( |∇ x η ν − | ) (cid:1) , (3) = (cid:0) N ν − N ν − (cid:1) f ( |∇ x η ν − | ) . We have, k (1) k H σ, s − . k N ν − N ν − k H σ, s − (cid:0) k N ν k H σ, s − + k N ν − k H σ, s − (cid:1) + k N ν − N ν − k H σ, s − (cid:0) k N ν k H σ, s − + k N ν − k H σ, s − (cid:1) According to (6.10) and (6.12) we have for µ = ν − , ν, (6.24) k N µ k H σ, s − . M ε + k b k H s − = O (1) , k N µ k H σ, s − . k e ψ µ k H s + 12 + k e η µ k H s + 12 + k b k H − . Moreover according to (6.17) we have, k N ν − N ν − k H σ, s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 + k b k H s − (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 , and k N ν − N ν − k H λh, s − . k e η ν − e η ν − k H σ, s + k e ψ ν − e ψ ν − k H s . It follows that,(6.25) k (1) k H σ, s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 + k b k H s − (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 . Now we have, k (2) k H σ, s − . k N ν k H σ, s − k f ( |∇ x η ν | ) − f ( |∇ x η ν − | ) k H σ, s − + k N ν k H σ, s − k N ν k H σ, s − k f ( |∇ x η ν | ) − f ( |∇ x η ν − | ) k H σ, s − . sing Proposition 8.7, (6.24) and the estimates, k|∇ x η ν | − |∇ x η ν − | k H σ, s − . k η ν − η ν − k H σ, s + 12 , k|∇ x η ν | − |∇ x η ν − | k H σ, s − . k η ν − η ν − k H σ, s , we obtain,(6.26) k (2) k H σ, s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 . The same method gives the estimate,(6.27) k (3) k H σ, s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 . Using (6.25),(6.26), (6.27) and (6.23) we deduce that, k H ν k H σ, s − . ν X µ = ν − (cid:16) k e η µ k H s + 12 + k e ψ µ k H s + 12 + k b k H s − ) (cid:17)(cid:0) k e η ν − e η ν − k H s + k e ψ ν − e ψ ν − k H s (cid:1) + k e η ν − e η ν − k H s + 12 + k e ψ ν − e ψ ν − k H s + 12 . Going back to (6.19) we obtain eventually,(6.28) | B ( t ) | . K (cid:16) K + k b k L ( R ,H s − (cid:17) M ν . Now we use (6.15), (6.18), (6.19), (6.20), (6.21)(6.28). We obtain,(6.29) M ν +1 . K (cid:16) K + k b k L ( R ,H s − (cid:17) M ν , which completes the proof of Proposition 6.4. (cid:3) Proof of Theorem 1.4.
It follows from Proposition 6.4 that there exist ε and K such that M ν ≤ δ ν M where δ <
1. Take
T < λhK and set X = C ([0 , T ] , H σ ( t ) , s ) ∩ L ((0 , T ) , H σ ( t ) , s + ).It follows that the sequence ( η ν , ψ ν ) converges in X × X to ( η, ψ ). It remains to prove that( η, ψ ) is a solution of system (1.1). According to (6.1) we can write,(6.30) η ν +1 ( t ) = u + Z t F ν ( τ ) dτ, F ν = G ( η ν )( ψ ν , b ) ,ψ ν +1 ( t ) = v + Z t G ν ( τ ) dτ, G ν = − gη ν − |∇ x ψ ν | + ( G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν ) |∇ x η ν | ) . It is enough to prove that R t F ν ( τ ) dτ converges in L ∞ ([0 , T ] , H σ ( t ) , s − ) to R t F ( τ ) dτ where F = G ( η )( ψ, b ) similarly for R t G ν ( τ ) dτ . We have with fixed t , A := (cid:13)(cid:13)(cid:13)(cid:13)Z t ( F ν ( τ ) − F ( τ )) dτ (cid:13)(cid:13)(cid:13)(cid:13) H σ ( t ) , s − = X Z d e σ ( t ) | ξ | h ξ i s − (cid:12)(cid:12)(cid:12)(cid:12)Z t ( \ F ν − F )( τ, ξ ) dτ (cid:12)(cid:12)(cid:12)(cid:12) . ince for 0 ≤ τ ≤ t we have σ ( t ) ≤ σ ( τ ) we deduce from the the Cauchy-Schwarz inequalitythat, A ≤ T Z t k ( F ν − F )( τ ) k H σ ( τ ) , s − dτ. Using Theorem 3.4 with η = η ν , η = η and the fact that the sequences k η ν k X and k ψ ν k X are uniformly bounded we deduce that there exists C = C ( k η k X , k ψ k X ) > I ≤ C ( k η ν − η k X + k ψ ν − ψ k X ) → , if ν → + ∞ . Now we have,(6.31) B : = (cid:13)(cid:13)(cid:13)(cid:13)Z t ( G ν − G )( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) H σ ( t ) , s − ≤ C ( B + B + B ) ,B = (cid:13)(cid:13)(cid:13)(cid:13)Z t ( η ν − η )( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) H σ ( t ) , s − ,B = (cid:13)(cid:13)(cid:13)(cid:13)Z t ( |∇ x ψ ν | − |∇ x ψ | )( τ ) dτ (cid:13)(cid:13)(cid:13)(cid:13) H σ ( t ) , s − ,B = (cid:13)(cid:13)(cid:13)(cid:13)Z t N ν ( τ )(1 + |∇ x η ν ( τ ) | ) − N ( τ )(1 + |∇ x η ( τ ) | ) (cid:13)(cid:13)(cid:13)(cid:13) H σ ( t ) , s − , where N ν = G ( η ν )( ψ ν , b ) + ∇ x η ν · ∇ x ψ ν , N = G ( η )( ψ, b ) + ∇ x η · ∇ x ψ. As for the term A we have, B ≤ C Z t k ( η ν − η )( τ ) k H σ ( t ) , s − dτ ≤ C k ( η ν − η )( τ ) k X → . The terms B et B can be estimated similarly using (6.19) with η ν − and ψ ν − replacedby η, ψ , together with the estimates (6.21) and (6.28). It follows that B tends to zero in L ∞ ([0 , T ] , H σ ( t ) , s − ), which implies that ( η, ψ ) is a solution of system (1.1).The uniqueness of the solution follows from the computation made in Proposition 6.4 wherewe replace ( e η ν , e η ν +1 ) by ( e η , e η ) and ( e ψ ν , e ψ ν +1 ) by ( e ψ , e ψ ) where ( η , ψ ) , ( η , ψ ) are the twosupposed solutions. (cid:3) Existence of a solution on a time interval of size ε − . In this section we prove Theorem 1.6 about the well-posedness of the Cauchy problem on largetime intervals. We shall construct solutions as limits of solutions to a sequence of approximatenonlinear systems. The analysis is in three different steps:(1) Firstly, we define approximate systems and prove that the Cauchy problem for thelatter are well-posed locally in time by means of an ODE argument.(2) Secondly, we prove that the solutions of the approximate systems are bounded on auniform time interval.(3) Thirdly, we prove that these approximate solutions converge to a solution of thewater-waves equations. .1. Approximate system.
Let us rewrite the water-wave system under the form(7.1) ∂ t f = T ( f ; b ) , where(7.2) f = (cid:18) ηψ (cid:19) , T ( f ) = G ( η )( ψ, b ) − gη − |∇ x ψ | + |∇ x η | ) ( G ( η )( ψ, b ) + ∇ x η · ∇ x ψ ) ! , where b = b ( x ) is a given function. We denote by f = ( η , ψ ) the initial data.To define the approximate systems, we use a version of Galerkin’s method based on Friedrichsmollifiers. To do so, following [6], it is convenient to use smoothing operators which areprojections and we consider, for n ∈ N \ { } , the operators J n defined by(7.3) d J n u ( ξ ) = b u ( ξ ) for | ξ | ≤ n, d J n u ( ξ ) = 0 for | ξ | > n. Notice that J n is a projection, J n = J n .Now we consider the following approximate Cauchy problems:(7.4) ( ∂ t f = J n (cid:0) T ( f ; b ) (cid:1) ,f | t =0 = J n f . The following lemma states that, for each n ∈ N , the Cauchy problem (7.4) is well-posedlocally in time. Lemma 7.1.
For all f = ( η , ψ ) ∈ L ( T d ) , for all b ∈ L ( T d ) and for all n ∈ N , thereexists T n > such that the Cauchy problem (7.4) has a unique maximal solution f n = ( η n , ψ n ) ∈ C ([0 , T n ); L ( T d ) ) . Moreover, f n is a smooth function which belongs to C ([0 , T n ); H λ,µ ( T d )) for any λ, µ ≥ .Moreover, either (7.5) T n = + ∞ or lim sup t → T n k f n ( t ) k L = + ∞ . Proof.
Fix f = ( η , ψ ) ∈ L ( T d ) and b ∈ L ( T d ). We begin by studying an auxiliaryCauchy problem which reads(7.6) ( ∂ t f = F n ( f ) where F n ( f ) = J n (cid:0) T ( J n f ; b (cid:1) ,f | t =0 = J n f . We will prove that the Cauchy problem is well-posed by using the classical fixed point argu-ment. Then we will prove that if f n solves (7.6), then it also solves the original problem (7.4).Notice that the operator J n is a smoothing operator: it is bounded from L ( T d ) into H µ ( T d )for any µ ≥
0. Consequently, it will be sufficient to exploit some rough estimates forthe Dirichlet-Neumann operator which can be proved by elementary variational estimates. amely, we just need to know that there exists k large enough and a non-decreasing function F : R + → R + , such that, k G ( η )( ψ, b ) k H − ≤ F (cid:0) k η k H k (cid:1)(cid:0) k ψ k H k + k b k H k (cid:1) , and k G ( η )( ψ, b ) − G ( η )( ψ, b ) k H − ≤ F (cid:0) k η k H k + k η k H k (cid:1)(cid:0) k ψ k H k + k b k H k (cid:1) k η − η k H k . It follows that that the operator f F n ( f ) is locally Lipschitz from L ( T d ) to itself. Conse-quently, the Cauchy-Lipschitz theorem implies that the Cauchy problem (7.6) has a uniquemaximal solution f n in C ([0 , T n ); L ( T d )). Since J n = J n , we check that the function( I − J n ) f n solves ∂ t ( I − J n ) f n = 0 , ( I − J n ) f n | t =0 = 0 . This shows that ( I − J n ) f n = 0, so J n f n = f n . Consequently, the fact that f n solves (7.6)implies that f n is also a solution to (7.4). In addition, since the Fourier transform of f n iscompactly supported, the function f n belongs to C ([0 , T n ); H λ,µ ( T d )) for any λ, µ ≥ (cid:3) Uniform estimates.
Let n ∈ N and denote by ( η n , ψ n ) the approximate solution asconstructed in the previous paragraph. Recall that(7.7) ∂ t η n = J n G ( η n )( ψ n , b ) ,∂ t ψ n = J n (cid:16) − gη n − |∇ x ψ n | + ( G ( η n )( ψ n , b ) + ∇ x η n · ∇ x ψ n ) |∇ x η n | ) (cid:17) ,η n | t =0 = J n η , ψ n | t =0 = J n ψ . In this paragraph we shall state the main estimates that we shall prove in this section. Toprove estimates in large times, we shall work with the unknowns ( ζ, V, B ) already introducedin Section 5. More precisely, we shall consider the unknowns associated to the approximatesystem (7.7): they are(7.8) B n = G ( η n ) ψ n + ∇ x η n · ∇ x ψ n |∇ x η n | , V n = ∇ x ψ n − B n ∇ x η n , together wirh ζ n = ∇ x η n , u n = √ gζ n + ia ( D x ) V n . Then u n is a solution of the equation,(7.9) ∂ t u n + iJ n (cid:0) g a ( D x ) (cid:1) u n = J n (cid:0) √ gF ,n + ia ( D x ) F ,n (cid:1) := F ,n where ,F ,n = G ( η n )( V n , − G (0)( V n ,
0) + G ( η n )(0 , ∇ x b ) − ( V n · ∇ x ) ζ n − (div V n ) ζ n ,F ,n = − ( V n · ∇ x ) V n − ( a n − g ) ζ n . We introduce,(7.10) U s ,n ( t ) = e σ ( t ) h D x i h D x i s − u n . Since e σ ( t ) | ξ | ≤ e σ ( t ) h ξ i ≤ e σ ( t ) e σ ( t ) | ξ | ≤ e λh e σ ( t ) | ξ | nd since ζ n and a ( D x ) V n are real valued functions, there exist two absolute constants0 < C < C such that, for all n and for all t ∈ [0 , T ] and µ = 0 or µ = we have,(7.11) k ζ n ( t ) k H σ ( t ) , s −
12 + µ + k a ( D x ) V n ( t ) k H σ ( t ) , s −
12 + µ ≤ C k U n ( t ) k H µ , k U n ( t ) k H µ ≤ C (cid:0) k ζ n ( t ) k H σ ( t ) , s −
12 + µ + k a ( D x ) V n ( t ) k H σ ( t ) , s −
12 + µ (cid:1) . Fix two real numbers h > λ ∈ [0 , η , ψ ) such that k η k H λh, s + 12 + k a ( D x ) ψ k H λh, s + k V k H λh, s + k B k H λh, s < + ∞ , where, as above, B = G ( η ) ψ + ∇ x η · ∇ x ψ |∇ x η | , V = ∇ x ψ − B ∇ x η . Notice that, when ψ = 0, the functions V and B vanish and hence the previous assumptionis satisfied whenever η ∈ H λh, s + .Recall that we denote by T n the lifespan of the approximate solution ( η n , ψ n ) and that wedenote by ε the constant determined by means of Theorem 3.1. We are now in position tostate our main Sobolev estimates. For this we introduce some notations. We set,(7.12) N s ( b ) = k b k L ∞ ( R ,H s + 12 ) + k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) ,M s ,n ( T ) = k η n k X ∞ , s + 12 T + k a ( D x ) ψ n k X ∞ , s T + k V n k X ∞ , s T + k B n k X ∞ , s T where X ∞ , s T = L ∞ ([0 , T ] , H σ ( · ) , s ) with σ ( t ) = λh − Kεt.
Proposition 7.2.
Assume that s > / d/ and set, (7.13) ε = k η k H λh, s + 12 + k a ( D x ) ψ k H λh, s + k V k H λh, s + k B k H λh, s + N s ( b ) . There exists a constant C ≥ such that, for all n ∈ N \ { } , for all T ≤ T n and for all K > the norm M s ,n ( T ) satisfies the following inequality: if M s ,n ( T ) ≤ ε then, (7.14) M s ,n ( T ) + 2 Kε Z T k U s ,n ( t ) k H dt ≤ C (cid:16) ε + M s ,n ( T ) Z T k U s ,n ( t ) k H dt + T M s ,n ( T ) + 1 Kε T M s ,n ( T ) + εK T M s ,n ( T ) (cid:17) . The proof of the previous proposition is postponed to § Corollary 7.3.
There exist four positive real numbers ε ∗ , c ∗ , C ∗ and K ∗ such that, for all n ∈ N \ { } , the following properties hold: if the initial norm ε (as defined by (7.13) ) satisfies ε ≤ ε ∗ , then for all n ∈ N \ { } , the lifespan is bounded from below by T n ≥ c ∗ ε , nd moreover M s ,n (cid:16) c ∗ ε (cid:17) ≤ C ∗ ε, where the norm M s ,n is as defined in (7.12) , with K replaced by K ∗ .Proof. Fix µ such that µ ≥
12 and µC ≥ . Then set, C ∗ = p µC ≥ , ε ∗ = ε C ∗ , c ∗ = 1 C ∗ ≤ , K ∗ ≥ max (cid:16) , µ C ∗ (cid:17) , where C is the constant whose existence is the main assertion of Proposition 7.2.Hereafter we assume that ε ≤ ε ∗ . Given n ∈ N \ { } , introduce the interval I n = h , min n T n , c ∗ ε oi . We want to prove that, for all n ∈ N \ { } , we have(7.15) ∀ T ∈ I n , M s ,n ( T ) ≤ C ∗ ε. Notice that if (7.15) holds, then it follows that T n ≥ c ∗ /ε . In particular, we can apply (7.15)for T = c ∗ /ε , which will give the wanted result.It remains to prove (7.15). To do so, introduce the set J n = { T ∈ I n : M s ,n ( T ) ≤ C ∗ ε } .With this notation, we want to prove that J n = I n . Notice that 0 ∈ J n since M s ,n (0) ≤ ε and C ∗ ≥
1. Since T M s ,n ( T ) is continuous, the set J n is closed. Hence, to conclude the proof,it is sufficient to prove that J n is open. This is turn will be a straightforward corollary of thefollowing claim:(7.16) ∀ T ∈ J n , M s ,n ( T ) ≤ C ∗ ε. Let us prove this claim. To do so, we will exploit (7.14). Since ε ≤ ε ∗ = ε/ (4 C ∗ ), notice thatif M s ,n ( T ) ≤ C ∗ ε , then we automatically obtain that M s ,n ( T ) ≤ ε . Then we are in positionto apply the estimate (7.14). We obtain, M s ,n ( T ) + 2 K ∗ ε Z T k U s ,n ( t ) k H dt ≤ µ C ∗ (cid:16) ε + C ∗ ε Z T k U s ,n ( t ) k H dt + c ∗ C ∗ ε + c ∗ C ∗ K ∗ ε + c ∗ C ∗ K ∗ ε (cid:17) . Since µ C ∗ ≤ K ∗ we can absorb the integral term in the right hand side by the left hand side.Moreover we have, c ∗ C ∗ = 1 , C ∗ ≥ , c ∗ ≤ , K ∗ ≥ . Therefore we obtain, M s ,n ( T ) ≤ µ C ∗ ε ≤ C ∗ ε . This completes the proof of the claim (7.16). (cid:3) .3. Proof of Proposition 7.2.
We fix d ≥ λ < h > s > / d/
2. To simplifynotations, the indexes n will be skipped: we fix an integer n in N \ { } and denote simply by( η, ψ ) the solutions to the approximate system (7.7). We shall denote by C many differentconstants, whose values may change from a line to another, and which depend only on theparameters which are considered fixed (that is d, λ, h and s ). In particular, these constantsare independent of T, K, ε and n .We set, I = [0 , T ] ,σ ( t ) = λh − Kεt, t ∈ [0 , T ] , ε > , K > ,a ( D x ) = | D x | tanh( h | D x | ) (= G (0)) ,X ∞ , s = L ∞ ( I, H σ ( · ) , s ) ,M s ( T ) = k η k X ∞ , s + 12 + k a ( D x ) ψ k X ∞ , s + k V k X ∞ , s + k B k X ∞ , s ,N s ( b ) = k b k L ∞ ( R ,H s + 12 ) + k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) . Recall (see (2.17)) that, E σ,µ = { u : e λz | D x | u ∈ C z ([ − h, , H σ,µ ( T d ) } ,F σ,µ = { u : e λz | D x | u ∈ L z (( − h, , H σ,µ ( T d ) } . In what follows we fix t ∈ [0 , T ] and we write σ ( t ) = λ ( t ) h where λ ( t ) = λ − Kεth ≤ λ < . Remember that we assume in Proposition 7.2 that, M s ( T ) ≤ ε ≤ . Recall that the function u = √ gζ + ia ( D x ) V which has been introduced above satisfies theequation, ∂ t u + i (cid:0) g a ( D x ) (cid:1) u = √ gF + ia ( D x ) F , where F = G ( η )( V, − G (0)( V,
0) + G ( η )(0 , ∇ x b ) − ( V · ∇ x ) ζ − (div V ) ζ,F = − ( V · ∇ x ) V − ( a − g ) ζ. We shall now estimate the terms F and F . Recall that σ ( t ) = λh − Kεt = λ ( t ) h. Estimate of F . We have, F = G ( η )( V, − G (0)( V,
0) + G ( η )(0 , ∇ x b ) − ( V · ∇ x ) ζ − (div V ) ζ. We shall estimate each term in F . All the estimates will be with fixed t , with constantsindependent of t . Therefore t will be omitted in what follows.First of all we have,(7.17) k G ( η )( V, − G (0)( V, k H σ, s − . M s ( T ) k U s k H . o see this, we use Theorem 3.4 with η = η, η = 0 , ψ = ψ = V, b = 0. Then, with thenotations in (3.12), we have, k G ( η )( V, − G (0)( V, k H σ, s − . λ k η k H σ, s + λ k η k H σ, s + 12 ≤ ( λ + λ ) M s ( T ) . Now, by definition of U s (see (7.10)), we have λ = k η k H σ, s + 12 k a ( D x ) V k H σ ( t ) , s − + k a ( D x ) V k H σ, s . M s ( T ) k U s k L + k U s k H ,λ = k a ( D x ) V k H σ, s − . k U s k L . This implies (7.17). Now, since ζ = ∇ x η , it follows from the product rule in Proposition 8.3that,(7.18) k ( V · ∇ x ) ζ k H σ, s − . k V k H σ, s − k ζ k H σ, s . M s ( T ) k U s k H , and,(7.19) k (div V ) ζ k H σ, s − . k V k H σ, s k ζ k H σ, s − . M s ( T ) k U s k L . Eventually, according to Theorem 3.1, we have,(7.20) k G ( η )(0 , ∇ x b ) k H σ, s − . k b k H s + 12 + M s ( T ) k b k H s − . This completes the analysis of F .7.3.2. Estimate of F = − ( V · ∇ x ) V − ( a − g ) ζ . The first term in F can be estimated asfollows. k a ( D x ) ( V · ∇ x ) V k H σ, s − . k ( V · ∇ x ) V k H σ, s − . k V k H σ, s − k V k H σ, s + 12 . k V k H σ, s − ( k V k H σ, s + k a ( D x ) V k H σ, s ) . Therefore,(7.21) k a ( D x ) ( V · ∇ x ) V k H σ, s − . M s ( T ) + M s ( T ) k U s k H . We estimate now the second term in F . We claim that,(7.22) k ( a − g ) k H σ, s − ≤ C (cid:0) k a ( D x ) V k H σ, s − + k b ( t ) k H s + 12 + k η k H s + 12 + k ∂ t b k H s − + k a ( D x ) ψ k H σ, s − (cid:1) . Recall that a = ∂ y P | Σ . Let Q be the variational solution (for fixed t ) of the problem,(7.23) ∆ x,y Q = 0 , in Ω , Q | Σ = gη + 12 ( B + | V | ) , ∂ y Q | y = − h = − ∂ t b. The pressure P is then defined in Ω by,(7.24) P ( t, x, y ) = Q ( t, x, y ) + gy − |∇ x,y φ ( t, x, y ) | . Since the function φ is the solution of the problem,(7.25) ∆ x,y φ = 0 , φ | Σ = ψ, ∂ y φ | y = − h = b, we have,(7.26) ∂ y ( P − gy ) | y = − h = − ∂ t b + ∇ x b · ( ∇ x φ h ) − b (∆ x φ h ) := P , here φ h = φ | y = − h . We are going to work in the ( x, z ) variables defined previously. Let us recall some notations. e φ ( x, z ) = φ ( x, ρ ( x, z )) , Λ = 1 ∂ z ρ ∂ z , Λ = ∇ x − ∇ x ρ∂ z ρ ∂ z . Set φ h ( x ) := φ ( x, − h ) = e φ ( x, − h ) := e φ h ( x )and, P ( x, z ) = P ( x, ρ ( x, z )) − gρ ( x, z ) . Since Λ ρ = 1 we have,(7.27) a − g = − (Λ P ) | z =0 . On the other hand, according to (7.23) (7.24) and (7.26) the pressure P satifies,∆ x,y ( P − gy ) = − (cid:12)(cid:12) ∇ x,y φ (cid:12)(cid:12) , ( P − gy ) | y = η ( x ) = − gη ( x ) ,∂ y ( P − gy ) | y = − h = − ∂ t b + ∇ x b · ( ∇ x φ h ) − b (∆ x φ h ) := P It follows that P is a solution of the problem,(7.28) (Λ + Λ ) P = G P| z =0 = − gη := P , ∂ z P| z = − h = − ( ∂ z ρ | z = − h ) P . where G is a linear combination of (Λ j Λ k e φ ) , ≤ j, k ≤ . From Theorem 2.10 we have,(7.29) k e λ ( z + h ) | D x | ∇ x,z Pk C ([ − h, ,H s − ) ≤ C (cid:0) k η k H σ, s + 12 (cid:1)(cid:0) k G k F λh, s − + k a ( D x ) P k H λh, s + k ( ∂ z ρ ) | z = − h P k H s − (cid:1) . Estimate of F in F λ, s − . According to Lemma 8.8 we have since s > d , k (Λ j Λ k e φ ) k F λh, s − ≤ C k Λ j Λ k e φ k E λh, s − k Λ j Λ k e φ k F λh, s − . We shall prove the following estimate.(7.30) X j,k =1 k Λ j Λ k e φ k E λh, s − + X j,k =1 k Λ j Λ k e φ k F λh, s − ≤ C ( k a ( D x ) V k H λh, s − + k b k H s − ) . Indeed since Λ and Λ commute we have (Λ + Λ )Λ e φ = 0. On the other hand, by definitionΛ e φ | z =0 = V . Now, Λ (Λ e φ ) | z = − h = Λ (Λ e φ ) | z = − h = ∇ x b. Indeed the right hand side is the image by our diffeomorphism of the quantity ∇ x ( ∂ y φ ) | y = − h = ∇ x [( ∂ y φ ) | y = − h ] = ∇ x b . Summing up U = Λ e φ is a solution of the problem,(Λ + Λ ) U = 0 , U | z =0 = V, ∂ z U | z = − h = ( ∂ z ρ ) | z = − h ∇ x b = ( e − h | D x | η ) ∇ x b. Using Corollary 2.7 we obtain when M s ( T ) ≤ ε, (7.31) k∇ x,z U k F λ, s − ≤ C (cid:0) k a ( D x ) V k H λh, s − + k b k H s − (cid:1) . ecall that Λ = ∂ z ρ and Λ = ∇ x − ∇ x ρ∂ z ρ . Since ∇ x ρ and ∂ z ρ − E and sinceΛ e φ = − Λ e φ we obtain, X j,k =1 k Λ j Λ k e φ k F λh, s − ≤ C ( k a ( D x ) V k H λh, s − + k b k H s − )Now using Lemma 2.12 we can write, k∇ x,z U k E λ, s − ≤ C k∇ x,z U k F λ, s − . Using same argumentas above and (7.31) we deduce that, X j,k =1 k Λ j Λ k e φ k E λh, s − ≤ C ( k a ( D x ) V k H λh, s − + k b k H s − ) , which proves (7.30). Therefore with the notation in (7.28) we have,(7.32) k G k F λ, s − ≤ C ( k a ( D x ) V k H λh, s − + k b k H s − )On the other hand,(7.33) kP k H λh, s + 12 ≤ C k η k H λh, s + 12 . Eventually let us estimate P = ∂ t b − ∇ x b · ( ∇ x φ h ) + b (∆ x φ h ) in H s − . We have, kP k H s − ≤ k ∂ t b k H s − + C k b k H s + 12 ( k∇ x φ h k H s − + k ∆ x φ h k H s − ) . Now, k∇ x φ h k H s − ≤ C ( k∇ x φ h k H s − + k ∆ x φ h k H s − ) , so that, kP k H s − ≤ k ∂ t b k H s − + C k b k H s + 12 ( k∇ x φ h k H s − + k ∆ x φ h k H s − ) . By Corollary 4.2 and Lemma 7.7 we have, k ∆ x φ h k H s − ≤ C (cid:0) k b k H s + 12 + k B k H (cid:1) . Theorem 2.10 and the fact that, X λh, s − ⊂ { u : e λ ( z + h ) | D x | u ∈ C ([ − h, , H s − ) } imply that, k∇ x φ h k H s − ≤ C (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s − (cid:1) . Therefore, kP k H s − ≤ k ∂ t b k H s − + C k b k H s + 12 (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s + 12 (cid:1) and eventually,(7.34) kP k H s − ≤ k ∂ t b k H s − + C (cid:0) k a ( D x ) ψ k H λh, s − + k b k H s + 12 (cid:1) . Then (7.22) follows from (7.29) to (7.34).Then we have,(7.35) k a ( D x ) ( a − g ) k H σ, s − . M s ( T ) + k b k H s + 12 + k ∂ t b k H s − . ndeed this follows from (7.22) since we have, k a ( D x ) V k H σ, s − + k a ( D x ) ψ ( t ) k H σ, s − ≤ CM s ( T ) . Now, k a ( D x ) (( a − g ) ζ ) k H σ, s − . k a ( D x ) ( a − g ) k H σ, s − k η k H σ, s + 12 , Therefore,(7.36) k a ( D x ) (( a − g ) ζ ) k H σ, s − . M s ( T ) + M s ( T ) k b k H s + 12 + M s ( T ) k ∂ t b k H s − . It follows from (7.21) and (7.36) that,(7.37) k a ( D x ) F k H σ, s − . M s ( T ) (cid:0) M s ( T ) + k b k H s + 12 + k ∂ t b k H s − + k U s k H (cid:1) . The a priori estimates.
In this paragraph we first bound the terms in M s ( T ) containing η and V . Recall for the reader’s convenience some notations. We have set, ζ = ∇ x η, u = √ gζ + ia ( D x ) V. Then u is solution of the equation (see (7.9)), ∂ t u + i (cid:0) g a ( D x ) (cid:1) u = √ gF + ia ( D x ) F := F . Set U s ( t ) = e σ ( t ) h D x i h D x i s − u ( t ) , σ ( t ) = λh − Kεt, where K is a large positive constant to be chosen. It follows that U s satisfies the equation, ∂ t U s + i ( g a ( D x )) U s + Kε h D x i U s = e σ ( t ) h D x i h D x i s − F . So we have, ddt k U s ( t ) k L = 2 (cid:0) U s ( t ) , ∂ t U s ( t ) (cid:1) L , = − Kε k h D x i U s ( t ) k L + 2 (cid:0) U s ( t ) , e σ ( t ) h D x i h D x i s − F ( t ) (cid:1) L , because the term 2Re i (cid:0) U s ( t ) , ( g a ( D x )) U s ( t ) (cid:1) L vanishes since the symbol a is real. Wededuce the estimate,(7.38) k U s ( t ) k L +2 Kε Z t k U s ( t ′ ) k H dt ′ = k U s (0) k L + 2 Z t (cid:0) U s ( t ′ ) , e σ ( t ′ ) h D x i h D x i s − F ( t ′ ) (cid:1) L dt ′ , where F = √ gF + ia ( D x ) F . Set,(7.39) A ( t ) = C √ g Z t | (cid:0) U s ( t ′ ) , e σ ( t ′ ) h D x i h D x i s − F ( t ′ ) (cid:1) L | dt ′ ,A ( t ) = C Z t | (cid:0) U s ( t ′ ) , e σ ( t ′ ) h D x i h D x i s − a ( D x ) F ( t ′ ) (cid:1) L | dt ′ . Estimate of A . et us write, by simplicity, K s = e σ ( t ′ ) h D x i h D x i s − . By (7.17),(7.18), (7.19), (7.20) we canwrite, | (cid:0) U s ( t ′ ) , K s ( G ( η ( t ′ )( V ( t ′ ) , − G (0)( V ( t ′ ) , (cid:1) L | . M s ( T ) k U s ( t ′ ) k L k U s ( t ′ ) k H , | (cid:0) U s ( t ′ ) , K s ( V ( t ′ ) · ∇ ) ζ ( t ′ )) (cid:1) L | . M s ( T ) k U s ( t ′ ) k H , | (cid:0) U s ( t ′ ) , K s (div V ( t ′ )) ζ ( t ′ )) (cid:1) L | . M s ( T ) k U s ( t ′ ) k L k U s ( t ′ ) k H , | (cid:0) U s ( t ′ ) , K s ( G ( η ( t ′ )(0 , ∇ x b ( t ′ )) (cid:1) L | . k U s ( t ′ ) k L k b ( t ′ ) k H s + 12 . It follows that,(7.40) | A ( t ) | . M s ( T ) Z t k U s ( t ′ ) k H dt ′ + Z t k U s ( t ′ ) k L k b ( t ′ ) k H s + 12 dt ′ Estimate of A .Using (7.37) we can write, | (cid:0) U s ( t ′ ) , K s a ( D x ) F ( t ′ ) (cid:1) L | . M s ( T ) k U s ( t ′ ) k H (cid:0) M s ( T )+ k b ( t ′ ) k H s + 12 + k ∂ t b ( t ′ ) k H s − + k U s ( t ′ ) k H (cid:1) It follows that,(7.41) | A ( T ) | . M s ( T ) Z t k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k b ( t ′ ) k H s + 12 k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k ∂ t b ( t ′ ) k H s − k U s ( t ′ ) k H dt ′ . Using (7.38), (7.39), (7.40), (7.41) we obtain with,(7.42) E s ( t ) = k U s ( t ) k L + 2 Kε Z t k U s ( t ′ ) k H dt ′ , the estimate,(7.43) E s ( t ) . k U s (0) k L + Z t k b ( t ′ ) k H s + 12 k U s ( t ′ ) k L dt ′ + M s ( T ) Z t k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k b ( t ′ ) k H s + 12 k U s ( t ′ ) k H dt ′ + M s ( T ) Z t k ∂ t b ( t ′ ) k H s − k U s ( t ′ ) k H dt ′ =: k U s (0) k L + X k =1 I k ( t ) . ow we estimate separately each term I k . We have, I ( t ) ≤ k U s k L ∞ ( I,L ) k b k L ( R ,H s + 12 ) ≤ δ k U s k L ∞ ( I,L ) + C δ k b k L ( R ,H s + 12 ) ,I ( t ) ≤ M s ( T ) Z t k U s ( t ′ ) k H dt ′ ,I ( t ) ≤ M s ( T ) √ T (cid:16) Z t k U s ( t ′ ) k H dt ′ (cid:17) ≤ δKε Z t k U s ( t ′ ) k H dt ′ + C δ Kε T M s ( T ) ,I ( t ) ≤ δKε Z t k U s ( t ′ ) k H dt ′ + C δ Kε T M s ( T ) k b k L ∞ ( R ,H s + 12 ) ,I ( t ) ≤ δKε Z t k U s ( t ′ ) k H dt ′ + C δ Kε T M s ( T ) k ∂ t b k L ∞ ( R ,H s − ) . Taking δ small enough and using the above estimates for the I k we can absorb the termscontaining δ by the left hand side of (7.43). We obtain,(7.44) E s ( t ) . k U s (0) k L + δ k U s k L ∞ ( I,L ) + M s ( T ) Z t k U s ( t ′ ) k H dt ′ + 1 Kε T M s ( T ) + 1 Kε T M s ( T ) k b k L ∞ ( R ,H s + 12 ) + 1 Kε T M s ( T ) k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) . Now according to (7.11) we have,(7.45) k U s (0) k L . k η k H λh, s + 12 + k V k H λh, s , k∇ η ( t ) k H σ ( t ) , s − + k a ( D x ) V ( t ) k H σ ( t ) , s − . k U s ( t ) k L . Moreover we have,(7.46) k η ( t ) k H σ ( t ) , s + 12 . k∇ η ( t ) k H σ ( t ) , s − + k η ( t ) k L , k V ( t ) k H σ ( t ) , s . k a ( D x ) V ( t ) k H σ ( t ) , s − + k V ( t ) k L . Using (7.42), (7.44),(7.45),(7.46), we obtain,(7.47) E s ( t ) + k η ( t ) k H σ ( t ) , s + 12 + k V ( t ) k H σ ( t ) , s . k η k H λh, s + 12 + k V k H λh, s + δ k U s ( t ) k L ∞ ( I,L ) + M s ( T ) Z t k U s ( t ′ ) k H dt ′ + 1 Kε M s ( T ) k b k L ∞ ( R ,H s + 12 ) + 1 Kε T M s ( T ) + 1 Kε T M s ( T ) k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) + k η ( t ) k L + k V ( t ) k L . To complete the proof of Proposition 7.2 we are left with two things. First we have to boundthe part in M s ( T ) containing B and ψ. Then we have to bound the low frequency norms ofthe unknowns. This is the object of the two next paragraphs.7.3.4.
Estimates of ψ and B . Lemma 7.4.
There exist
C > , such that if M s ( T ) ≤ ε ≤ we have, k B ( t ) k H σ, s ≤ C (cid:0) k V ( t ) k H σ, s + k b k L ∞ ( R ,H s − ) + k B ( t ) k H (cid:1) . roof. We fix t ∈ [0 , T ] and we take σ ≤ λh . We start from the identity G ( η )( B, − ∆ x φ h ) = G ( η )( B,
0) + G ( η )(0 , − ∆ x φ h ) = − div V proved in (5.9).We have seen that G (0)( B,
0) = | D x | tanh( h | D x | ) B . We write,(7.48) G (0)( B,
0) = − div V + G ( η )(0 , ∆ x φ h ) + ( G (0) − G ( η ))( B,
0) := F. Let χ ∈ C ∞ ( R d ) be such that χ ( ξ ) = 1 if | ξ | ≤ χ ( ξ ) = 0 if | ξ | ≥
2. We have k χ ( D ) B k H σ, s ≤ C k B k L . On the other hand it follows from (7.48) that,(1 − χ ( ξ )) b B ( ξ ) = 1 − χ ( ξ ) | ξ | tanh( h | ξ | ) b F ( ξ ) , so that, k (1 − χ ( D )) B k H σ, s ≤ C k F k H σ, s − . First of all we have k div V k H σ, s − . k V k H σ, s . Using Theorem 3.4, and Remark 3.5 we have, k ( G (0) − G ( η ))( B, k H σ, s − ≤ CM s ( T ) k B k H σ, s ≤ Cε k B k H σ, s . Eventually from Theorem 3.1, Remark 3.2, Corollary 4.2 applied with µ = s − we obtain, k G ( η )(0 , ∆ x φ h ) k H σ, s − . k ∆ φ h k H s − . k b k H s − + k B k H . It follows that, k (1 − χ ( D )) B k H σ, s ≤ C (cid:0) k V k H σ, s + k b k H s − + k B k H + ε k B k H σ, s . (cid:1) . It follows from the above estimate of χ ( D ) B that, k B k H σ, s ≤ C (cid:0) k V k H σ, s + k b k H s − + k B k H + ε k B k H σ, s . (cid:1) . Taking ε sufficiently small we obtain the desired result. (cid:3) Lemma 7.5.
There exists
C > such that for all t ∈ (0 , T ) we have, k a ( D x ) ψ ( t ) k H σ, s ≤ C (cid:0) k V ( t ) k H σ, s + k B ( t ) k H σ, s k η ( t ) k H σ, s + k a ( D x ) ψ ( t ) k L (cid:1) . Proof.
We write, for fixed t ∈ [0 , T ] and σ ≤ λh, k a ( D x ) ψ k H σ, s = X | ξ |≤ h ξ i s e σ | ξ | | ξ | tanh( h | ξ | ) | b ψ ( ξ ) | + X | ξ |≥ h ξ i s e σ | ξ | | ξ | tanh( h | ξ | ) | b ψ ( ξ ) | := I + I . The integral I is estimated by Ce h k a ( D x ) ψ k L . Then we write, I ≤ C X | ξ |≥ | ξ | h ξ i s e σ | ξ | || ξ | b ψ ( ξ ) | ≤ C k∇ x ψ k H σ, s − . Eventually we notice that ∇ x ψ = V + B ∇ x η , so that, since s − > d , we obtain, k∇ x ψ ( t ) k H σ, s − . k V ( t ) k H σ, s − k B k H σ, s − k η k H σ, s + 12 , which proves the lemma. (cid:3) orollary 7.6. There exists
C > such that for every t ∈ (0 , T ) we have, (7.49) E s ( t ) + k η ( t ) k H σ ( t ) , s + 12 + k a ( D x ) ψ k H σ ( t ) , s + k V ( t ) k H σ ( t ) , s + k B ( t ) k H σ ( t ) , s . k η k H λh, s + 12 + k V k H λh, s + δ k U s k L ∞ ( I,L ) + M s ( T ) Z t k U s ( t ′ ) k H , dt ′ + 1 Kε T M s ( T ) k b k L ∞ ( R ,H s + 12 ) + 1 Kε T M s ( T ) + 1 Kε T M s ( T ) k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s + 12 ) + k b k L ∞ ( R ,H s − ) + k η ( t ) k L + k a ( D x ) ψ ( t ) k L + k V ( t ) k L + k B ( t ) k H . Low frequency estimates of η, V, ψ, B . Lemma 7.7.
Assume s > d and M s ( T ) ≤ ε ≤ . There exists
C > such that for every t ∈ (0 , T ) , (7.50) k η ( t ) k H s − + k a ( D x ) ψ ( t ) k H s − ≤ C (cid:16) k η k H s − + k a ( D x ) ψ k H s − + T M s ( T ) + k b k L ( I,H s − ) (cid:17) , (7.51) k V ( t ) k H s − + k B ( t ) k H s − ) ≤ C (cid:16) k η k H s − + k a ( D x ) ψ k H s − + T M s ( T ) + k b k L ( R ,H s − ) + k b k L ∞ ( R ,H s − ) (cid:17) . Proof.
We start from system (1.1) which we write as,(7.52) ∂ t η − G (0) ψ = f := ( G ( η ) − G (0))( ψ,
0) + G ( η )(0 , b ) ,∂ t ψ − gη = f := 12 |∇ x ψ | − ( G ( η )( ψ, b ) + ∇ x η · ∇ x ψ ) |∇ x η | ) , where G (0) ψ = G (0)( ψ,
0) = a ( D x ) ψ = | D x | tanh( h | D x | ) ψ. We set u = gη + ig a ( D x ) ψ . Then u is a solution of the equation,(7.53) ∂ t u + i ( ga ( D x )) u = gf + i ( ga ( D x )) f . Computing ddt k u ( t ) k H s − on the interval I = [0 , T ] we obtain the inequality,(7.54) k u ( t ) k H s − ≤ C (cid:16) k u k H s − + Z t k f ( t ′ ) k H s − dt ′ + Z t k f ( t ′ ) k H s − dt ′ (cid:17) . Let us estimate f . First of all Corollary 3.6 gives, k ( G ( η ( t )) − G (0))( ψ ( t ) , k H s − ≤ k ( G ( η ( t )) − G (0))( ψ ( t ) , k H σ ( t ) , s − ≤ CM s ( T ) . It follows that,(7.55) Z t k ( G ( η ( t )) − G (0)( ψ ( t ′ ) , k H s − dt ′ ≤ CT M s ( T ) . ow, from Theorem 3.1 we get,(7.56) Z t k G ( η ( t ′ ))(0 , b ( t ′ )) k H s − dt ′ ≤ Z t k G ( η ( t ′ ))(0 , b ( t )) k H σ ( t ) , s − dt ′ , ≤ C k b k L ( I,H s − ) . It follows from (7.55), (7.56) that,(7.57) Z t k f ( t ′ ) k H s − dt ′ ≤ C (cid:0) T M s ( T ) + k b k L ( I,H s − ) (cid:1) Let us estimate f . Using Remark 2.16 we can write, k|∇ x ψ ( t ) | k H s − ≤ C k∇ x ψ ( t ) k H s − ≤ C k∇ x ψ ( t ) k H σ ( t ) , s − ≤ C ′ k a ( D x ) ψ ( t ) k H σ ( t ) , s . Therefore, k|∇ x ψ | k L ∞ ( I,H s − ) ≤ CM s ( T ) . It follows that,(7.58) Z T k|∇ x ψ ( t ) | k H s − dt ≤ CT M s ( T ) . Now we can write, e f = ( G ( η )( ψ, b ) + ∇ x η · ∇ x ψ ) |∇ x η | ) =: U (cid:0) − g ( |∇ x η | ) (cid:1) ,U = 12 ( G ( η )( ψ, b ) + ∇ x η · ∇ x ψ ) , g ( t ) = t t . Then, k e f k H s − ≤ C k U k H s − (cid:0) k g ( |∇ η | ) k H s − (cid:1) . By the product laws in the usual Sobolev spaces we can write, k U k H s − . k G ( η )( ψ, b ) k H s − + k η k H s + 12 k a ( D x ) ψ k H s . k G ( η )( ψ, b ) k H s − + M s ( T ) , k g ( |∇ η | ) k H s − . k η k H s + 12 . M s ( T ) . Using Theorem 3.1 we have, k G ( η )( ψ, b ) k H s − . (cid:0) k a ( D x ) ψ k H σ, s + k b k H s − + k η k H s + 12 ( k a ( D x ) ψ k H σ, s − + k b k H s − ) (cid:1) , . M s ( T ) + k b k H s − . Therefore, k e f ( t ) k H s − ≤ C (cid:0) M s ( T ) + k b ( t ) k H s − (cid:1) . It follows that,(7.59) Z t k e f ( t ) k H s − dt . T M s ( T ) + k b k L ( I,H s − ) . Using (7.54),(7.57), (7.58), (7.59) we obtain, k u ( t ) k H s − . k u k H s − + T M s ( T ) + k b k L ( R ,H s − ) . sing (7.54) and the definition of u we get,(7.60) k η ( t ) k H s − + k a ( D x ) ψ ( t ) k L ∞ ( I,H s − ) . k η k H s − + k a ( D x ) ψ k H s − + T M s ( T ) + k b k L ( R ,H s − ) + k b k L ( R ,H s − ) . Let us estimate B with fixed t . By definition we can write, B = G ( η )( ψ, b ) + ∇ x ψ · ∇ x η |∇ x η | = W (cid:0) − g ( |∇ x η | (cid:1) ,W = G ( η )( ψ, b ) + ∇ x ψ · ∇ x η, g ( t ) = t t . Now, we use as before the product laws in the usual Sobolev spaces. Since s > d/
2, itfollows from Remark 3.2 applied with δ = 1 and λ = 0, that, k G ( η )( ψ, b ) k H s − . k a ( D x ) ψ k H s − + k b k H s − . Therefore, k W k H s − . k a ( D x ) ψ k H s − + k b k H s − + k a ( D x ) ψ k H s − k η k H s − . Moreover, k g ( |∇ η | ) k H s . k η k H s − . ε ≤ . It follows that, k B k H s − . k a ( D x ) ψ k H s − + k b k H s − . Using (7.60) we deduce that, It follows that,(7.61) k B ( t ) k H s − . (cid:0) k η k H s − + k a ( D x ) ψ k H s − + T M s ( T ) + k b k L ( R ,H s − ) + k b k L ( R ,H s − ) + k b k L ∞ ( R ,H s − ) (cid:1) Now we have, V = ∇ x ψ − B ∇ x η . It follows that, k V k L ∞ ( I,H s − ) . k a ( D x ) ψ k L ∞ ( I,H s − ) + k B k L ∞ ( I,H s − ) k η k L ∞ ( I,H s − ) . Since k η k L ∞ ( I,H s − ) ≤ ε ≤ k V ( t ) k H s − . (cid:0) k η k H s − + k a ( D x ) ψ k H s − + T M s ( T ) + k b k L ( R ,H s − ) + k b k L ( R ,H s − ) + k b k L ∞ ( R ,H s − ) (cid:1) (cid:3) orollary 7.8. There exists
C > such that for every t ∈ (0 , T ) we have, (7.63) E s ( t ) + k η ( t ) k H σ ( t ) , s + 12 + k a ( D x ) ψ k H σ ( t ) , s + k V ( t ) k H σ ( t ) , s + k B ( t ) k H σ ( t ) , s . k η k H λh, s + 12 + k V k H λh, s + k a ( D x ) ψ k H λh, s + δ k U s k L ∞ ( I,L ) + M s ( T ) Z t k U s ( t ′ ) k H , dt ′ + T M s ( T ) + 1 Kε T M s ( T ) + 1 Kε T M s ( T ) k b k L ∞ ( R ,H s + 12 ) + 1 Kε T M s ( T ) k ∂ t b k L ∞ ( R ,H s − ) + k b k L ( R ,H s − ) + k b k L ( R ,H s + 12 ) + k b k L ∞ ( R ,H s − ) . Proof.
This follows from Corollary 7.6 and Lemma 7.7. (cid:3)
End of the proof of Proposition 7.2.
Acccording to the definition of E s ( T ) (see (7.42)) takingthe supremum of both members with respect to t in (0 , T ) and δ small enough we can absorbthe term δ k U s k L ∞ ( I,L ) by the left hand side. Moreover by definition of ε we have, k η k H λh, s + 12 + k a ( D x ) ψ k H λh, s + k V k H λh, s + k B k H λh, s ≤ ε, k b k L ( R ,H s + 12 ) + k b k L ∞ ( R ,H s + 12 ) + k ∂ t b k L ∞ ( R ,H s − ) ≤ ε. By H¨older’s inequality we also have k b k L ( R ,H s + 12 ) ≤ ε. Therefore using Corollary 7.8 one canfind C > M s ( T ) + 2 Kε Z T k U s ( t ) k H dt ≤ C (cid:16) ε + M s ( T ) Z T k U s ( t ) k H dt + T M s ( T ) + 1 Kε T M s ( T ) + εK T M s ( T ) (cid:17) . This completes the proof. (cid:3)
End of the proof.
We are now in position to complete the proof of Theorem 1.6.
Uniqueness.
Without source term (that is when b = 0), the uniqueness of smooth solutionsis a well-known result. When b is non-trivial, we notice that the uniqueness result asserted byProposition 1.4 implies the uniqueness of the solutions satisfying the regularity assumptionsin Theorem 1.6. Indeed, if we consider an initial data ( η , ψ ) satisfying assumption (1.4),and two possible solutions ( η , ψ ) and ( η , ψ ), satisfying the Cauchy problem (1.1), withthe same initial data ( η , ψ ), and satisfying the regularity result (1.5), then they are bothsolutions satisfying trivially (1.3). Passage to the limit.
So it remains to prove the existence part of the result. For the readerconvenience, let us recall that we have proved the existence of approximate solutions ( η n ψ n ) o the Cauchy problem(7.64) ∂ t η n = J n G ( η n )( ψ n , b ) ,∂ t ψ n = J n (cid:16) − gη n − |∇ x ψ n | + ( G ( η n )( ψ n , b ) + ∇ x η n · ∇ x ψ n ) |∇ x η n | ) (cid:17) ,η n | t =0 = J n η , ψ n | t =0 = J n ψ , where J n is a truncation in frequency space defined in (7.3). In this paragraph, we shall provethat one can extract a sub-sequence of (( η n ′ , ψ n ′ )) which converges weakly to a solution ofthe water-wave system (thanks to the uniqueness of the solution to the water-wave system,this will imply that the whole sequence converges, without extraction of a sub-sequence).This part relies on classical arguments from functional analysis, but since we work in analyticspaces and since the problem is nonlinear and nonlocal, some verifications are needed.Recall that there exist four positive real numbers ε ∗ , c ∗ , C ∗ and K ∗ such that, for all n ∈ N \ { } , the following properties hold: if the initial norm ε (as defined by (7.13)) satisfies ε ≤ ε ∗ , then for all n ∈ N \ { } , the lifespan is bounded from below by T n ≥ c ∗ ε and moreover(7.65) M s ,n (cid:16) c ∗ ε (cid:17) ≤ C ∗ ε, where the norm M s ,n ( T ) is defined by(7.66) M s ,n ( T ) = k η n k X ∞ , s + 12 T + k a ( D x ) ψ n k X ∞ , s T + k V n k X ∞ , s T + k B n k X ∞ , s T where X ∞ , s T = L ∞ ([0 , T ] , H σ, s ) with σ ( t ) = λh − K ∗ εt. Let us notice that Theorem 3.1 implies that k G ( η n )( ψ n , b ) k X ∞ , s − c ∗ /ε ≤ C ′∗ ε, for some constant C ′∗ independent of ε and n . Then, by using the product rule given bypoint ii ) in Proposition 8.3, as we already did repeatedly in the previous paragraph, we inferfrom the equations (7.64) that(7.67) k ∂ t η n k X ∞ , s − c ∗ /ε + k ∂ t ψ n k X ∞ , s − c ∗ /ε ≤ C ′′∗ ε. Now, using the Arzela-Ascoli’s theorem and the compact embedding of H λh, s ( T d ) in H λh, s ′ ( T d )for s ′ < s , it follows that there is a sub-sequence (( η n ′ , ψ n ′ )) and a limit ( η, ψ ) such that( η n ′ , ψ n ′ ) converges to ( η, ψ ) in X ∞ , s ′ c ∗ /ε . Now, the contraction result for the Dirichlet-to-Neumann operator given by Theorem 3.4 implies that the sequence ( G ( η n ′ )( ψ n ′ , b )) convergesto G ( η )( ψ, b ). It follows that the limit ( η, ψ ) ∈ X ∞ , s ′ c ∗ /ε solves the water-waves equations. . Appendix. Some properties of the H σ, s spaces. Characterization.
In this paragraph, we prove Theorem 1.1, whose statement is re-called below, together with the fact that functions in H σ, s ( T d ) are the traces on T d of holo-moprhic functions in S σ = { ( x, y ) ∈ T d × R d : | y | < σ } where | y | = (cid:16) d X j =1 y j (cid:17) . Recall that, given U : S σ → C , we denote by U y the function from T d to C defined by x U ( x + iy ). Theorem 8.1.
Let σ > and s ∈ R .1. Let u ∈ H σ, s ( T d ) . There exists U ∈ H ol ( S σ ) such that U = u and sup | y | <σ k U y k H s x ( T d ) ≤ k u k H σ, s .
2. Let U ∈ H ol ( S σ ) such that M := sup | y | <σ k U y k H s x ( T d ) < + ∞ . Set u = U . Then,(i) If d = 1 , then u belongs to H σ, s ( T d ) and k u k H σ, s ≤ M . (ii) If d ≥ , then u belongs to H δ, s ( T d ) for any δ < σ and there exists a constant C δ > such that k u k H δ, s ≤ C δ M .3. Let U ∈ H ol ( S σ ) be such that, M := sup | y | <σ k U y k H s ′ x ( T d ) < + ∞ with s ′ > s + d − . Then the function u = U belongs to H σ, s ( T d ) and there exists a constant C > such that k u k H σ, s ≤ CM . First step: existence of an holomorphic extension . Let us prove statement (1). Fix σ > s ∈ R and consider a function u ∈ H σ, s ( T d ). We want to prove that there exists U ∈ H ol ( S σ ) such that U = u andsup | y | <σ k U y k H s x ( T d ) ≤ k u k H σ, s . We begin by observing that, for any z ∈ S σ , the function Z d → C , ξ e iz · ξ b u ( ξ ) belongs to ℓ ( Z d ). To see this, we write that(8.1) | e iz · ξ b u ( ξ ) | = e − y · ξ | b u ( ξ ) | = (cid:2) h ξ i − s e − σ | ξ | e − y · ξ (cid:3) × (cid:2) h ξ i s e σ | ξ | | b u ( ξ ) | (cid:3) := f ( ξ ) × f ( ξ ) , and then conclude since | f ( ξ ) | ≤ h ξ i − s e − ( σ −| y | ) | ξ | ∈ ℓ ( Z d ) and f ∈ ℓ ( Z d ), by assumptionon u . So we can define the function(8.2) U ( z ) = (2 π ) − d X ξ ∈ Z d e iz · ξ b u ( ξ ) . The previous inequality implies that, for any ε > z ∈ S σ − ε X ξ ∈ Z d | e iz · ξ b u ( ξ ) | < + ∞ . ince the function z e iz · ξ b u ( ξ ) is holomorphic, we deduce that U ∈ H ol ( S σ ).We next observe that the Fourier inversion formula implies that U = F b u = u . In addition,we have U ( x + iy ) = F ( e − y · ξ b u ), hence c U y = e − y · ξ b u which in turn implies that k U y k H s ( T d ) = X ξ ∈ Z d h ξ i s e − y · ξ | b u ( ξ ) | ≤ X ξ ∈ Z d h ξ i s e σ | ξ | | b u ( ξ ) | = k u k H σ, s . This completes the proof of statement (1).
Second step: a Sobolev estimate . For later purpose, let us prove an additional estimate.Consider a real number s > d/ e − y · ξ | b u ( ξ ) | = h ξ i − s e − y · ξ e − σ | ξ | e σ | ξ | h ξ i s | b u ( ξ ) | . Now, compared to the previous proof, we see that the factor h ξ i − s e − y · ξ e − σ | ξ | is summablein ξ , uniformly for | y | ≤ σ , thanks to the assumption s > d/
2. Then, the Cauchy-Schwarzinequality implies that there exists a positive constant C = C ( d, s ) such that,(8.3) k U k L ∞ ( S σ ) = sup | y | <σ k U y k L ∞ ( T d ) ≤ C k u k H σ, s . Third step: trace of an holomorphic function . In this step, we initiate the proof of thevarious points in statement (2). Consider a function U ∈ H ol ( S σ ) such that(8.4) M := sup | y | <σ k U y k H s x ( T d ) < + ∞ . We want to study the regularity of the trace u = U .First, observe the assumption (8.4) implies that u = U ∈ H s ( T d ).Now, let ψ = 1 | k |≤ . Given λ >
0, we introduce the functions ψ λ ( ξ ) = ψ ( ξλ ) and ϕ λ = F ψ λ .Set, F λ ( z ) = (2 π ) − d X ξ ∈ T d e iz · ξ ψ λ ( ξ ) b u ( ξ ) . This function is holomorphic in S σ . Indeed, the integrand is holomorphic and, for z ∈ S σ , | e iz · ξ ψ λ ( ξ ) b u ( ξ ) | = e − (Im z ) · ξ ψ λ ( ξ ) | b u ( ξ ) | ≤ e σ | ξ | ψ λ ( ξ ) | b u ( ξ ) | ≤ e σλ ψ λ ( ξ ) | b u ( ξ ) | , ≤ (cid:2) e σλ h ξ i − s ψ λ ( ξ ) (cid:3) × (cid:2) h ξ i s | b u ( ξ ) | ] ∈ ℓ ( Z d ) . Notice that, F λ ( x + iy ) = F ( e − y · ξ ψ λ ( ξ ) b u ( ξ ))( x ) . For z = x + iy ∈ S σ we set, V λ ( z ) = U y ⋆ ϕ λ ( x ) = Z T d U ( x + iy − t ) ϕ λ ( t ) dt. This function is holomorphic in the strip S σ . In addition, V λ | y =0 = u ⋆ ϕ λ = F ( ψ λ b u ) = (2 π ) − d X ξ ∈ Z d e ix · ξ ψ λ b u = F λ | y =0 . y uniqueness for analytic functions, this implies that V λ = F λ in S σ . By taking the Fouriertransform of the previous identity, we obtain F U y ( ξ ) ψ λ ( ξ ) = e − y · ξ ψ λ ( ξ ) b u ( ξ ) . By letting λ goes to + ∞ , we infer that F U y ( ξ ) = e − y · ξ b u ( ξ ). Consequently, k U y k H s ( T d ) = X ξ ∈ Z d e − y · ξ h ξ i s | b u ( ξ ) | . Then the assumption on U implies that,(8.5) sup | y | <σ X ξ ∈ Z d e − y · ξ h ξ i s | b u ( ξ ) | = M < + ∞ . This is the key ingredient to prove the three statements in point (2).
Fourth step: trace of an holomorphic function in dimension one .Assume that d = 1. Set v ( ξ ) = h ξ i s | b u ( ξ ) | . For any real number 0 < b < σ , the inequality(8.5) applied with y = − b (resp. y = b ) implies that + ∞ X ξ =0 e bξ v ( ξ ) ≤ M , resp. X ξ = −∞ e − bξ v ( ξ ) ≤ M . It follows that P ξ ∈ Z e b | ξ | v ( ξ ) ≤ M . Fatou’s lemma then implies that, when b goes to σ ,we have, X ξ ∈ Z e σ | ξ | v ( ξ ) ≤ M , which proves statement (2i). Fifth step: arbitrary dimension .Let us prove statement (2ii). We now assume that d ≥ δ < σ .We can write δ = (1 − ε ) σ for some ε >
0. Then there exists N = N ( ε ) and ω , . . . , ω N ∈ S d − such that, Z d \ { } = N [ j =1 Γ j where Γ j = n ξ ∈ Z d : (cid:12)(cid:12)(cid:12)(cid:12) ξ | ξ | − ω j (cid:12)(cid:12)(cid:12)(cid:12) < ε o . Notice that, ξ ∈ Γ j ⇒ (cid:12)(cid:12)(cid:12)(cid:12) ξ | ξ | (cid:12)(cid:12)(cid:12)(cid:12) + | ω j | − ξ | ξ | · ω j < ε ⇒ (cid:18) − ε (cid:19) | ξ | ≤ ξ · ω j . Consider 0 < b < δ and set e b = b − ε / < σ . We have X Z d e b | ξ | v ( ξ ) ≤ N X j =1 X Γ j e b | ξ | v ( ξ ) ≤ N X j =1 X Γ j e e b ( ξ · ω j ) v ( ξ ) . Since the vector y j = e bω j satisfies | y j | := e b | ω j | < σ , the key estimate (8.5) implies that, X Z d e b | ξ | v ( ξ ) ≤ N M . s above, we conclude by using Fatou’s lemma, which implies that, X Z d e δ | ξ | v ( ξ ) ≤ N M , which concludes the proof of statement (2ii). Sixth step: arbitrary dimension, sharp estimate .We now prove statement (3), which gives a smaller loss in analyticity. Namely, we assumethat, U ∈ H ol ( S σ ) is such that, M := sup | y | <σ k U y k H s ′ x < + ∞ with s ′ > s + d − . Our goal is to prove that, the function u = U belongs to H σ, s and there exists a constant C > k u k H σ, s ≤ CM .Let b < σ and s ′ > s . By replacing s by s ′ in (8.5) we have,(8.6) sup | y | <σ X Z d e − y · ξ h ξ i s ′ | b u ( ξ ) | = sup | y | <σ k U y k H s ′ = M < + ∞ . Set v s ( ξ ) = h ξ i s | b u ( ξ ) | and consider a real number R such that, σR ≫
1. We first noticethat,(8.7) X | ξ |≤ R e b | ξ | v s ( ξ ) ≤ e bR X v s ( ξ ) ≤ e bR X h ξ i d − v s ( ξ ) ≤ e bR M . Define ℓ as the largest integer such that, ℓ ≤ R and then, for ℓ ≥ ℓ , introduce thedyadic rings, C ℓ = { ξ ∈ Z d : 12 2 ℓ ≤ | ξ | ≤ ℓ +1 } . Write,(8.8) X | ξ | >R e b | ξ | v s ( ξ ) ≤ + ∞ X ℓ = ℓ X C ℓ e b | ξ | v s ( ξ ) =: + ∞ X ℓ = ℓ I ℓ . Fix ℓ ≥ ℓ and set δ ℓ = b ℓ ≪
1. Let ω be an arbitrary point on the sphere S d − andintroduce, Ω ω = { ω ∈ S d − : ω · ω > − δ ℓ } = { ω ∈ S d − : | ω − ω | < p δ ℓ } . The sets Ω ω have a ( d − − dimensional measure independent of ω . Moreover, there existstwo positive constants c , c independent of the dimension such that, c δ d − ℓ ≤ µ (Ω ω ) ≤ c δ d − ℓ . There exists ω , . . . , ω N ℓ ∈ S d − where N ℓ ∼ C d δ − d − ℓ such that S d − = ∪ N ℓ j =1 Ω ω j . Set, C ℓ,j = (cid:26) ξ ∈ C ℓ : ξ | ξ | ∈ Ω ω j (cid:27) . hen one can split the dyadic ring, C ℓ as C ℓ = ∪ N ℓ j =1 C ℓ,j , to obtain I ℓ ≤ N ℓ X j =1 X C ℓ,j e b | ξ | v s ( ξ ) := N ℓ X j =1 I ℓ,j . If ξ belongs to C ℓ,j one has, ξ | ξ | · ω j > − δ ℓ , so | ξ | < ξ · ω j + | ξ | δ ℓ . Then we write, I ℓ ≤ N ℓ N ℓ X j =1 X C ℓ,j e bω j ) · ξ +2 b | ξ | δ ℓ N ℓ v s ( ξ ) . Recall that, δ ℓ = 1 / ( b ℓ ). Consequently if ξ in C ℓ,j , we have, 2 b | ξ | δ ℓ ≤
4. Then one has e b | ξ | δ ℓ ≤ e . Moreover, N ℓ ≤ C d δ − d − ℓ ≤ C d ( b ℓ ) d − ≤ C d (2 b ) d − | ξ | d − and, ℓ ≤ c log | ξ | .Remembering that v s ( ξ ) = h ξ i s | b u ( ξ ) | , we deduce that, for any ε > I ℓ ≤ C ′ d b d − ℓ ε N ℓ N ℓ X j =1 X C ℓ,j e bω j ) · ξ | ξ | d − (log | ξ | ) ε h ξ i s | b u ( ξ ) | . Now, we use (8.6) to infer that, for ε small enough (so that s + ε < s ′ ) we have,(8.9) I ℓ ≤ C ′ d b d − ℓ ε M . Since P ℓ ≥ ℓ − − ε < + ∞ , it follows from (8.8) that, X | ξ | >R e b | ξ | h ξ i s | b u ( ξ ) | ≤ CM . Using (8.7) we obtain eventually, X Z d e b | ξ | h ξ i s | b u ( ξ ) | ≤ CM . Again, we conclude the proof thanks to Fatou’s lemma. This completes the proof of Theo-rem 8.1.Notice that, if u is radial, then one can remove the factor (log | ξ | ) ε in (8.9) and hence itis sufficient to assume that M = sup | y | <σ k U y k H s + d − x < + ∞ . Notice also that, in this case,the above assumption on U y is optimal to insure that u ∈ H σ, s .8.2. An interpolation lemma.Lemma 8.2.
Let s ∈ R and h > . There exists C > such that for all σ ≥ , if f isin L (( − h, , H σ, s + ) and ∂ z f is in L (( − h, , H σ, s − ) , then f belongs to C ([ − h, , H σ, s ) together with the estimate, sup z ∈ [ − h, k f ( z, · ) k H σ, s ≤ C (cid:0) k f k L (( − h, , H σ, s + 12 ) + k ∂ z f k L (( − h, , H σ, s − ) (cid:1) . .3. Nonlinear estimates.Proposition 8.3. i) Consider three real numbers s , s , s such that s + s ≥ , s ≤ min { s , s } , s < s + s − d . Then there exists
C > such that, for all σ ≥ , (8.10) k u u k H σ, s ≤ C k u k H σ, s k u k H σ, s . ii) For all s > d/ , there exists C > such that, for all σ ≥ , k u u k H σ, s ≤ C k u k H σ, s k u k H σ, s . iii) For all s > d/ and all t ≥ , there exists C > such that, for all σ ≥ , k u u k H σ,t ≤ C k u k H σ, s k u k H σ,t + C k u k H σ, s k u k H σ,t . Proof.
We shall use the following lemma (see Theorem 8 . . Lemma 8.4 (H¨ormander) . Consider three real numbers s , s , s such that s + s ≥ , s ≤ min { s , s } , s ≤ s + s − d , with the last inequality strict if s or s or − s is equal to d/ .For ξ, ζ in R d , define F ( ξ, ζ ) = h ξ i s h ξ − ζ i s h ζ i s , and then set T F ( f, g )( ξ ) = X ζ ∈ Z d F ( ξ, ζ ) f ( ξ − ζ ) g ( ζ ) , when f and g are continuous functions with compact support. Then there exists a positiveconstant C such that k T F ( f, g ) k ℓ ≤ C k f k ℓ k g k ℓ . Let us prove statement i ). Set U = h D i s e σ | D | ( u u ). Then, b U ( ξ ) = h ξ i s e σ | ξ | X Z d c u ( ξ − ζ ) c u ( ζ ) . Since | ξ | ≤ | ξ − η | + | η | we can write, (cid:12)(cid:12) b U ( ξ ) (cid:12)(cid:12) ≤ X Z d F ( ξ, ζ ) f ( ξ − ζ ) g ( ζ ) , where F ( ξ, ζ ) = h ξ i s h ξ − ζ i s h ζ i s , f ( ξ ) = h ξ i s e σ | ξ | | c u ( ξ ) | , g ( ζ ) = h ζ i s e σ | ζ | | c u ( ζ ) | . Then (8.10) follows from Lemma 8.4 and Plancherel’s identity. Statement ii ) is an immediatecorollary of the first point applied with s = s = s = s . (cid:3) orollary 8.5. Consider two real numbers s > d/ and s ≥ . Let P be a polynomial ofdegree m ≥ such that P (0) = 0 and let u ∈ H σ, s ∩ H σ, s . Then P ( u ) ∈ H σ, s and, k P ( u ) k H σ, s ≤ Q ( k u k H σ, s ) k u k H σ, s where Q is a polynomial of degree m − . Proof.
This is an immediate consequence of statement ii ) in Proposition 8.3. (cid:3) Proposition 8.6.
Consider three real numbers s > d/ , s ≥ , M > and let f be aholomorphic function in the ball { z ∈ C : | z | < M } , such that f (0) = 0 . There exists ε > such that for all σ > , if u ∈ H σ, s ∩ H σ, s satisfies k u k H σ, s ≤ ε , then f ( u ) belongs to H σ, s .Moreover, there exists C > depending only on f, s , s , ε such that, k f ( u ) k H σ, s ≤ C k u k H σ, s . Proof.
Set C s = 2 s C ( d, s ). It follows from Proposition 8.3 an from an induction that for all n ≥ , (8.11) k u n k H σ, s ≤ (2 C max( s , s ) ) n − k u k n − H σ, s k u k H σ, s . For | z | < M we can write f ( z ) = P + ∞ n =1 a n z n where a n is such that | a n | ≤ K n , K >
0. Weshall show that the series P a n u n is normaly convergent in H σ, s . Indeed according to (8.11)and the hypothesis we have, k a n u n k H σ, s ≤ K (2 C max( s , s ) Kε ) n − k u k H σ, s . We have just to take ε small enough, so that 2 C max( s , s ) Kε <
1. Since k u k L ∞ ( R d ) ≤ C ( d, s ) k u k H σ, s , taking moreover ε such that C ( d, s ) ε < M we will have, k f ( u ) k H σ, s ≤ K (cid:16) + ∞ X n =1 (2 KC max( s , s ) ) n − k u k n − H σ, s (cid:17) k u k H σ, s . This completes the proof. (cid:3)
Corollary 8.7.
Consider three real numbers s > d/ , t ≥ , M > and let f be aholomorphic function in the ball { z ∈ C : | z | < M } . There exists ε > , C > such that forall σ > , if u , u ∈ H σ,t ∩ H σ, s satisfy k u j k H σ, s ≤ ε , for j = 1 , then, k f ( u ) − f ( u ) k H σ,t ≤ C (cid:16) k u − u k H σ,t + ( k u k H σ,t + k u k H σ,t ) k u − u k H σ, s (cid:17) . Proof.
Set g ( z ) = f ′ ( z ) − f ′ (0) . Then g is holomorphic in the set { z ∈ C : | z | < M } and g (0) = 0 . Then one can write, f ( u ) − f ( u ) = f ′ (0)( u − u ) + ( u − u ) Z g ( λu + (1 − λ ) u ) dλ. Since k λu + (1 − λ ) u k H σ, s ≤ λ k u k H σ, s + (1 − λ ) k u k H σ, s ≤ ε , we can apply proposition 8.6 with s = t and s = s to g and write, k g ( λu + (1 − λ ) u ) k H σ,t ≤ C ( k u k H σ,t + k u k H σ,t ) , k g ( λu + (1 − λ ) u ) k H σ, s ≤ C. sing Proposition 8.3 and the above inequalities we get, k f ( u ) − f ( u ) k H σ,t ≤ | f ′ (0) |k u − u k H σ,t + C (cid:0) k u − u k H σ,t + ( k u k H σ,t k + u k H σ,t ) k u − u k H σ, s (cid:1) . This completes the proof. (cid:3)
Recall (see Definition 2.3) that for λ ∈ [0 ,
1] and µ ∈ R , we have introduced the spaces, E λ,µ = { u : e λz | D x | u ∈ C ([ − h, , H λh,µ ( R d )) } ,F λ,µ = { u : e λz | D x | u ∈ L ( I h , H λh,µ ( R d )) } , I h = ( − h, , X λ,µ = E λ,µ ∩ F λ,µ + , equipped with their natural norms.We have several estimates using these norms. Lemma 8.8. i) Consider three real numbers s , s , s such that s + s ≥ , s ≤ min { s , s } , s < s + s − d . Then there exists
C > such that, for any λ ∈ [0 , , k u u k F λ, s ≤ C k u k E λ, s k u k F λ, s . ii) For all s > d/ , there exists C > such that, for any λ ∈ [0 , , k u u k F λ, s ≤ C k u k E λ, s k u k F λ, s , k u u k E λ, s ≤ C k u k E λ, s k u k E λ, s . iii) Let s > d/ and t ≥ . There exists C > such that, (8.12) k u u k F λ,t ≤ C k u k E λ, s k u k F λ,t + C k u k F λ,t k u k E λ, s . iv) For any µ ∈ R and any λ ∈ [0 , , there exists a constant C > such that k u k E λ,µ ≤ C k ∂ z u k F λ,µ − + k u k F λ,µ + 12 . Proof. i ) We first use Proposition 8.3 with fixed z and σ = λ ( z + h ), then we we bound the L norm in z of the products by the L ∞ and L norms. The proof of statement ii ) and iii )are similar and iv ) follows directly from Lemma 8.2. (cid:3) Estimates on the coefficients.
In this paragraph, we prove some elementary estimatesfor the the derivatives of the functions ρ and ψ introduced in Section 2.1. Lemma 8.9.
For all t ∈ R there exists C > such that, k ∂ z ρ − k L ∞ ( I h , H λh,t ) + k∇ x ρ k L ∞ ( I h , H λh,t ) + (cid:13)(cid:13) ∇ x,z ρ (cid:13)(cid:13) L ∞ ( I h , H λh,t − ) ≤ C k η k H λh,t +1 , k ∂ z ρ − k L ( I h , H λh,t + 12 ) + k∇ x ρ k L ( I h , H λh,t + 12 ) + (cid:13)(cid:13) ∇ x,z ρ (cid:13)(cid:13) L ( I h , H λh,t − ) ≤ C k η k H λh,t +1 . roof. We have,(8.13) ∂ z ρ ( x, z ) = 1 + 1 h e z | D x | η ( x ) + 1 h ( z + h ) e z | D x | | D x | η ( x ) , ∇ x ρ ( x, z ) = 1 h ( z + h ) e z | D x | ∇ x η ( x ) . Then the first set of estimates follow from the fact that, since z ≤
0, we have e z | ξ | ≤ e z | D x | is bounded from H λh, s ( R d ) to itself for any s . To prove the secondset of estimates, we use the special choice for ρ involving the operator e z | D x | . Notice that,using Fubini, Z I h X Z d | ξ | e z | ξ | (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) dz = X Z d (cid:18)Z I h | ξ | e z | ξ | dz (cid:19) (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) ≤ X Z d (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) . Consequently, by applying this result with f = h D x i s e λh | D x | u and using the Plancherel iden-tity, we get (cid:13)(cid:13)(cid:13) | D x | e z | D x | u (cid:13)(cid:13)(cid:13) L ( I h , H λh, s ) . k u k H λh, s + 12 . Using again (8.13), we complete the proof of the lemma. (cid:3)
Definition 8.10.
For k = 1 , let E k be the set of functions ϕ ( z, η ) defined on, A = I h × { η ∈ H λh, s + : k η k H λh, s ≤ ε ≤ } , such that for every d < t ≤ s − k there exists C > satisfying, k ϕ ( · , η ) k L ∞ ( I h , H λh,t ) ≤ C k η k H λh,t + k , k ϕ ( · , η ) k L ( I h , H λh,t +1 ) ≤ C k η k H λh,t + k + 12 . Lemma 8.11. (1) E k is an algebra and E ⊂ E .(2) If f is a holomorphic function in a ball { Z ∈ C : | Z | < M } such that f (0) = 0 and if ϕ ∈ E k where ε is small enough then f ( ϕ ) ∈ E k . (3) We have, ∇ x ρ ∈ E , ∂ z ρ − ∈ E , ∇ x,z ρ ∈ E , |∇ x ρ | − ∂ z ρ∂ z ρ ∈ E . Proof.
Statement (1) follows from point ii ) in Proposition 8.3 applied with s = t and frompoint iii ) applied s = t and t replaced by t + 1. Statement (2) is a consequence of Proposition8.6. The first three claims in point (3) follow directly from Lemma 8.9. For the last one wenotice that, q := ∂ z ρ − h e z | D x | η + 1 h ( z + h ) e z | D x | | D x | η ∈ E . Then we can write, 1 + |∇ x ρ | ∂ z ρ = 1 − q q + |∇ x ρ | − |∇ x ρ | q q , and hence the desired result follows from the previous statements. (cid:3) We shall need the following extension of Lemma 8.11. We begin by a definition. efinition 8.12. For k = 1 , let F k be the set of functions ϕ ( z, η ) defined on, A = I h × { η ∈ H λh, s + : k η k H λh, s ≤ ε ≤ } , such that ϕ ( z,
0) = 0 and such that, for all d < t ≤ s − k there exists C > such that thefunction Φ( z, η , η ) = ϕ ( z, η ) − ϕ ( z, η ) satisfies the two following estimates k Φ( · , η , η ) k L ( I h , H λh,t +1 ) ≤ C (cid:16) k η − η k H λh,t + k + 12 + X j =1 k η j k H λh,t + k + 12 k η − η k H λh,t + k (cid:17) , k Φ( · , η , η ) k L ∞ ( I h , H λh,t ) ≤ C k η − η k H λh,t + k . Lemma 8.13. (1) For k = 1 , , F k is an algebra and F ⊂ F . (2) As functions of ( z, η ) , the functions, ∇ x ρ, |∇ x ρ | , q = ∂ z ρ − , f ( q ) = q q , |∇ x ρ | ∂ z ρ − , belong to F , and the functions, ∇ x ρ, ∇ x ∂ z ρ, ∂ z ρ belong to F . Proof. (1) It is obvious that F ⊂ F so it is sufficient to prove that if ϕ, θ ∈ F k then, ϕ θ ∈ F k . Taking η = η and η = 0, we first notice that the hypotheses imply that, forevery η ∈ H λh, s + such that k η k H λh, s ≤ ε ≤ , every d < t ≤ s − k we have,(8.14) k θ ( · , η ) k L ( I h , H λh,t +1 ) . k η k H λh,t + k + 12 , k θ ( · , η ) k L ∞ ( I h , H λh,t ) . , together with similar estimates for ϕ. Then we write,( ϕ θ )( z, η ) − ( ϕ θ )( z, η ) = ( ϕ ( z, η ) − ϕ ( z, η )) θ ( z, η ) + ( θ ( z, η ) − θ ( z, η )) ϕ ( z, η )= A + B. The terms A and B are handled in the same way. Using point iii ) in Proposition 8.3 with t = t + 1 , s = t > d , we can write, k A k L ( I h , H λh,t +1 ) . k ϕ ( · , η ) − ϕ ( · , η ) k L ( I h , H λh,t +1 ) k θ ( · , η ) k L ∞ ( I h , H λh,t ) + k ϕ ( · , η ) − ϕ ( · , η ) k L ∞ ( I h , H λh,t ) k θ ( · , η ) k L ( I h , H λh,t +1 ) . Then using the hypotheses and (8.14) we can write, k A k L ( I h , H λh,t +1 ) . k η − η k H λh,t + k + 12 + X j =1 k η j k H λh,t + k + 12 k η − η k H λh,t + k . On the other hand using point ii ) in Proposition 8.3 with s replaced by t we obtain, k A k L ∞ ( I h , H λh,t ) . k ϕ ( z, η ) − ϕ ( z, η ) k L ∞ ( I h , H λh,t ) k θ ( z, η ) k L ∞ ( I h , H λh,t ) . k η − η k H λh,t + k . Which completes the proof of the first statement.
2) For the three first functions this follows from Lemma 8.9 and statement (1). For f ( q ) thisfollows from Corollary 8.7 applied for fixed z ∈ I h . Now one can write,(8.15) 1 + |∇ x ρ | ∂ z ρ = 1 + A, A = − f ( q ) − |∇ x ρ | f ( q ) + |∇ x ρ | ∈ F . The last claim follows from Lemma 8.9. (cid:3)
Lemma 8.14.
For all µ ∈ R , there exists C > such that for all σ ≥ and all ψ such that a ( D x ) ψ ∈ H σ,µ , there holds k∇ x,z ψ k L ( I h , H σ,µ ) ≤ C k a ( D x ) ψ k H σ,µ , k ∂ z ψ k L ( I h , H σ,µ − ) ≤ C k a ( D x ) ψ k H σ,µ , k∇ x,z ψ k L ∞ ( I h , H σ,µ − ) ≤ C k a ( D x ) ψ k H σ,µ . Proof.
To obtain the first two estimates, we use the Fourier transform formula to express ψ in terms of ψ (see (2.12)), and then the wanted estimates follow from arguments similar tothe ones used in the proof of Lemma 8.9. The third estimate is then a consequence of theinterpolation argument given by Lemma 8.2. (cid:3) Some estimates on the remainder.
Consider a function η = η ( x ) and defined ρ = ρ ( x, z ) as in Section 2.1. In this paragraph, we prove several estimates for the action of theoperator R defined by,(8.16) R = a∂ z + b ∆ x + c · ∇ x ∂ z − d∂ z , where ,a = 1 + |∇ x ρ | ∂ z ρ − , b = ∂ z ρ − , c = − ∇ x ρ,d = 1 + |∇ x ρ | ∂ z ρ ∂ z ρ + ∂ z ρ ∆ x ρ − ∇ x ρ · ∇ x ∂ z ρ. Proposition 8.15.
Consider two real numbers s > d/ and λ ∈ [0 , . There exist ε > and a constant C > such that for all η ∈ H λh, s + ( R d ) satisfying k η k H λh, s ≤ ε , the twofollowing properties hold: k Ru k F λ, s − ≤ Cε (cid:0) k∇ x,z u k F λ, s − + (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ, s − (cid:1) , (8.17) k Ru k F λ, s − ≤ Cε (cid:0) k∇ x,z u k F λ, s + (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ, s − (cid:1) + C k η k H λh, s + 12 k ∂ z u k E λ, s − , (8.18) k Ru k F λ, s − ≤ Cε (cid:0) k∇ x,z u k F λ, s + (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ, s − (cid:1) (8.19) + C k η k H λh, s + 12 (cid:0) k ∂ z u k F λ, s − + k ∂ z u k F λ, s − (cid:1) . Proof.
We deduce from Lemma 8.11 that, a, b, c ∈ E , d ∈ E . tep 1. We begin by studying a∂ z u + b ∆ x u + c · ∇ x ∂ z u . We will prove an estimate whichholds for both cases. Namely, we claim that for all s − ≤ ν ≤ s we have,(8.20) (cid:13)(cid:13) a∂ z u + b ∆ x u + c · ∇ x ∂ z u (cid:13)(cid:13) F λ,ν − ≤ Cε (cid:13)(cid:13) ∇ x,z u (cid:13)(cid:13) F s ,ν − . Consider the first term a∂ z u . Since d < ν − ≤ s − ii )in Lemma 8.8 implies that(8.21) (cid:13)(cid:13) a∂ z u (cid:13)(cid:13) F λ,ν − ≤ C k a k E λ, s − (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ,ν − ≤ C k a k L ∞ ( I h , H λh, s − ) (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ,ν − , ≤ Cε (cid:13)(cid:13) ∂ z u (cid:13)(cid:13) F λ,ν − , since a ∈ E . By the same way,(8.22) k b ∆ x u k F λ,ν − ≤ C k b k E λ, s − k ∆ x u k F λ,ν − ≤ C k b k L ∞ ( I h , H λh, s − ) k ∆ x u k F λ,ν − , ≤ Cε k∇ x u k F λ,ν , since b ∈ E and similarly,(8.23) k c · ∇ x ∂ z u k F λ, s − ≤ Cε k ∂ z u k F λ,ν . Step 2.
We now estimate the F λ, s − norm of d∂ z u , using parallel arguments to those usedabove. Firstly, since s − > d/ s − ≥
0, the product rule given by Lemma 8.8, with s = s − , s = s − , s = s − k d∂ z u k F λ, s − ≤ C k d k E λ, s − k ∂ z u k F λ, s − ≤ Cε k ∂ z u k F λ, s − , since d ∈ E . Taking ν = s − Step 3.
We now estimate the F λ, s − norm of d∂ z u . We proceed as in the previous step,but now the term d is estimated in F λ, s − . This is here the only place where we are takingadvantage of the special definition of ρ (the so-called smoothing diffeomorphism). Namely wewrite,(8.25) k d∂ z u k F λ, s − ≤ C k d k F λ, s − k ∂ z u k E λ, s − ≤ C k d k L ( I h , H λ, s − ) k ∂ z u k E λ, s − ≤ C k η k H λh, s + 12 k ∂ z u k E λ, s − . Using (8.21), (8.22), (8.25) with ν = s we complete the proof of (8.18).To obtain (8.19) we use Lemma 8.8 iii ) with s = s − . We obtain, k d∂ z u k F λ, s − ≤ C ( k d k E λ, s − k ∂ z u k F λ, s − + k d k F λ, s − k ∂ z u k E λ, s − ) . Since d ∈ E we have, k d k E λ, s − ≤ C k η k H λh, s ≤ Cε, k d k F λ, s − ≤ C k η k H λh, s + 12 . On the other hand by the interpolation lemma (see Lemma 8.8, iv )) we have, k ∂ z u k E λ, s − ≤ C ( k ∂ z u k F λ, s − + k ∂ z u k F λ, s − ) . Therefore, k d∂ z u k F λ, s − ≤ Cε k ∂ z u k F λ, s − + C k η k H λh, s + 12 ( k ∂ z u k F λ, s − + k ∂ z u k F λ, s − ) . This completes the proof of (8.19). (cid:3) orollary 8.16. Let ψ be the lifting of the function ψ defined in (2.11) . Under the assump-tions of Lemma 8.15, there exists a constant C > such that (8.26) k Rψ k F λ, s − ≤ Cε (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh, s − , and (8.27) k Rψ k F λ, s − ≤ Cε k a ( D x ) ψ k H λh, s + C k η k H λh, s + 12 k a ( D x ) ψ k H λh, s − . Proof.
Notice that (cid:13)(cid:13) ∇ x,z ψ (cid:13)(cid:13) F λ,µ ≤ (cid:13)(cid:13) ∇ x,z ψ (cid:13)(cid:13) L z ( I h , H λh,µ ) for any µ in R . Then Lemma 2.1implies that, for any real number µ , (cid:13)(cid:13) ∇ x,z ψ (cid:13)(cid:13) F λ,µ . (cid:13)(cid:13) a ( D x ) ψ (cid:13)(cid:13) H λh,µ . Then (8.26) and (8.27) follow from Proposition 8.15. (cid:3)
References [1] Thomas Alazard, Pietro Baldi, and Daniel Han-Kwan. Control of water waves.
J. Eur. Math. Soc. (JEMS) ,20(3):657–745, 2018.[2] Thomas Alazard, Nicolas Burq, and Claude Zuily. On the water-wave equations with surface tension.
Duke Math. J. , 158(3):413–499, 2011.[3] Thomas Alazard, Nicolas Burq, and Claude Zuily. The water-wave equations: from Zakharov to Euler.In
Studies in phase space analysis with applications to PDEs , volume 84 of
Progr. Nonlinear DifferentialEquations Appl. , pages 1–20. Birkh¨auser/Springer, New York, 2013.[4] Thomas Alazard, Nicolas Burq, and Claude Zuily. On the Cauchy problem for gravity water waves.
Invent.Math. , 198(1):71–163, 2014.[5] Thomas Alazard and Jean-Marc Delort. Global solutions and asymptotic behavior for two dimensionalgravity water waves.
Ann. Sci. ´Ec. Norm. Sup´er. (4) , 48(5):1149–1238, 2015.[6] Thomas Alazard and Omar Lazar. Paralinearization of the Muskat Equation and Application to theCauchy Problem.
Arch. Ration. Mech. Anal. , 237(2):545–583, 2020.[7] Thomas Alazard and Guy M´etivier. Paralinearization of the Dirichlet to Neumann operator, and regularityof three-dimensional water waves.
Comm. Partial Differential Equations , 34(10-12):1632–1704, 2009.[8] Serge Alinhac and Guy M´etivier. Propagation de l’analyticit´e des solutions de syst`emes hyperboliquesnon-lin´eaires.
Invent. Math. , 75(2):189–204, 1984.[9] Borys Alvarez-Samaniego and David Lannes. Large time existence for 3-D water-waves and asymptotics.
Invent. Math. , 171(3):485–541, 2008.[10] Mohamed-Salah Baouendi and Charles Goulaouic. Remarks on the abstract form of nonlinear Cauchy-Kovalevsky theorems.
Comm. Partial Differential Equations , 2(11):1151–1162, 1977.[11] Jacob Bedrossian, Nader Masmoudi, and Cl´ement Mouhot. Landau damping: paraproducts and Gevreyregularity.
Ann. PDE , 2(1):Art. 4, 71, 2016.[12] Jerry L. Bona, Zoran Gruji´c, and Henrik Kalisch. Global solutions of the derivative Schr¨odinger equationin a class of functions analytic in a strip.
J. Differential Equations , 229(1):186–203, 2006.[13] Jerry L. Bona, David Lannes, and Jean-Claude. Saut. Asymptotic models for internal waves.
J. Math.Pures Appl. (9) , 89(6):538–566, 2008.[14] Angel Castro, Diego C´ordoba, Charles Fefferman, Francisco Gancedo, and Javier G´omez-Serrano. Finitetime singularities for the free boundary incompressible Euler equations.
Ann. of Math. (2) , 178(3):1061–1134, 2013.[15] Walter Craig. An existence theory for water waves and the Boussinesq and Korteweg-de Vries scalinglimits.
Comm. Partial Differential Equations , 10(8):787–1003, 1985.[16] Walter Craig and Catherine Sulem. Numerical simulation of gravity waves.
J. Comput. Phys. , 108(1):73–83, 1993.[17] Francisco Gancedo, Rafael Granero-Belinch´on, and Stefano Scrobogna. Surface tension stabilization ofthe rayleigh-taylor instability for a fluid layer in a porous medium. arXiv: Analysis of PDEs , 2019.
18] Pierre Germain, Nader Masmoudi, and Jalal Shatah. Global solutions for the gravity water waves equationin dimension 3.
Ann. of Math. (2) , 175(2):691–754, 2012.[19] Emmanuel Grenier, Toan Trong Nguyen, and Igor Rodnianski. Landau damping for analytic and Gevreydata. arXiv:2004.05979 , 2020.[20] Nakao Hayashi. Analyticity of solutions of the Korteweg-de Vries equation.
SIAM J. Math. Anal. ,22(6):1738–1743, 1991.[21] A. Alexandrou Himonas and Gerson Petronilho. Evolution of the radius of spatial analyticity for theperiodic BBM equation.
Proc. Amer. Math. Soc. , 148(7):2953–2967, 2020.[22] Lars H¨ormander.
Lectures on nonlinear hyperbolic differential equations , volume 26 of
Math´ematiques &Applications (Berlin) [Mathematics & Applications] . Springer-Verlag, Berlin, 1997.[23] John Hunter, Mihaela Ifrim, and Daniel Tataru. Two dimensional water waves in holomorphic coordinates.
Comm. Math. Phys. , 346(2):483–552, 2016.[24] Tatsuo Iguchi. A long wave approximation for capillary-gravity waves and an effect of the bottom.
Comm.Partial Differential Equations , 32(1-3):37–85, 2007.[25] Tatsuo Iguchi. Isobe-Kakinuma model for water waves as a higher order shallow water approximation.
J.Differential Equations , 265(3):935–962, 2018.[26] Alexandru D. Ionescu and Fabio Pusateri. Global solutions for the gravity water waves system in 2-D.
Invent. Math. , 199(3):653–804, 2015.[27] Tadayoshi Kano. Une th´eorie trois-dimensionnelle des ondes de surface de l’eau et le d´eveloppement deFriedrichs. II.
J. Math. Kyoto Univ. , 26(2):157–175, 1986.[28] Tadayoshi Kano and Takaaki Nishida. Sur les ondes de surface de l’eau avec une justification math´ematiquedes ´equations des ondes en eau peu profonde.
J. Math. Kyoto Univ. , 19(2):335–370, 1979.[29] Tadayoshi Kano and Takaaki Nishida. Water waves and Friedrichs expansion. In
Recent topics in non-linear PDE (Hiroshima, 1983) , volume 98 of
North-Holland Math. Stud. , pages 39–57. North-Holland,Amsterdam, 1984.[30] Tadayoshi Kano and Takaaki Nishida. A mathematical justification for Korteweg-de Vries equation andBoussinesq equation of water surface waves.
Osaka J. Math. , 23(2):389–413, 1986.[31] Tosio Kato and Ky¯uya Masuda. Nonlinear evolution equations and analyticity. I.
Ann. Inst. H. Poincar´eAnal. Non Lin´eaire , 3(6):455–467, 1986.[32] Igor Kukavica and Vlad Vicol. On the radius of analyticity of solutions to the three-dimensional Eulerequations.
Proc. Amer. Math. Soc. , 137(2):669–677, 2009.[33] Sergei Kuksin and Nikolai Nadirashvili. Analyticity of solutions for quasilinear wave equations and otherquasilinear systems.
Proc. Roy. Soc. Edinburgh Sect. A , 144(6):1155–1169, 2014.[34] David Lannes. Well-posedness of the water-waves equations.
J. Amer. Math. Soc. , 18(3):605–654 (elec-tronic), 2005.[35] David Lannes.
Water waves: mathematical analysis and asymptotics , volume 188 of
Mathematical Surveysand Monographs . American Mathematical Society, Providence, RI, 2013.[36] Tokio Matsuyama and Michael Ruzhansky. On the Gevrey well-posedness of the Kirchhoff equation.
J.Anal. Math. , 137(1):449–468, 2019.[37] Cl´ement Mouhot and C´edric Villani. On Landau damping.
Acta Math. , 207(1):29–201, 2011.[38] Louis Nirenberg. An abstract form of the nonlinear Cauchy-Kowalewski theorem.
J. Differential Geometry ,6:561–576, 1972.[39] Takaaki Nishida. A note on a theorem of Nirenberg.
J. Differential Geometry , 12(4):629–633 (1978), 1977.[40] Lev Vasil’evich Ovsjannikov. A nonlinear Cauchy problem in a scale of Banach spaces.
Dokl. Akad. NaukSSSR , 200:789–792, 1971.[41] Lev Vasil’evich Ovsjannikov. To the shallow water theory foundation.
Arch. Mech. (Arch. Mech. Stos.) ,26:407–422, 1974.[42] Lev Vasil’evich Ovsjannikov. Cauchy problem in a scale of Banach spaces and its application to the shallowwater theory justification. In
Applications of methods of functional analysis to problems in mechanics(Joint Sympos., IUTAM/IMU, Marseille, 1975) , pages 426–437. Lecture Notes in Math., 503. 1976.[43] Olivier Pierre. Analytic current-vortex sheets in incompressible magnetohydrodynamics.
J. Math. FluidMech. , 20(3):1269–1315, 2018.[44] Jean-Claude Saut.
Asymptotic models for surface and internal waves . Publica¸c˜oes Matem´aticas do IMPA.[IMPA Mathematical Publications]. Instituto Nacional de Matem´atica Pura e Aplicada (IMPA), Rio deJaneiro, 2013. 29o Col´oquio Brasileiro de Matem´atica. [29th Brazilian Mathematics Colloquium].
45] Guido Schneider and C. Eugene Wayne. On the validity of 2D-surface water wave models.
GAMM Mitt.Ges. Angew. Math. Mech. , 25(1-2):127–151, 2002.[46] Sigmund Selberg and Daniel Oliveira da Silva. Lower bounds on the radius of spatial analyticity for theKdV equation.
Ann. Henri Poincar´e , 18(3):1009–1023, 2017.[47] Catherine Sulem and Pierre-Louis Sulem. Finite time analyticity for the two- and three-dimensionalRayleigh-Taylor instability.
Trans. Amer. Math. Soc. , 287(1):127–160, 1985.[48] Achenef Tesfahun. Asymptotic lower bound for the radius of spatial analyticity to solutions of KdVequation.
Commun. Contemp. Math. , 21(8):1850061, 33, 2019.[49] Xuecheng Wang. Global solution for the 3-D gravity water waves system above a flat bottom.
Adv. Math. ,346:805–886, 2019.[50] Sijue Wu. Almost global wellposedness of the 2-D full water wave problem.
Invent. Math. , 177(1):45–135,2009.[51] Sijue Wu. Global wellposedness of the 3-D full water wave problem.
Invent. Math. , 184(1):125–220, 2011.[52] Vladimir E. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid.
Journalof Applied Mechanics and Technical Physics , 9(2):190–194, 1968.[53] Fan Zheng. Long-term regularity of 3-D gravity water waves. arXiv:1910.01912 , 2019.[54] Hui Zhu. Control of three dimensional water waves.
Arch. Ration. Mech. Anal. , 236(2):893–966, 2020., 236(2):893–966, 2020.