Full justification of a new Green-Naghdi system for internal waves propagation over large topography variation
FFull justification of a new Green-Naghdi system for internalwaves propagation over large topography variation
Ralph Lteif ∗ February 25, 2021
Abstract
We consider here asymptotic models describing the evolution of internal waves propagatingbetween a flat rigid-lid and a variable topography. In this paper, we derive and fully justify (in thesense of consistency, well-posedness and convergence) a new Green-Naghdi model in the Camassa-Holm regime taking into account large amplitude topography variations, thus relaxing the smallnessassumption on the topographic variation parameter used in [Communications on Pure & AppliedAnalysis, 2015, 14 (6): 2203-2230] and hence improving the result of the aforementioned paper.
Ocean water is not uniform when it comes to mass density. In fact, the temperature and salinity of waterin the ocean vary according to depth which create a stratification effect dividing the water into layerswith different densities. The disturbance of these layers by tidal flows over variable topography generatesinternal waves. The absorbed solar radiation make the upper surface water warmer with a lower densitylying above a colder denser water. Internal waves play an important role in underwater biological life andnavigation, thus understanding their behavior is very essential. In this paper, we are interested in theone-dimensional motion of internal waves located between two layers of fluids with different densities.Simplifying assumptions on the nature of the fluids are commonly used in oceanography. Namely, thefluids are supposed to be homogeneous, immiscible, inviscid and affected only by gravitational force.Moreover, the fluids are assumed to be irrotational and incompressible.The mathematical aspects of the internal waves have been the subject of many studies in the liter-ature. The evolution equations describing the two-layers flow are easy to derive. These equations arecommonly called the “ full Euler system ”. In this paper, we omit the detailed derivation of this system forthe sake of readability, instead, we briefly recall the system in section 2. For more details, the interestedreader can see for instance [2, 4, 9]. Solving the “ full Euler system ” mathematically is very difficult dueto the free boundary problem i.e the domain is moving with time. Indeed, the interface deformationboundary function is one of the unknowns. To overcome this problem, many researchers searched forapproximate solutions of the exact system. To this end, the derivation of simpler asymptotic modelshad attract a lot of attention. These approximate models are derived in much simpler settings wheredimensionless variables and unknowns are introduced, allowing a fair description of the exact behaviorof the full system in particular physical regimes. In this paper, we derive an asymptotic model for thetwo-layers flow over strong variations in bottom topography using an additional smallness assumptionon the interface deformation. More precisely, we restrict ourselves to the Camassa-Holm regime wherewe consider medium amplitude deformations at the interface level.The two-layers flow has been widely studied in the literature setting an important theoretical frame-work. Many asymptotic models describing the evolution of the interface between two layers of fluidbounded from above by a flat rigid lid and from below by a flat topography have been previously derived
Mathematics subject classification.
Key words and phrases.
Internal waves, asymptotic model, full justification, variable topography, medium amplitude.. ∗ Applied and Computational Mathematics, Lebanese American University. ( [email protected] ) a r X i v : . [ m a t h . A P ] F e b nd studied, see for instance [18, 17, 5, 6, 4, 12, 11] and references therein. Two-layers flow over variabletopography has been also investigated. Many two-layers systems describing the propagation of internalwaves over variable topography have been derived and studied in some significant works [19, 2, 8, 3, 10].These models were proved to be consistent with the exact system, however they are not supported with arigorous justification ( i.e well posedness, stability, convergence). More recently, a full justification resultfor a new Green-Naghdi type model in the Camassa-Holm regime (medium amplitude deformation atthe interface) dealing with medium amplitude topography variations has been obtained in [16].In this paper, we derive and fully justify a new Green-Naghdi model in the same scaling of theCamassa-Holm regime but taking into account large amplitude topography variations, thus improvingthe result obtained in [16]. In fact, large amplitude topography variations is a quite more reasonableassumption in oceanography, yet arise serious difficulties. Indeed, relaxing any smallness assumption onthe parameter β prompt some terms that cannot be controlled by the energy norm associated to ourmodel and thus do not always allow its full justification. Consequently, we establish a specially designedmodel allowing to deal with these terms and possessing a satisfying hyperbolic symmetrizabe quasilinearstructure. This structure is suitable to the application of the hyperbolic systems classical theory andin particular studying energy estimates and hence allowing its full justification. The obtained modelis valid under some key restrictions on the bottom deformation, see (2.8) and remark 3.14. Moreover,one can easily deduce a direct result of full justification of the model in the one-layer case with unevenbottoms and a small amount of surface tension.The paper is organized as follows: the full Euler system is briefly introduced in section 2. In section 3,we precisely derive the new asymptotic model starting from the original Green-Naghdi model. In section4, we state some preliminary results on the properties of the specially designed symmetric differentialoperator. In section 5, we present the linear analysis of the asymptotic model. Section 6 contains themain components of the full justification of the asymptotic model. Notations.
In the following, C denotes any nonnegative constant whose exact expression is of noimportance.The notation a (cid:46) b means that a ≤ C b and we write A = O ( B ) if A ≤ C B .We denote by C ( λ , λ , . . . ) a nonnegative constant depending on the parameters λ , λ ,. . . and whosedependence on the λ j is always assumed to be nondecreasing.We use the condensed notation A s = B s + (cid:104) C s (cid:105) s>s , to express that A s = B s if s ≤ s and A s = B s + C s if s > s .Let p be any constant with 1 ≤ p < ∞ and denote L p = L p ( R ) the space of all Lebesgue-measurablefunctions f with the standard norm | f | L p = (cid:0) (cid:90) R | f ( x ) | p dx (cid:1) /p < ∞ . The real inner product of any functions f and f in the Hilbert space L ( R ) is denoted by( f , f ) = (cid:90) R f ( x ) f ( x ) dx. The space L ∞ = L ∞ ( R ) consists of all essentially bounded, Lebesgue-measurable functions f withthe norm | f | L ∞ = ess sup | f ( x ) | < ∞ . Let k ∈ N , we denote by W k, ∞ = W k, ∞ ( R ) = { f ∈ L ∞ , | f | W k, ∞ < ∞} , where | f | W k, ∞ = (cid:88) α ∈ N ,α ≤ k | ∂ αx f | L ∞ .For any real constant s ≥ H s = H s ( R ) denotes the Sobolev space of all tempered distributions f with the norm | f | H s = | Λ s f | L < ∞ , where Λ is the pseudo-differential operator Λ = (1 − ∂ x ) / .For a given µ >
0, we denote by H s +1 µ ( R ) the space H s +1 ( R ) endowed with the norm (cid:12)(cid:12) · (cid:12)(cid:12) H s +1 µ ≡ (cid:12)(cid:12) · (cid:12)(cid:12) H s + µ (cid:12)(cid:12) · (cid:12)(cid:12) H s +1 . For any function u = u ( t, x ) and v ( t, x ) defined on [0 , T ) × R with T >
0, we denote the inner product,the L p -norm and especially the L -norm, as well as the Sobolev norm, with respect to the spatial variable2 , by ( u, v ) = ( u ( t, · ) , v ( t, · )), | u | L p = | u ( t, · ) | L p , | u | L = | u ( t, · ) | L , and | u | H s = | u ( t, · ) | H s , respectively.We denote L ∞ ([0 , T ); H s ( R )) the space of functions such that u ( t, · ) is controlled in H s , uniformly for t ∈ [0 , T ): (cid:13)(cid:13) u (cid:13)(cid:13) L ∞ ([0 ,T ); H s ( R )) = ess sup t ∈ [0 ,T ) | u ( t, · ) | H s < ∞ . Finally, C k ( R ) denote the space of k -times continuously differentiable functions.For any closed operator T defined on a Banach space X of functions, the commutator [ T, f ] is definedby [
T, f ] g = T ( f g ) − f T ( g ) with f , g and f g belonging to the domain of T . The same notation is usedfor f an operator mapping the domain of T into itself. In what follows, we derive breifly the governing equations of the two-layers flow commonly known as the full Euler system . One can see [2, 4, 9, 10] for more details. The main interest of this system is in itsreduced form that is of two evolution equations coupling two unknowns so-called Zakharov’s canonicalvariables.We restrict our study to horizontally one-dimensional case. Let ζ ( t, x ) and b ( x ) be two functions thatrepresent respectively the interface and bottom variation from their rest place (resp. ( x, , ( x, − d )).Moreover, we assume a flat and rigid top surface. The domain of study of the upper fluid is denoted byΩ t while the one of the lower fluid is denoted by Ω t , see figure 1.Figure 1: Domain of studyWe assume that there exists h > h , h ≥ h >
0, where h and h stand for the heightsof the top and low fluids respectively. That is the two domains are supposed to stay connected.Now let us specify the assumptions on the nature and domain of the fluids. This type of reasonablehypothesis is commonly used in oceanography when determining the governing equations of two-layersflow.First, we consider both fluids are homogeneous, therefore the mass density of the top and low fluidsdenoted by ρ and ρ respectively are constant. Each layer of fluid is incompressible so the correspondingvelocity field has a zero divergence. Assuming irrotational flows, there exists velocity potentials denotedby φ i ( i = 1 ,
2) that satisfy the Laplace equation. Assuming ideal fluids with no viscosity, one obtainstwo Bernoulli equations. The surface, interface and bottom are all assumed to be bounding surfaces,that is to say no particle of fluid can cross the surface, interface or bottom. One may close the set ofequations using one last condition on the pressure, more precisely we assume the continuity of the stress3ensor at the interface. At this stage, one obtains: ∆ x,z φ i = 0 in Ω ti , i = 1 , ,ρ i ∂ t φ i + ρ i |∇ x,z φ i | = − P i − ρ i gz in Ω ti , i = 1 , ,∂ n φ = n t . ∇ x,z φ = ∂ z φ = 0 on Γ t ,∂ t ζ = (cid:112) | ∂ x ζ | ∂ n φ = (cid:112) | ∂ x ζ | ∂ n φ on Γ ,∂ n φ = n b . ∇ x,z φ = 1 (cid:112) | ∂ x b | ( − ∂ x b.∂ x φ + ∂ z φ ) = 0 = 0 on Γ b , (cid:98) P ( t, x ) (cid:99) = σ∂ x (cid:16) (cid:112) | ∂ x ζ | ∂ x ζ (cid:17) on Γ , (2.1)where Γ t ≡ { ( x, z ) , z = d } , Γ ≡ { ( x, z ) , z = ζ ( t, x ) } , Γ b ≡ { ( x, z ) , z = − d + b ( x ) } . (cid:98) P ( t, x ) (cid:99) def = lim χ → (cid:16) P ( t, x, ζ ( t, x ) + χ ) − P ( t, x, ζ ( t, x ) − χ ) (cid:17) and ∂ n = n. ∇ x,z is the upward normalderivative in the direction of the vector n under consideration. We denote by n t = (0 , T , n ζ =1 (cid:112) | ∂ x ζ | ( − ∂ x ζ, T and n b = 1 (cid:112) | ∂ x b | ( − ∂ x b, T the unit outward normal vectors at the toprigid surface, interface and bottom respectively.Here σ denotes the surface (or interfacial) tension coefficient. Studying both theoretical and nu-merical aspects of system (2.1) remains difficult as the domain is one of the unknowns. At this point,extra assumptions are made on some parameters with no dimensions in order to obtain reduced asymp-totic models suitable for numerical implementation. Thus, we introduce the following dimensionlessparameters: γ = ρ ρ , (cid:15) ≡ ad , β ≡ a b d , µ ≡ d λ , δ ≡ d d , bo = g ( ρ − ρ ) d σ , where we denote by a (resp. a b ) the maximal elevation of the internal wave (resp. bottom topography)and λ the horizontal length scale of the wave at the interface.Written in its dimensionless form, system (2.1) can be reduced to two equations with two unknowns( ζ, ψ ≡ φ ( t, x, ζ ( t, x )), see [20, 7]. ∂ t ζ − µ G µ ψ = 0 ,∂ t (cid:16) H µ,δ ψ − γ∂ x ψ (cid:17) + ( γ + δ ) ∂ x ζ + (cid:15) ∂ x (cid:16) | H µ,δ ψ | − γ | ∂ x ψ | (cid:17) = µ(cid:15)∂ x N µ,δ − µ γ + δ bo ∂ x (cid:0) k ( (cid:15) √ µζ ) (cid:1) (cid:15) √ µ , (2.2)where we denote: N µ,δ ≡ (cid:0) µ G µ ψ + (cid:15) ( ∂ x ζ ) H µ,δ ψ (cid:1) − γ (cid:0) µ G µ ψ + (cid:15) ( ∂ x ζ )( ∂ x ψ ) (cid:1) µ | (cid:15)∂ x ζ | ) . The system (2.2) is the so-called full Euler system . We define below the two Dirichlet-Neumann operators G µ ψ and H µ,δ ψ . Definition 2.3
Let ζ, b ∈ H t +1 ( R ) , t > / , such that there exists h > with h ≡ − (cid:15)ζ ≥ h > and h ≡ δ + (cid:15)ζ − βb ≥ h > , and let ψ ∈ L ( R ) , ∂ x ψ ∈ H / ( R ) . Then we define: G µ ψ ≡ G µ [ (cid:15)ζ ] ψ ≡ (cid:112) µ | (cid:15)∂ x ζ | (cid:0) ∂ n φ (cid:1) | z = (cid:15)ζ = − µ(cid:15) ( ∂ x ζ )( ∂ x φ ) | z = (cid:15)ζ + ( ∂ z φ ) | z = (cid:15)ζ ,H µ,δ ψ ≡ H µ,δ [ (cid:15)ζ ] ψ ≡ ∂ x (cid:0) φ | z = (cid:15)ζ (cid:1) = ( ∂ x φ ) | z = (cid:15)ζ + (cid:15) ( ∂ x ζ )( ∂ z φ ) | z = (cid:15)ζ , here, φ and φ are uniquely deduced from ( ζ, ψ ) as solutions of the following Laplace’s problems: (cid:0) µ∂ x + ∂ z (cid:1) φ = 0 in Ω ≡ { ( x, z ) ∈ R , (cid:15)ζ ( x ) < z < } ,∂ z φ = 0 on Γ t ≡ { ( x, z ) ∈ R , z = 1 } ,φ = ψ on Γ ≡ { ( x, z ) ∈ R , z = (cid:15)ζ } , (2.4) (cid:0) µ∂ x + ∂ z (cid:1) φ = 0 in Ω ≡ { ( x, z ) ∈ R , − δ + βb ( x ) < z < (cid:15)ζ } ,∂ n φ = ∂ n φ on Γ ,∂ z φ = 0 on Γ b ≡ { ( x, z ) ∈ R , z = − δ + βb ( x ) } . (2.5) Definition 2.6 (Regimes)
Firstly, the commonly known shallow water regime is defined. We considerthe two-layers to be of similar depths: P SW ≡ (cid:110) ( µ, (cid:15), δ, γ, β, bo) : 0 < µ ≤ µ max , ≤ (cid:15) ≤ , δ ∈ ( δ min , δ max ) , ≤ γ < , ≤ β ≤ β max , bo min ≤ bo ≤ ∞ (cid:111) , (2.7) with given ≤ µ max , δ − , δ max , bo − , β max < ∞ .Moreover, the model (3.19) is valid under the following extra restrictions: P CH ≡ (cid:26) ( µ, (cid:15), δ, γ, β, bo) ∈ P SW : (cid:15) ≤ M √ µ, β ≤ β max and ν (b) ≡ γδ δ ( γ + δ − γδβ b) − ≥ ν (cid:27) , (2.8) with given ≤ M, ν − < ∞ .We proceed by denoting without difficulty M SW ≡ max (cid:8) µ max , δ − , δ max , bo − , β max (cid:9) , M CH ≡ max (cid:8) M SW , M , ν − (cid:9) . In this section, we are interested in the construction and full justification of a new asymptotic modeldescribing the two-layers flow in the Camassa-Holm regime ( µ (cid:28) , (cid:15) ≤ M √ µ with M >
0) takinginto account large amplitude topography variations. To this end, we start our analysis by recallingthe commonly know Green-Naghdi model. The latter model is obtained after plugging the asymptoticexpansions of the Dirichlet-Neumann operators given in [8, 10] into the system (2.2) and after straight-forward computations while neglecting all terms of order µ . At this point, the commonly known “shearmean velocity” variable v is introduced coupling the upper and lower layer vertically integrated horizontalvelocities u and u , see [10]: v ≡ u − γu , (3.1) u ( t, x ) = 1 h ( t, x ) (cid:90) (cid:15)ζ ( t,x ) ∂ x φ ( t, x, z ) dz and u ( t, x ) = 1 h ( t, x ) (cid:90) (cid:15)ζ ( t,x ) − δ + βb ( x ) ∂ x φ ( t, x, z ) dz, where φ and φ are the solutions to the Laplace’s problems (2.4)-(2.5).In this paper, for the sake of simplicity, we do not give the asymptotic expansions of the Dirichlet-Neumann operators nor the detailed derivation of the Green-Naghdi model. Instead, we recall below5he Green-Naghdi system and refer to [8, 10] for more details: ∂ t ζ + ∂ x (cid:16) h h h + γh v (cid:17) = 0 ,∂ t (cid:16) v + µ Q [ h , h ] v (cid:17) + ( γ + δ ) ∂ x ζ + (cid:15) ∂ x (cid:16) h − γh ( h + γh ) v (cid:17) = µ(cid:15)∂ x (cid:0) R [ h , h ] v (cid:1) + µ γ + δ bo ∂ x ζ, (3.2)where we denote h = 1 − (cid:15)ζ and h = 1 δ + (cid:15)ζ − βb , as well as Q [ h , h ] v = T [ h , βb ] (cid:16) h vh + γh (cid:17) − γ T [ h , (cid:16) − h vh + γh (cid:17) , = − h ∂ x (cid:16) h ∂ x (cid:0) h vh + γh (cid:1)(cid:17) + 12 h β (cid:104) ∂ x (cid:16) h ( ∂ x b ) h vh + γh (cid:17) − h ( ∂ x b ) ∂ x (cid:0) h vh + γh (cid:1)(cid:105) + β ( ∂ x b ) (cid:0) h vh + γh (cid:1) − γ (cid:104) h ∂ x (cid:16) h ∂ x (cid:0) h vh + γh (cid:1)(cid:17)(cid:105) . R [ h , h ] v = 12 (cid:16) − h ∂ x ( h vh + γh ) + β ( ∂ x b )( h vh + γh ) (cid:17) − γ (cid:16) h ∂ x ( − h vh + γh ) (cid:17) − ( h vh + γh ) T [ h , βb ] (cid:16) h vh + γh (cid:17) + γ ( − h vh + γh ) T [ h , (cid:16) − h vh + γh (cid:17) , with, T [ h, b ] V ≡ − h ∂ x ( h ∂ x V ) + 12 h [ ∂ x ( h ( ∂ x b ) V ) − h ( ∂ x b )( ∂ x V )] + ( ∂ x b ) V. The Green-Naghdi model (3.2) has been proved to be consistent with the full Euler system (2.2) in [10].However, it was not supported with a rigorous justification result (i.e well posedness, stability, con-vergence). In this paper, starting from the Green-Naghdi model (3.2), we construct a new equivalentmodel (having the same precision) using an extra assumption on the interface deformations (that is weassume (cid:15) = O ( √ µ )) taking into account large amplitude bottom variations. The resulting model enjoysa hyperbolic quasilinear structure allowing its full justification. The construction of the aforementionedmodel is detailed in the section below. In this section, we construct a new Green-Naghdi model in the Camassa-Holm regime ( (cid:15) = O ( √ µ ))but taking into account large variations in topography, thus relaxing any smallness assumption on theamplitude topographic variations parameter β . This is more reasonable in ocean beds. In fact, wesuppose that there exists β max < ∞ such that β = O (1) with β ∈ [0 , β max ] . Using the following asymptotic expansion: 11 − X = 1 + X + O ( X ) with X (cid:28)
1, one gets: h h + γh = δγ + δ − γδβb + O ( (cid:15) ) ,h h + γh = 1 − δβbγ + δ − γδβb + O ( (cid:15) ) ,h h + γh = δγ + δ − γδβb (cid:16) − (cid:15)ζ + (cid:15)ζδ (1 − γ ) γ + δ − γδβb + O ( (cid:15) ) (cid:17) , h + γh = δγ + δ − γδβb (cid:16) δ − + (cid:15)ζ − βb + (1 − δβb ) (cid:15)ζ (1 − γ ) γ + δ − γδβb + O ( (cid:15) ) (cid:17) . After replacing these functions by their corresponding approximations in Q [ h , h ] v and R [ h , h ] v , onegets the following: Q [ h , h ] v = − λ ( b ) ∂ x v + (cid:15) (cid:18) θ ( b ) v∂ x ζ + (cid:0) θ ( b ) + ( γ − g ( b ) (cid:1) ∂ x ζ∂ x v + (cid:0) θ ( b ) + 23 ( γ − g ( b ) (cid:1) ζ∂ x v (cid:19) + β (cid:18) α ( b ) v∂ x b + 2 α ( b ) ∂ x b∂ x v + (cid:0) γ f ( b ) + 23 δ − f ( b ) (cid:1) b∂ x v (cid:19) + (cid:15)β (cid:16)(cid:0) θ ( b ) − α ( b ) (cid:1) ζv∂ x b + (cid:0) θ ( b ) − α ( b ) (cid:1) ζ∂ x b∂ x v (cid:17) + (cid:15)β (cid:18)(cid:0) θ ( b ) − α ( b ) + 13 ( δ − − βb ) f (cid:48) ( b ) − γ g (cid:48) ( b ) (cid:1) ∂ x ζ∂ x bv (cid:19) + β (cid:18) η ( b )( ∂ x b ) v + 2 γ f (cid:48) ( b ) b∂ x b∂ x v + γ f (cid:48) ( b ) b∂ x bv − f ( b ) b ∂ x v (cid:19) + (cid:15)β (cid:0) η ( b )( ∂ x b ) ζv (cid:1) + β (cid:16) γ f (cid:48)(cid:48) ( b )( ∂ x b ) bv (cid:17) + O ( (cid:15) ) , (3.3) R [ h , h ] v = (1 − γ ) g ( b ) (cid:18)
12 ( ∂ x v ) + 13 v∂ x v (cid:19) + s ( b ) v + t ( b ) v∂ x v + O ( (cid:15) ) , (3.4)with λ ( b ) = 1 + γδ δ ( γ + δ − γδβb ) , f ( b ) = δ ( γ + δ − γδβb ) , g ( b ) = 1 − δβb ( γ + δ − γδβb ) ,θ ( b ) = 13 ( δ − − βb ) f ( b ) −
13 ( δ − − βb ) (1 − γ ) f ( b ) − γ f ( b ) − γ f ( b ) g ( b )(1 − γ ) ,α ( b ) = −
13 ( δ − − βb ) f (cid:48) ( b ) + ( δ − − βb )2 f ( b ) − γ δ − f (cid:48) ( b ) + γ f ( b ) ,θ ( b ) = 13 ( δ − − βb ) f (cid:48) ( b ) −
13 ( δ − − βb ) (1 − γ )2 f ( b ) f (cid:48) ( b ) − ( δ − − βb )2 f ( b )+ ( δ − − βb )2 )(1 − γ ) f ( b ) + f ( b )2 − γ f (cid:48) ( b ) + 2 γ g (cid:48) ( b ) −
23 ( δ − − βb ) f (cid:48) ( b ) ,α ( b ) = γ − γ ) (cid:16) g ( b ) f (cid:48) ( b ) + g (cid:48) ( b ) f ( b ) (cid:17) ,η ( b ) = −
13 ( δ − − βb ) f (cid:48)(cid:48) ( b ) + ( δ − − βb ) f (cid:48) ( b ) − γ δ − f (cid:48)(cid:48) ( b ) + 2 γ f (cid:48) ( b ) ,η ( b ) = 13 ( δ − − βb ) f (cid:48)(cid:48) ( b ) − ( δ − − βb ) f (cid:48) ( b ) + 2( δ − − βb )(1 − γ ) f (cid:48) ( b ) f ( b ) −
13 ( δ − − βb ) (1 − γ )4( f (cid:48) ( b )) −
13 ( δ − − βb ) (1 − γ )2 f ( b ) f (cid:48)(cid:48) ( b ) + ( δ − − βb )3 f (cid:48)(cid:48) ( b )+ 2 f (cid:48) ( b ) − ( δ − − βb ) f (cid:48)(cid:48) ( b ) − γ f (cid:48)(cid:48) ( b ) − γ (1 − γ )3 f (cid:48)(cid:48) ( b ) g ( b ) − γ (1 − γ )3 2 f (cid:48) ( b ) g (cid:48) ( b ) − γ (1 − γ )3 f ( b ) g (cid:48)(cid:48) ( b ) + 2 γ g (cid:48)(cid:48) ( b ) − f (cid:48) ( b ) , ( b ) = 12 ( δ − − βb ) ( ∂ x f ( b )) − δ − − βb ) ∂ x ( f ( b )) f ( b ) β∂ x b + ( δ − − βb ) f ( b ) ∂ x ( f ( b ))+ β ∂ x b ) f ( b ) − β δ − − βb ) ∂ x bf ( b ) − γ g ( b ) ∂ x ( g ( b )) − γ ∂ x ( g ( b ))) ,t ( b ) = 53 ( δ − − βb ) f ( b ) ∂ x ( f ( b )) − δ − − βb ) f ( b ) β∂ x b − γ g ( b ) ∂ x ( g ( b )) . The key ingredient for constructing a simplified asymptotic Green-Naghdi model is plugging the ex-pansions of Q [ h , h ] v and R [ h , h ] v into system (3.2). In fact, all terms of order O ( µ(cid:15) ) are nowneglected. However, new topographic terms appear in the expansion of Q [ h , h ] v due to the largetopography variation assumption. These terms are actually accompanied with derivative terms on ζ that cannot be controlled by the energy norm we have in mind. To this end, we will construct in whatfollows an equivalent model that can deal with these terms.Firstly, we introduce a second order symmetric differential operator, T [ (cid:15)ζ, βb ] V = q ( (cid:15)ζ, βb ) V + µ(cid:15)βκ ∂ x ζ∂ x bV − µ∂ x (cid:16) νq ( (cid:15)ζ, βb ) ∂ x V (cid:17) , (3.5)where q ( X, Y ) ≡ κ X + ω Y and q ( X, Y ) ≡ κ X + ω Y + η XY and ν, κ , κ , κ , ω , ω , η are functions of b to be determined in an appropriate way treating all terms that cannot be controlledby the intended energy norm. For the sake of readability, we omit here and for the rest of the paper thedependence of these functions on b , thus one can write: T [ (cid:15)ζ, βb ] ∂ t v − q ( (cid:15)ζ, βb ) ∂ t (cid:16) v + µ Q [ h , h ] v (cid:17) + q ( (cid:15)ζ, βb ) µ γ + δ bo ∂ x ζ = µ(cid:15)βκ ∂ x ζ∂ x b∂ t v − µ∂ x ν∂ x ∂ t v − µν∂ x ∂ t v − µ∂ x ( ν(cid:15)κ ζ∂ x ∂ t v ) − µ∂ x ( νβω b∂ x ∂ t v ) − µ∂ x ( ν(cid:15)βη ζb∂ x ∂ t v ) − µ∂ t Q [ h , h ] v − µ(cid:15)κ ζ∂ t Q [ h , h ] v − µβω b∂ t Q [ h , h ] v + µ γ + δ bo ∂ x ζ + µ(cid:15)κ ζ γ + δ bo ∂ x ζ + µβω b γ + δ bo ∂ x ζ. (3.6)A proper choice of ν ought to treat the first order ( µ∂ x ∂ t v ) terms by canceling them out. In fact, thesecond equation of system (3.2) gives ∂ t v = − ( γ + δ ) ∂ x ζ − (cid:15) ∂ x (cid:16) ( f ( b ) − γg ( b ) ) | v | (cid:17) + O ( (cid:15) , µ ) . Equivalently, one has µ γ + δ bo ∂ x ζ = − µ bo ∂ x ∂ t v − µ(cid:15) ∂ x (cid:16) ( f ( b ) − γg ( b ) ) | v | (cid:17) + O ( µ(cid:15) , µ ) . Lets consider now equation (3.6). Substituting the term µ γ + δ bo ∂ x ζ of the this same equation by itsabove expression and applying the time partial derivative to the expansion of Q [ h , h ] v in (3.3), thusone defines ν = λ ( b ) − . (3.7)Moreover, all terms including ζ and its derivatives can be canceled with an appropriate choice of thefunctions κ , κ , κ , ω , ω and η . Indeed, using (3.2) two approximations hold: ∂ t v = − ( γ + δ ) ∂ x ζ + O ( (cid:15), µ ) and ∂ t ζ = − ∂ x ( g ( b ) v ) + O ( (cid:15), µ ). Thus, one can set κ = (1 + βω b ) (cid:0) θ − α + 13 ( δ − − βb ) f (cid:48) ( b ) − γ g (cid:48) ( b ) (cid:1) , (3.8)so that ( ∂ x ζ ) ∂ x b terms are withdrawn.A suitable choice of κ can cancel the ζ∂ x ζ terms, κ = (cid:0) − θ − ( γ − g ( b ) (cid:1)(cid:0) βω b (cid:1) ν − βb (cid:0) γ f ( b ) + δ − f ( b ) (cid:1) + β b f ( b ) . (3.9)8 and η are determined as a solution of the system below so that ∂ x ζ∂ x ζ and ζ∂ x b∂ x ζ terms arewithdrawn: νκ + νβbη = − (cid:0) θ + ( γ − g ( b ) (cid:1) (1 + βω b ) , (3.10) ν (cid:48) κ + νκ (cid:48) + ( ν + ν (cid:48) βb ) η + νβbη (cid:48) = ( − θ + α )(1 + βω b ) − βbκ γ f (cid:48) ( b ) − κ α. (3.11)Treating the b∂ x ζ terms is done through determining the function ω as follows, ω = νω + γ f ( b ) + δ − f ( b ) − βb f ( b ) ν − βb (cid:0) γ f ( b ) + δ − f ( b ) (cid:1) + β b f ( b ) . (3.12)Solving the following first order linear differential equation leads to the determination of the function ω that can cancel the ∂ x b∂ x ζ terms.( ν + βbν (cid:48) ) ω + βbνω (cid:48) = − ν (cid:48) − βbω α − β b ω γ f (cid:48) ( b ) − α − γ f (cid:48) ( b ) βb. (3.13) Remark 3.14
A first order linear differential equation has the following form: y (cid:48) + n ( x ) y = p ( x ) , where y = y ( x ) and y (cid:48) = dydx , where n ( x ) and p ( x ) must be continuous functions. The general solution is given by: y = Ce − F ( x ) + e − F ( x ) (cid:90) e F ( x ) p ( x ) dx, where F ( x ) = (cid:90) n ( x ) dx, and C is an arbitrary constant . Using (3.12) , the differential equation (3.13) can be easily rewritten as: ω (cid:48) + (cid:32) βb + ν (cid:48) ν + (2 α + βb γ f (cid:48) ( b )) ν − βb (cid:0) γ f ( b ) + δ − f ( b ) (cid:1) + β b f ( b ) (cid:33) ω = − ν (cid:48) βbν − (cid:16) α + βb γ f (cid:48) ( b ) (cid:17)(cid:16) γ f ( b ) + δ − f ( b ) − βb f ( b ) (cid:17) ν (cid:2) ν − βb (cid:0) γ f ( b ) + δ − f ( b ) (cid:1) + β b f ( b ) (cid:3) − αβbν − γ ν f (cid:48) ( b ) , (3.15) The continuity of the functions n ( x ) and p ( x ) is necessary in order to solve (3.15) but requires certainlimitation conditions consisting of additional assumptions on the bottom deformation function b ( x ) . Inwhat follows, we briefly present these conditions: • If (cid:2) ( γδ bo + 2 δ bo − γδ ) − δ bo((1 + γδ )bo − δ ( γ + δ )) (cid:3) > then: ν ( b ) (cid:54) = 0 , βb (cid:54) = 0 and βb (cid:54) = 2 δ bo + γδ bo − γδ ± (cid:112) ( γδ bo + 2 δ bo − γδ ) − δ bo((1 + γδ )bo − δ ( γ + δ ))2 δ bo . • If (cid:2) ( γδ bo + 2 δ bo − γδ ) − δ bo((1 + γδ )bo − δ ( γ + δ )) (cid:3) < then: ν ( b ) (cid:54) = 0 , βb (cid:54) = 0 . • If (cid:2) ( γδ bo + 2 δ bo − γδ ) − δ bo((1 + γδ )bo − δ ( γ + δ )) (cid:3) = 0 then: ν ( b ) (cid:54) = 0 , βb (cid:54) = 0 and βb (cid:54) = 2 δ bo + γδ bo − γδ δ bo . (H0)Moreover, ∂ x v terms remain in (3.6), as well as in ∂ x (cid:0) R [ h , h ] v (cid:1) . These terms cannot be controlled bythe energy space considered hereinafter. For this reason, and in order to cancel these terms, we introduce9 new function of b to be determined to this end, named ς and embedded in the term T [ (cid:15)ζ, βb ]( (cid:15)ςv∂ x v ).In fact, one has T [ (cid:15)ζ, βb ]( (cid:15)ςv∂ x v ) + µ(cid:15)q ( (cid:15)ζ, βb ) ∂ x (cid:16) R [ h , h ] v (cid:17) = q ( (cid:15)ζ, βb )( (cid:15)ςv∂ x v ) + µ(cid:15)βκ ∂ x ζ∂ x b ( (cid:15)ςv∂ x v ) − µ∂ x (cid:16) νq ( (cid:15)ζ, βb ) ∂ x ( (cid:15)ςv∂ x v ) (cid:17) + µ(cid:15)q ( (cid:15)ζ, βb ) ∂ x (cid:16) R [ h , h ] v (cid:17) . (3.16)Adding (3.6) to (3.16), and fixing ς as follows ν (1 + βbω ) ς = (1 − γ ) g ( b ) βω b ) − f ( b ) − γg ( b ) ) + θ ( b ) g ( b )(1 + βω b )+ νβbω ( f ( b ) − γg ( b ) ) + (1 + βω b ) (cid:2) βb ( γ f ( b ) + 23 δ − f ( b ))( f ( b ) − γg ( b ) ) (cid:3) − (1 + βω b ) (cid:2) β b f ( b )( f ( b ) − γg ( b ) ) (cid:3) − βw bλ ( b )( f ( b ) − γg ( b ) ) , (3.17)while dropping all terms of order O ( µ , µ(cid:15) ) yields the following approximation: T ( ∂ t v + (cid:15)ςv∂ x v ) − q ( (cid:15)ζ, βb ) ∂ t (cid:16) v + µ (cid:0) Q [ h , h ] v (cid:1)(cid:17) + q ( (cid:15)ζ, βb ) µ γ + δ bo ∂ x ζ + µ(cid:15)q ( (cid:15)ζ, βb ) ∂ x (cid:16) R [ h , h ] v (cid:17) = q ( (cid:15)ζ, βb )( (cid:15)ςv∂ x v ) + µ [ A ] v∂ x v + µ [ B ]( ∂ x v ) + µ [ C ] v∂ x v + µ [ D ] ∂ x (cid:0) ( ∂ x v ) (cid:1) + µ [ E ] v + µ [ F ] ∂ x ζ + O ( µ , µ(cid:15) ) . (3.18)We would like to mention that the terms A , B , C , D , E , and F are functions depending on ζ ( t, x ) and b ( x ). We do not try to give in here their exact expressions for the sake of simplicity. However, thesefunctions are detailed at the end of this paper in Appendix A. In fact, their exact expressions are ofno interest for our present purpose. Although these functions are extensive but they remain easy tocontrol.Now, the last step to get the new equivalent asymptotic model is to multiply the second equationof (3.2) by q ( (cid:15)ζ, βb ) and include the estimate (3.18). Consequently, one gets the below system: ∂ t ζ + ∂ x (cid:18) h h h + γh v (cid:19) = 0 , T [ (cid:15)ζ, βb ] ( ∂ t v + (cid:15)ςv∂ x v ) + ( γ + δ ) q ( (cid:15)ζ, βb ) ∂ x ζ + (cid:15) q ( (cid:15)ζ, βb ) ∂ x (cid:18) h − γh ( h + γh ) | v | (cid:19) − q ( (cid:15)ζ, βb )( (cid:15)ςv∂ x v )= µ [ A ] v∂ x v + µ [ B ]( ∂ x v ) + µ [ C ] v∂ x v + µ [ D ] ∂ x (cid:0) ( ∂ x v ) (cid:1) + µ [ E ] v + µ [ F ] ∂ x ζ. (3.19) Remark 3.20
It is worth mentioning that one can easily recover the same model obtained and fully justi-fied in [11] just after setting β = 0 in (3.19) . Moreover, one can also recover the same model obtained andfully justified in [16] just after setting β = O ( √ µ ) and dropping all terms of order O ( µ , µ(cid:15) , µ(cid:15)β, µβ ) in (3.19) . T The strong ellipticity of the operator T [ (cid:15)ζ, βb ], defined in (3.5) and recalled below is necessary to obtainthe well-posedness and continuity of the inverse T − : T [ (cid:15)ζ, βb ] V = q ( (cid:15)ζ, βb ) V + µ(cid:15)βκ ∂ x ζ∂ x bV − µ∂ x (cid:16) νq ( (cid:15)ζ, βb ) ∂ x V (cid:17) , (4.1)with ν, κ , κ , κ , ω , ω are functions of b . 10n what follows, we search for sufficient conditions to guarantee the ellipticity property of the oper-ator T . First, we assume that ν ( b ) > ν ( b ) = λ ( b ) − ≥ ν > . Second, we assume the non-zero depth condition on both respectively upper and lower layer of fluid: ∃ h > , such that inf x ∈ R h ≥ h > , inf x ∈ R h ≥ h > , (H1)Briefly, since (cid:15) ≤ (cid:15) max , thus (cid:15)ζ ≤ (cid:15) | ζ | L ∞ ≤ (cid:15) max | ζ | L ∞ < min(1 , δ max ) < − (cid:15)ζ ) > , and (cid:15)ζ − βb ≥ − (cid:15) max | ζ | L ∞ − β max | b | L ∞ > − min(1 , δ max ) > − δ max > − δ so ( 1 δ + (cid:15)ζ − βb ) >
0. Thus, onecan easily say that the condition (cid:15) max | ζ | L ∞ + β max | b | L ∞ < min(1 , δ max ) is adequate to define h > µ, (cid:15), δ, γ, β, bo) ∈ P CH .Equivalently, we introduce the condition ∃ h > , such that inf x ∈ R (cid:16) q ( (cid:15)ζ, βb ) + µ(cid:15)βκ ∂ x ζ∂ x b (cid:17) ≥ h > x ∈ R q ( (cid:15)ζ, βb ) ≥ h > . (H2) Lemma 4.2
Let ζ ∈ L ∞ , b ∈ W , ∞ and (cid:15) max = min( M √ µ max , be such that there exists h > with max( | κ | L ∞ , | κ | L ∞ , , δ max ) (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + max( | ω | L ∞ , | ω | L ∞ , δ max ) β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + | η | L ∞ (cid:15) max | ζ | L ∞ β max | b | L ∞ + µ max (cid:15) max β max (cid:16)(cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ (cid:12)(cid:12) ∂ x ζ (cid:12)(cid:12) L ∞ | ∂ x b (cid:12)(cid:12) L ∞ ) ≤ − h . Then there exists h , h > such that (H1) - (H2) hold for any ( µ, (cid:15), δ, γ, β, bo) ∈ P CH . .Proof . One can easily check that, (cid:15)ζ ≤ (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ ≤ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ ≤ − h − (cid:15)ζ ≥ h inf x ∈ R h ≥ h ,δ max (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + δ max β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ ≤ − h − (cid:16) δ max (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + δ max β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ (cid:17) ≥ h − δ(cid:15)ζ − δβb ≥ h −
11 + δ(cid:15)ζ − δβb ≥ h δ + (cid:15)ζ − βb ≥ h inf x ∈ R h ≥ h and in the same way, | κ | L ∞ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + | ω | L ∞ β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + µ max (cid:15) max β max (cid:16)(cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ (cid:12)(cid:12) ∂ x ζ (cid:12)(cid:12) L ∞ | ∂ x b (cid:12)(cid:12) L ∞ (cid:17) ≤ − h κ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + ω β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + µ max (cid:15) max β max (cid:16)(cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ ∂ x ζ∂ x b (cid:17) ≤ − h − κ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ − ω β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ − µ max (cid:15) max β max (cid:16)(cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ ∂ x ζ∂ x b (cid:17) ≥ h − κ (cid:15)ζ + ω βb + µ(cid:15)β (cid:16) κ ∂ x ζ∂ x b (cid:17) ≥ h −
11 + κ (cid:15)ζ + ω βb + µ(cid:15)β (cid:16) κ ∂ x ζ∂ x b (cid:17) ≥ h inf x ∈ R (cid:16) q ( (cid:15)ζ, βb ) + µ(cid:15)βκ ∂ x ζ∂ x b (cid:17) ≥ h , κ | L ∞ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + | ω | L ∞ β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + | η | L ∞ (cid:15) max | ζ | L ∞ β max | b | L ∞ ≤ − h κ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + ω β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + η (cid:15) max ζβ max | b | L ∞ ≤ − h − (cid:16) κ (cid:15) max (cid:12)(cid:12) ζ (cid:12)(cid:12) L ∞ + ω β max (cid:12)(cid:12) b (cid:12)(cid:12) L ∞ + η (cid:15) max ζβ max | b | L ∞ (cid:17) ≥ h − κ (cid:15)ζ + ω βb + η (cid:15)ζβb ≥ h −
11 + κ (cid:15)ζ + ω βb + η (cid:15)ζβb ≥ h inf x ∈ R (cid:0) q ( (cid:15)ζ, βb ) (cid:1) ≥ h . (cid:3) Definition 4.3
We define by H µ ( R ) the space H ( R ) endowed with the norm | · | H µ that is equivalent tothe H ( R ) -norm but not uniformly with respect to µ , defined as ∀ v ∈ H ( R ) , | v | H µ = | v | L + µ | ∂ x v | L . We define also ( H µ ( R )) (cid:63) the space H − ( R ) the dual space of H µ ( R ) . Lets now assert the strong ellipticity result of the operator T , Lemma 4.4
Let ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and ζ ∈ W , ∞ ( R ) , b ∈ W , ∞ ( R ) such that (H2) is satisfied.Then the operator T [ (cid:15)ζ, βb ] : H µ ( R ) −→ ( H µ ( R )) (cid:63) is uniformly continuous and coercive. More precisely, there exists c > such that ( T u, v ) ≤ c | u | H µ | v | H µ ; (4.5)( T v, v ) ≥ c | v | H µ (4.6) with c = C ( M CH , h − , (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) W , ∞ , β (cid:12)(cid:12) b (cid:12)(cid:12) W , ∞ ) .The constant c also depends on (cid:12)(cid:12) ν (cid:12)(cid:12) L ∞ , (cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ , (cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ , (cid:12)(cid:12) κ (cid:12)(cid:12) L ∞ , (cid:12)(cid:12) ω (cid:12)(cid:12) L ∞ , (cid:12)(cid:12) ω (cid:12)(cid:12) L ∞ .Moreover, the following estimates hold:Let s > and s ≥ ,(i) If ζ ∈ H s ( R ) ∩ H s ( R ) and b ∈ H s +2 ( R ) ∩ H s +2 ( R ) and u ∈ H s +1 ( R ) and v ∈ H ( R ) , then: (cid:12)(cid:12)(cid:0) Λ s T [ (cid:15)ζ, βb ] u, v (cid:1)(cid:12)(cid:12) ≤ C (cid:16) ( (cid:15) | ζ | H s + β | b | H s ) (cid:12)(cid:12) u (cid:12)(cid:12) H s +1 µ + (cid:10) ( (cid:15) | ζ | H s + β | b | H s +2 ) (cid:12)(cid:12) u (cid:12)(cid:12) H s µ (cid:11) s>s (cid:17) (cid:12)(cid:12) v (cid:12)(cid:12) H µ . (4.7) (ii) If ζ ∈ H s +1 ∩ H s ( R ) and b ∈ H s +3 ∩ H s +2 ( R ) , u ∈ H s ( R ) and v ∈ H ( R ) , then: (cid:12)(cid:12)(cid:0)(cid:2) Λ s , T [ (cid:15)ζ, βb ] (cid:3) u, v (cid:1)(cid:12)(cid:12) ≤ max( (cid:15), β ) C (cid:16) ( | ζ | H s + | b | H s ) (cid:12)(cid:12) u (cid:12)(cid:12) H sµ + (cid:10) ( | ζ | H s + | b | H s +2 ) (cid:12)(cid:12) u (cid:12)(cid:12) H s µ (cid:11) s>s +1 (cid:17)(cid:12)(cid:12) v (cid:12)(cid:12) H µ , (4.8) where C = C ( M CH , h − ) .Proof . We define first the bilinear form a ( u, v ) = (cid:0) T u , v (cid:1) = (cid:0) (1+ (cid:15)κ ζ + βω b ) u , v (cid:1) + µ(cid:15)β (cid:0) κ ∂ x ζ∂ x b u , v (cid:1) + µ (cid:0) ν (1+ (cid:15)κ ζ + βω b + (cid:15)βη ζb ) ∂ x u , ∂ x v (cid:1) , where (cid:0) · , · ) denotes the L -based inner product. One can easily check that (cid:12)(cid:12) a ( u, v ) (cid:12)(cid:12) ≤ sup x ∈ R | (cid:15)κ ζ + βω b + µ(cid:15)βκ ∂ x ζ∂ x b | (cid:0) u , v (cid:1) + µ sup x ∈ R | ν (1+ (cid:15)κ ζ + βω b + (cid:15)βη ζb ) | (cid:0) ∂ x u , ∂ x v (cid:1) .
12y Cauchy-Schwarz inequality, one can easily obtain (4.5).Let us now prove the H µ ( R )-coercivity of a ( · , · ), inequality (4.6): (cid:0) T v , v (cid:1) = (cid:16) (cid:2) q ( (cid:15)ζ, βb ) + µ(cid:15)βκ ∂ x ζ∂ x b (cid:3) v , v (cid:17) + µ (cid:16) νq ( (cid:15)ζ, βb ) ∂ x v , ∂ x v (cid:17) . Since (H2) is satisfied one has, (cid:0) T v , v (cid:1) ≥ h | v | L + µν h | ∂ x v | L ≥ min( h , ν h ) (cid:16) | v | L + µ | ∂ x v | L (cid:17) . (4.9)Finally one can easily deduce that the inequality (4.9) is now: (cid:0) T v , v (cid:1) ≥ h min(1 , ν ) | v | H µ . We prove the product and commutator higher-order estimates of the Lemma,Making use of, κ ∂ x ζ∂ x bu = ∂ x ( κ uζ∂ x b ) − ∂ x ( κ u ) ζ∂ x b − κ uζ∂ x b , one has (cid:0) Λ s T u, v (cid:1) = (cid:0) Λ s { q ( (cid:15)ζ, βb ) u } , v (cid:1) + µ(cid:15)β (cid:0) Λ s { κ ∂ x ζ∂ x bu } , v (cid:1) + µ (cid:0) Λ s { νq ( (cid:15)ζ, βb ) ∂ x u } , ∂ x v (cid:1) = (cid:0) Λ s { (1 + (cid:15)κ ζ + βω b ) u } , v (cid:1) − µ(cid:15)β (cid:0) Λ s { κ uζ∂ x b } , ∂ x v (cid:1) − µ(cid:15)β (cid:0) Λ s { ∂ x ( κ u ) ζ∂ x b } , v (cid:1) − µ(cid:15)β (cid:0) Λ s { κ uζ∂ x b } , v (cid:1) + µ (cid:0) Λ s { ν (1 + (cid:15)κ ζ + βω b + (cid:15)βη ζb ) ∂ x u } , ∂ x v (cid:1) . Estimate (4.7) is now a straightforward consequence of Cauchy-Schwarz inequality and product estimatesin Sobolev spaces.As for the commutator estimates, using κ ∂ x ζ∂ x b = ∂ x ( κ ζ∂ x b ) − ∂ x ( κ ) ζ∂ x b − κ ζ∂ x b , and the fact that ∂ x (cid:16) [Λ s , f ] g (cid:17) = [Λ s , ∂ x f ] g + [Λ s , f ] ∂ x g ,one has (cid:0) [Λ s , T ] u , v (cid:1) = (cid:15) (cid:0) [Λ s , κ ζ ] u , v (cid:1) + β (cid:0) [Λ s , ω b ] u , v (cid:1) + µ(cid:15)β (cid:0) [Λ s , κ ∂ x ζ∂ x b ] u , v (cid:1) + µ (cid:0) [Λ s , ν ] ∂ x u , ∂ x v (cid:1) + µ(cid:15) (cid:0) [Λ s , νκ ζ ] ∂ x u , ∂ x v (cid:1) + µβ (cid:0) [Λ s , νω b ] ∂ x u, ∂ x v (cid:1) + µ(cid:15)β (cid:0) [Λ s , νη ζb ] ∂ x u, ∂ x v (cid:1) = (cid:15) (cid:0) [Λ s , κ ζ ] u , v (cid:1) + β (cid:0) [Λ s , ω b ] u , v (cid:1) − µ(cid:15)β (cid:0) [Λ s , κ ζ∂ x b ] u , ∂ x v (cid:1) − µ(cid:15)β (cid:0) [Λ s , κ ζ∂ x b ] ∂ x u , v (cid:1) − µ(cid:15)β (cid:0) [Λ s , ∂ x κ ζ∂ x b ] u, v (cid:1) − µ(cid:15)β (cid:0) [Λ s , κ ζ∂ x b ] u, v (cid:1) + µ (cid:0) [Λ s , ν ] ∂ x u , ∂ x v (cid:1) + µ(cid:15) (cid:0) [Λ s , νκ ζ ] ∂ x u , ∂ x v (cid:1) + µβ (cid:0) [Λ s , νω b ] ∂ x u, ∂ x v (cid:1) + µ(cid:15)β (cid:0) [Λ s , νη ζb ] ∂ x u, ∂ x v (cid:1) . Estimates (4.8) follow, using again Cauchy-Schwarz inequality and the commutator estimates. (cid:3)
The invertibility of T is asserted in the Lemma below. Lemma 4.10
Let ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and ζ ∈ W , ∞ ( R ) , b ∈ W , ∞ ( R ) such that (H2) is satisfied.Hence the operator T [ (cid:15)ζ, βb ] : H ( R ) −→ L ( R ) is bijective.(i) ( T [ (cid:15)ζ, βb ]) − : L → H µ ( R ) is continuous. More precisely, one has (cid:107) T − (cid:107) L ( R ) → H µ ( R ) ≤ c , with c = C ( M CH , h − , (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) W , ∞ , β (cid:12)(cid:12) b (cid:12)(cid:12) W , ∞ ) .(ii) Additionally, if ζ ∈ H s +1 ( R ) and b ∈ H s +3 ( R ) with s > , then one has for any < s ≤ s + 1 , (cid:107) T − (cid:107) H s ( R ) → H s +1 µ ( R ) ≤ c s +1 . iii) If ζ ∈ H s ( R ) and b ∈ H s +2 ( R ) with s ≥ s + 1 , s > , then one has (cid:107) T − (cid:107) H s ( R ) → H s +1 µ ( R ) ≤ c s where c ¯ s = C ( M CH , h − , (cid:15) | ζ | H ¯ s , β | b | H ¯ s +2 ) , thus uniform with respect to ( µ, (cid:15), δ, γ, β, bo) ∈ P CH .Proof. We use the Lax-Milgram theorem to prove that the operator T has an inverse. The bilinear form: a ( u, v ) = (cid:0) T u , v (cid:1) = (cid:0) (1+ (cid:15)κ ζ + βω b + µ(cid:15)βκ ∂ x ζ∂ x b ) u , v (cid:1) + µ (cid:0) ν (1+ (cid:15)κ ζ + βω b + (cid:15)βη ζb ) ∂ x u , ∂ x v (cid:1) is proved to be continuous and uniformly coercive on H µ ( R ) in the previous Lemma. For any µ >
0, thedual of H µ ( R ) is H − ( R ), of whom L ( R ) is a subspace, and one has (cid:0) f, g (cid:1) ≤ (cid:12)(cid:12) f (cid:12)(cid:12) H µ (cid:12)(cid:12) g (cid:12)(cid:12) L , independentlyof µ >
0. Hence, by Lax-Milgram Lemma, for all f ∈ L ( R ), there exists a unique u ∈ H µ ( R ) such that,for all v ∈ H µ ( R ) a ( u, v ) = ( f, v );equivalently, there is a unique variational solution to the equation T u = f. (4.11)By definition of T , one get that µν (1 + (cid:15)κ ζ + βω b + (cid:15)βη ζb ) ∂ x u = (1+ (cid:15)κ ζ + βω b + µ(cid:15)βκ ∂ x ζ∂ x b ) u − µ∂ x (cid:0) νq ( (cid:15)ζ, βb ) (cid:1) ∂ x u − f. (4.12)Using condition (H2), and since u ∈ H ( R ), ζ ∈ W , ∞ ( R ) and f ∈ L ( R ), we deduce that ∂ x u ∈ L ( R ),and thus u ∈ H ( R ). We proved that T [ (cid:15)ζ, βb ] : H ( R ) −→ L ( R ) is bijective. (cid:3) We skip the proof of the remaining estimates since they can be proved using the same techniquesas in [11] adapted to the new operator T [ (cid:15)ζ, βb ] depending on b .Finally, let us introduce the following technical estimate, which is used several times in the subse-quent sections. We do not try to give the proof of the estimate since it is a direct adaptation of theproof given in [11]. Corollary 4.13
Let ( µ, (cid:15), δ, γ, β, bo) ∈ P CH , ζ ∈ H s and b ∈ H s +2 ( R ) with s ≥ s + 1 , s > , suchthat (H2) is satisfied with C = C ( M CH , h − , (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) . Assume moreover that u ∈ H s − ( R ) andthat v ∈ H ( R ) . Then one has (cid:12)(cid:12)(cid:0) (cid:2) Λ s , T − [ (cid:15)ζ, βb ] (cid:3) u , T [ (cid:15)ζ, βb ] v (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:0) (cid:2) Λ s , T [ (cid:15)ζ, βb ] (cid:3) T − u , v (cid:1)(cid:12)(cid:12) ≤ max( (cid:15), β ) C (cid:12)(cid:12) u (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H µ (4.14) Let us recall the system (3.19). ∂ t ζ + ∂ x (cid:18) h h h + γh v (cid:19) = 0 , T [ (cid:15)ζ, βb ] ( ∂ t v + (cid:15)ςv∂ x v ) + ( γ + δ ) q ( (cid:15)ζ, βb ) ∂ x ζ + (cid:15) q ( (cid:15)ζ, βb ) ∂ x (cid:18) h − γh ( h + γh ) | v | (cid:19) − q ( (cid:15)ζ, βb )( (cid:15)ςv∂ x v )= µ [ A ] v∂ x v + µ [ B ]( ∂ x v ) + µ [ C ] v∂ x v + µ [ D ] ∂ x (cid:0) ( ∂ x v ) (cid:1) + µ [ E ] v + µ [ F ] ∂ x ζ, (5.1)with h = 1 − (cid:15)ζ , h = 1 /δ + (cid:15)ζ − βb , q ( X, Y ) = 1 + κ X + ω Y , q ( X, Y ) = 1 + κ X + ω Y + η XY ,where κ , κ , κ , ω , ω , η , ς defined in (3.8),(3.9),(3.10),(3.11),(3.12),(3.13),(3.17), and T [ (cid:15)ζ, βb ] V = q ( (cid:15)ζ, βb ) V + µ(cid:15)βκ ∂ x ζ∂ x bV − µ∂ x (cid:16) νq ( (cid:15)ζ, βb ) ∂ x V (cid:17) .
14n order to ease the reading, we define the function H : X → (1 − X )( δ − + X − βb )1 − X + γ ( δ − + X − βb ) , and G : X → (cid:16) (1 − X )1 − X + γ ( δ − + X − βb ) (cid:17) . One can easily check that H ( (cid:15)ζ ) = h h h + γh , H (cid:48) ( (cid:15)ζ ) = h − γh ( h + γh ) and G ( (cid:15)ζ ) = (cid:16) h h + γh (cid:17) . One can rewrite, ∂ t ζ + H ( (cid:15)ζ ) ∂ x v + (cid:15)∂ x ζH (cid:48) ( (cid:15)ζ ) v − β∂ x bG ( (cid:15)ζ ) v = 0 , T (cid:16) ∂ t v + (cid:15) ς∂ x ( v ) (cid:17) + ( γ + δ ) q ( (cid:15)ζ, βb ) ∂ x ζ + (cid:15)q ( (cid:15)ζ, βb ) (cid:0) H (cid:48) ( (cid:15)ζ ) − ς (cid:1) v∂ x v + (cid:15)q ( (cid:15)ζ, βb ) ∂ x (cid:0) H (cid:48) ( (cid:15)ζ )2 (cid:1) v = µ [ A ] v∂ x v + µ [ B ]( ∂ x v ) + µ [ C ] v∂ x v + µ [ D ] ∂ x (cid:0) ( ∂ x v ) (cid:1) + µ [ E ] v + µ [ F ] ∂ x ζ. (5.2)with ∂ x ( H (cid:48) ( (cid:15)ζ )2 ) = − γ(cid:15)∂ x ζ ( h + h ) + γβ∂ x bh ( h + h )( h + γh ) . The equations can be written after applying T − to the second equation in (5.2) as ∂ t U + A [ U ] ∂ x U + B [ U ] = 0 , (5.3)with A [ U ] = (cid:18) (cid:15)H (cid:48) ( (cid:15)ζ ) v H ( (cid:15)ζ ) T − ( Q ( (cid:15)ζ, βb ) · + (cid:15) Q ( (cid:15)ζ, βb, v ) · ) T − ( Q [ (cid:15)ζ, βb, v ] · ) + (cid:15)ςv (cid:19) , (5.4) B [ U ] = − β∂ x bG ( (cid:15)ζ ) v T − (cid:16) γ(cid:15)βq ( (cid:15)ζ, βb ) h ( h + h ) ∂ x b ( h + γh ) v − µ [ E ] v (cid:17) (5.5)where Q ( (cid:15)ζ, βb ) , Q ( (cid:15)ζ, βb, v ) are defined as Q ( (cid:15)ζ, βb ) = ( γ + δ ) q ( (cid:15)ζ, βb ) − µ [ F ] , Q ( (cid:15)ζ, βb, v ) = − γq ( (cid:15)ζ, βb ) ( h + h ) ( h + γh ) v (5.6)and the operator Q [ (cid:15), βb, v ] defined by Q [ (cid:15)ζ, βb, v ] f ≡ (cid:16) (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) − µ [ A ] (cid:17) vf − µ [ B ] ∂ x vf − µ [ C ] v∂ x f − µ [ D ] ∂ x ( ∂ x vf ) . (5.7)System (5.3) enjoys a quasilinear hyperbolic structure. We start by studying the properties andthe energy estimates for the linearized system to conclude finally the well-posedness of the initial valueproblem of the system. The linearization is around some reference state U = ( ζ, v ) (cid:62) as follows: (cid:26) ∂ t U + A [ U ] ∂ x U + B [ U ] = 0; U | t =0 = U . (5.8) First, we introduce a pseudo-symmetrizer of the system: Z [ U ] = Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) 00 T [ (cid:15)ζ, βb ] . (5.9)The pseudo-symmetrizer is defined and positive under the assumption below: ∃ h > Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) ≥ h > . (H3)Now, we define the energy space: 15 efinition 5.10 For given s ≥ and µ, T > , we denote by X s the vector space H s ( R ) × H s +1 µ ( R ) endowed with the norm ∀ U = ( ζ, v ) ∈ X s , | U | X s ≡ | ζ | H s + | v | H s + µ | ∂ x v | H s , while X sT stands for the space of U = ( ζ, v ) such that U ∈ C ([0 , T max( (cid:15), β ) ]; X s ) and ∂ t U ∈ L ∞ ([0 , T max( (cid:15), β ) ] × R ) , endowed with the canonical norm (cid:107) U (cid:107) X sT ≡ sup t ∈ [0 ,T/ max( (cid:15),β )] | U ( t, · ) | X s + ess sup t ∈ [0 ,T/ max( (cid:15),β )] ,x ∈ R | ∂ t U ( t, x ) | . The energy of the initial value problem (5.8) is now given naturally by: E s ( U ) = (Λ s U, Z [ U ]Λ s U ) = (Λ s ζ, Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) Λ s ζ ) + (cid:0) Λ s v, T [ (cid:15)ζ, βb ]Λ s v (cid:1) . (5.11)The energy of the pseudo-symmetrizer is equivalent to X s under the additional assumption given in(H3). We assert this equivalency in the Lemma below. We omit in here the proof of this Lemma sinceit is proved using the same techniques as in [16, Lemma 5.2]. Lemma 5.12
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH , s ≥ , U ∈ W , ∞ ( R ) and b ∈ W , ∞ ( R ) , satisfying (H1) , (H2) , and (H3) . Then E s ( U ) is equivalent to the | · | X s -norm.More precisely, there exists c = C ( M CH , h − , h − , h − , (cid:15) | U | W , ∞ , β | b | W , ∞ ) > such that c E s ( U ) ≤ (cid:12)(cid:12) U (cid:12)(cid:12) X s ≤ c E s ( U ) . To complete this section we assert some useful general estimates concerning our new operators.
Lemma 5.13
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH , and let U = ( ζ u , u ) (cid:62) ∈ W , ∞ , b ∈ W , ∞ satisfies (H1) , (H2) and (H3) . Then for any V, W ∈ X , one has (cid:12)(cid:12)(cid:12) (cid:16) Z [ U ] V , W (cid:17) (cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12) V (cid:12)(cid:12) X (cid:12)(cid:12) W (cid:12)(cid:12) X , (5.14) with C = C ( M CH , h − , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) W , ∞ , β (cid:12)(cid:12) b (cid:12)(cid:12) W , ∞ ) .Moreover, if U ∈ X s , b ∈ H s +2 , V ∈ X s − with s ≥ s + 1 , s > / , then one has (cid:12)(cid:12)(cid:12)(cid:16) (cid:2) Λ s , Z [ U ] (cid:3) V , W (cid:17)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12) V (cid:12)(cid:12) X s − (cid:12)(cid:12) W (cid:12)(cid:12) X (5.15) (cid:12)(cid:12)(cid:12)(cid:16) (cid:2) Λ s , Z − [ U ] (cid:3) V , Z [ U ] W (cid:17)(cid:12)(cid:12)(cid:12) ≤ C (cid:12)(cid:12) V (cid:12)(cid:12) H s − × H s − (cid:12)(cid:12) W (cid:12)(cid:12) X (5.16) with C = C ( M CH , h − , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) .Proof . The Lemma 5.13 is proved using Cauchy-Schwartz inequality, Lemma 4.10 and Corollary 4.13.We do not detail the proof, and refer to [11, Lemma 6.4]. In this section we aim at establishing an X s energy estimate regarding our linear system. The well-posedness and stability of the nonlinear system is made possible using linear analysis by considering amodified system of the form: (cid:26) ∂ t U + A [ U ] ∂ x U + B [ U ] = F ; U | t =0 = U . (5.17)where we added a right-hand-side F , whose properties will be precised in the following Lemmas.We begin by asserting a basic X s ( s > /
2) energy estimate.16 emma 5.18 ( X s energy estimate) Set ( µ, (cid:15), δ, γ, β, bo) ∈ P CH , and s ≥ s + 1 , s > / . Let U =( ζ, v ) (cid:62) and U = ( ζ, v ) (cid:62) be such that U, U ∈ L ∞ ([0 , T / max( (cid:15), β )]; X s ) , ∂ t U ∈ L ∞ ([0 , T / max( (cid:15), β )] × R ) , b ∈ H s +3 and U satisfies (H1) , (H2) , and (H3) uniformly on [0 , T / max( (cid:15), β )] , and such that sys-tem (5.17) holds with a right hand side, F , with (cid:0) Λ s F, Z [ U ]Λ s U (cid:1) ≤ C F max( (cid:15), β ) (cid:12)(cid:12) U (cid:12)(cid:12) X s + f ( t ) (cid:12)(cid:12) U (cid:12)(cid:12) X s , where C F is a constant and f is an integrable function on [0 , T / max( (cid:15), β )] .Then there exists λ, C = C ( (cid:13)(cid:13) U (cid:13)(cid:13) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +3 , C F ) such that the following energy estimate holds: E s ( U )( t ) ≤ e max( (cid:15),β ) λt E s ( U ) + (cid:90) t e max( (cid:15),β ) λ ( t − t (cid:48) ) (cid:0) f ( t (cid:48) ) + max( (cid:15), β ) C (cid:1) dt (cid:48) , (5.19) The constants λ and C are independent of p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH , but depend on M CH , h − , h − ,and h − . Remark 5.20
In this Lemma, and in the proof below, the norm (cid:13)(cid:13) U (cid:13)(cid:13) X sT is to be understood as an essentialsup: (cid:107) U (cid:107) X sT ≡ ess sup t ∈ [0 ,T/ max( (cid:15),β )] | U ( t, · ) | X s + ess sup t ∈ [0 ,T/ max( (cid:15),β )] ,x ∈ R | ∂ t U ( t, x ) | . Proof .Let us multiply the system (5.17) on the right by Λ s Z [ U ]Λ s U , and integrate by parts. One obtains (cid:0) Λ s ∂ t U, Z [ U ]Λ s U (cid:1) + (cid:0) Λ s A [ U ] ∂ x U, Z [ U ]Λ s U (cid:1) + (cid:0) Λ s B [ U ] , Z [ U ]Λ s U (cid:1) = (cid:0) Λ s F, Z [ U ]Λ s U (cid:1) , (5.21)from which we deduce, using the symmetry property of Z [ U ], as well as the definition of E s ( U ):12 ddt E s ( U ) = 12 (cid:0) Λ s U, (cid:2) ∂ t , Z [ U ] (cid:3) Λ s U (cid:1) − (cid:0) Z [ U ] A [ U ] ∂ x Λ s U, Λ s U (cid:1) − (cid:0)(cid:2) Λ s , A [ U ] (cid:3) ∂ x U, Z [ U ]Λ s U (cid:1) − (cid:0) Λ s B [ U ] , Z [ U ]Λ s U (cid:1) + (cid:0) Λ s F, Z [ U ]Λ s U (cid:1) . (5.22)We now estimate each of the different components of the r.h.s of the above identity. • Estimate of (cid:0) Λ s B [ U ] , Z [ U ]Λ s U (cid:1) , (cid:0) Λ s B [ U ] , Z [ U ]Λ s U (cid:1) = (cid:16) Λ s (cid:0) − G ( (cid:15)ζ ) vβ∂ x b (cid:1) , Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) Λ s ζ (cid:17) + (cid:16) Λ s T − (cid:0) γ(cid:15)βq ( (cid:15)ζ, βb ) h ( h + h ) v ∂ x b ( h + γh ) − µ [ E ] v (cid:1) , T [ (cid:15)ζ, βb ]Λ s v (cid:17) . Using Cauchy-Schwarz inequality, Lemma (4.4) and Lemma (4.10) one has, | (cid:0) Λ s B [ U ] , Z [ U ]Λ s U (cid:1) | ≤ βC (cid:0)(cid:13)(cid:13) U (cid:13)(cid:13) X sT , (cid:107) b (cid:107) H s +3 (cid:1) | U | X s ≤ max( (cid:15), β ) C | U | X s . (5.23) • Estimate of (cid:0) Z [ U ] A [ U ] ∂ x Λ s U, Λ s U (cid:1) . Now we have, Z [ U ] A [ U ] = (cid:15) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) Q [ (cid:15)ζ, βb, v ] + (cid:15) T [ (cid:15)ζ, βb ]( ςv. )
17o that, (cid:0) Z [ U ] A [ U ] ∂ x U, U (cid:1) = (cid:16) (cid:15) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v∂ x ζ, ζ (cid:17) + (cid:16) Q ( (cid:15)ζ, βb ) ∂ x v, ζ (cid:17) + (cid:16) (cid:15) Q ( (cid:15)ζ, βb, v ) ∂ x v, ζ (cid:17) + (cid:16) Q ( (cid:15)ζ, βb ) ∂ x ζ, v (cid:17) + (cid:16) (cid:15) Q ( (cid:15)ζ, βb, v ) ∂ x ζ, v (cid:17) + (cid:16) Q [ (cid:15)ζ, βb, v ] ∂ x v, v (cid:17) + (cid:15) (cid:16) T [ (cid:15)ζ, βb ]( ςv∂ x v ) , v (cid:17) . One deduces that, (cid:0) Z [ U ] A [ U ] ∂ x U, U (cid:1) = − (cid:16) (cid:15)∂ x (cid:0) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v (cid:1) ζ, ζ (cid:17) − (cid:16) ∂ x (cid:0) Q ( (cid:15)ζ, βb ) (cid:1) ζ, v (cid:17) − (cid:15) (cid:16) ∂ x (cid:0) Q ( (cid:15)ζ, βb, v ) (cid:1) ζ, v (cid:17) + (cid:16) Q [ (cid:15)ζ, βb, v ] ∂ x v, v (cid:17) + (cid:15) (cid:16) T [ (cid:15)ζ, βb ]( ςv∂ x v ) , v (cid:17) . One make use of the identities below, • (cid:16) Q [ (cid:15)ζ, βb, v ] ∂ x v, v (cid:17) = (cid:16) (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v∂ x v, v (cid:17) − µ (cid:16) [ A ] v∂ x v, v (cid:17) − µ (cid:16) [ B ] ∂ x v∂ x v, v (cid:17) − µ (cid:16) [ C ] v∂ x v, v (cid:17) − µ (cid:16) [ D ] ∂ x ( ∂ x v∂ x v ) , v (cid:17) = − (cid:16) (cid:15)∂ x (cid:2) q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v (cid:3) v, v (cid:17) − µ (cid:16) [ A ] v∂ x v, v (cid:17) − µ (cid:16) [ B ] ∂ x v∂ x v, v (cid:17) + µ (cid:16) ∂ x (cid:0) [ C ] vv (cid:1) , ∂ x v (cid:17) + µ (cid:16) ∂ x (cid:0) [ D ] v (cid:1) , ∂ x v∂ x v (cid:17) . • (cid:16) T [ (cid:15)ζ, βb ]( ςv∂ x v ) , v (cid:17) = (cid:16) q ( (cid:15)ζ, βb ) ςv∂ x v + ( µκ (cid:15)β∂ x ζ∂ x b ) ςv∂ x v − µ∂ x ( νq ( (cid:15)ζ, βb ) ∂ x ( ςv∂ x v )) , v (cid:17) = − (cid:16) ∂ x ( q ( (cid:15)ζ, βb ) ςv ) v, v (cid:17) + µ(cid:15)β (cid:16) ( κ ∂ x ζ∂ x bςv ) ∂ x v, v (cid:17) + µ (cid:16) νq ( (cid:15)ζ, βb ) ∂ x ( ςv∂ x v ) , ∂ x v (cid:17) = − (cid:16) ∂ x ( q ( (cid:15)ζ, βb ) ςv ) v, v (cid:17) + µ(cid:15)β (cid:16) ( κ ∂ x ζ∂ x bςv ) ∂ x v, v (cid:17) + µ (cid:16) νq ( (cid:15)ζ, βb )( ∂ x ς ) v∂ x v, ∂ x v (cid:17) + µ (cid:16) νq ( (cid:15)ζ, βb ) ς ( ∂ x v ) ∂ x v, ∂ x v (cid:17) − µ (cid:16) ∂ x ( νq ( (cid:15)ζ, βb ) ςv ) ∂ x v, ∂ x v (cid:17) . Using Cauchy-Schwartz inequality we obtain, (cid:12)(cid:12)(cid:12)(cid:0) Z [ U ] A [ U ] ∂ x U, U (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C (cid:16)(cid:13)(cid:13) U (cid:13)(cid:13) L ∞ + (cid:13)(cid:13) ∂ x U (cid:13)(cid:13) L ∞ + (cid:13)(cid:13) b (cid:13)(cid:13) W , ∞ (cid:17)(cid:12)(cid:12) U (cid:12)(cid:12) X . (5.24)Thanks to Sobolev embedding, one has for s > s + 1 , s > / C ( (cid:107) U (cid:107) L ∞ + (cid:107) ∂ x U (cid:107) L ∞ ) ≤ C ( (cid:13)(cid:13) U (cid:13)(cid:13) X sT ) , One can use the L estimate derived in (5.24), applied to Λ s U . One deduces (cid:12)(cid:12)(cid:12)(cid:0) Z [ U ] A [ U ] ∂ x Λ s U, Λ s U (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C (cid:16)(cid:13)(cid:13) U (cid:13)(cid:13) X sT + (cid:13)(cid:13) b (cid:13)(cid:13) W , ∞ (cid:17)(cid:12)(cid:12) U (cid:12)(cid:12) X s ≤ max( (cid:15), β ) C (cid:12)(cid:12) U (cid:12)(cid:12) X s . (5.25) • Estimate of (cid:0)(cid:2) Λ s , A [ U ] (cid:3) ∂ x U, Z [ U ]Λ s U (cid:1) . Using the definition of A [ · ] and Z [ · ] in (5.4) and (5.9), one18as (cid:0)(cid:2) Λ s , A [ U ] (cid:3) ∂ x U, Z [ U ]Λ s U (cid:1) = (cid:16) [Λ s , (cid:15)H (cid:48) ( (cid:15)ζ ) v ] ∂ x ζ + [Λ s , H ( (cid:15)ζ )] ∂ x v , Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) Λ s ζ (cid:17) + (cid:16) [Λ s , T − (cid:0) Q ( (cid:15)ζ, βb, v ) . (cid:1) ] ∂ x ζ , T Λ s v (cid:17) + (cid:16) [Λ s , T − ( Q [ (cid:15)ζ, βb, v ] · ) + (cid:15)ςv ] ∂ x v, T Λ s v (cid:17) ≡ B + B + B . Here and in the following, we denote T ≡ T [ (cid:15)ζ, βb ] and Q ( (cid:15)ζ, βb, v ) = Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) . − Control of B = (cid:16) [Λ s , (cid:15)H (cid:48) ( (cid:15)ζ ) v ] ∂ x ζ + [Λ s , H ( (cid:15)ζ )] ∂ x v , Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) Λ s ζ (cid:17) .From Cauchy-Schwarz inequality, one has | B | ≤ (cid:12)(cid:12)(cid:12) [Λ s , (cid:15)H (cid:48) ( (cid:15)ζ ) v ] ∂ x ζ + [Λ s , H ( (cid:15)ζ )] ∂ x v (cid:12)(cid:12)(cid:12) L (cid:12)(cid:12)(cid:12) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) Λ s ζ (cid:12)(cid:12)(cid:12) L . Since s ≥ s + 1, we can use the commutator estimate mainly due to the Kato-Ponce [13], and recentlyimproved by Lannes [14] (see Theorems 3 and 6), to get (cid:12)(cid:12)(cid:12) [Λ s , (cid:15)H (cid:48) ( (cid:15)ζ ) v ] ∂ x ζ + [Λ s , H ( (cid:15)ζ )] ∂ x v (cid:12)(cid:12)(cid:12) L (cid:46) (cid:0) | ∂ x ( (cid:15)H (cid:48) ( (cid:15)ζ )) | H s − + | ∂ x ( H ( (cid:15)ζ )) | H s − (cid:1) | ∂ x U | H s − (cid:46) max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:107) b (cid:107) H s ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . It follows, using that Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) ∈ L ∞ since ζ , b satisfies (H1): | B | ≤ max( (cid:15), β ) C (cid:12)(cid:12) U (cid:12)(cid:12) X s . − Control of B = (cid:16) [Λ s , T − ( Q ( (cid:15)ζ, βb, v ) · )] ∂ x ζ , T Λ s v (cid:17) .By symmetry of T , one has B = (cid:16) T [Λ s , T − ( Q ( (cid:15)ζ, βb, v ) · )] ∂ x ζ , Λ s v (cid:17) . Now, one can check that, by definition of the commutator, T [Λ s , T − ( Q ( (cid:15)ζ, βb, v ) · )] ∂ x ζ = T Λ s T − Q ( (cid:15)ζ, βb, v ) ∂ x ζ − Q ( (cid:15)ζ, βb, v )Λ s ∂ x ζ = T Λ s T − Q ( (cid:15)ζ, βb, v ) ∂ x ζ − Λ s TT − ( Q ( (cid:15)ζ, βb, v ) ∂ x ζ )+Λ s ( Q ( (cid:15)ζ, βb, v ) ∂ x ζ ) − Q ( (cid:15)ζ, βb, v )Λ s ∂ x ζ = − (cid:2) Λ s , T (cid:3) T − ( Q ( (cid:15)ζ, βb, v ) ∂ x ζ ) + (cid:2) Λ s , Q ( (cid:15)ζ, βb, v ) (cid:3) ∂ x ζ We can now use Corollary 4.13, and deduce (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , T ] T − ( Q ( (cid:15)ζ, βb, v ) ∂ x ζ ) , Λ s v (cid:17)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) Q ( (cid:15)ζ, βb, v ) ∂ x ζ (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ ≤ max( (cid:15), β ) C ( (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ . From the commutator estimate and the explicit definition of Q = Q + (cid:15) Q in (5.6), one has (cid:12)(cid:12) (cid:2) Λ s , Q ( (cid:15)ζ, βb, v ) (cid:3) ∂ x ζ (cid:12)(cid:12) L = (cid:12)(cid:12)(cid:12)(cid:2) Λ s , ( γ + δ )( (cid:15)κ ζ + βω b ) − µ [ F ] − γ(cid:15) q ( (cid:15)ζ, βb ) ( h + h ) ( h + γh ) v (cid:3) ∂ x ζ (cid:12)(cid:12)(cid:12) L ≤ max( (cid:15), β ) C ( (cid:12)(cid:12) U (cid:12)(cid:12) H s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) ∂ x ζ (cid:12)(cid:12) H s − ,
19o that we finally get | B | ≤ max( (cid:15), β ) C (cid:12)(cid:12) U (cid:12)(cid:12) X s . − Control of B = (cid:16) [Λ s , T − ( Q [ (cid:15)ζ, βb, v ] · ) + (cid:15)ςv ] ∂ x v, T Λ s v (cid:17) .Let us first use the definition of Q [ (cid:15)ζ, βb, v ] (5.7) to expand: B = (cid:16) [Λ s , T − ( (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v · )] ∂ x v, T Λ s v (cid:17) − µ (cid:16) [Λ s , T − ([ A ] v · )] ∂ x v, T Λ s v (cid:17) − µ (cid:16) [Λ s , T − ([ B ] ∂ x v · )] ∂ x v, T Λ s v (cid:17) − µ (cid:16) [Λ s , T − ([ C ] v∂ x · )] ∂ x v, T Λ s v (cid:17) − µ (cid:16) [Λ s , T − ([ D ] ∂ x ( ∂ x v · ))] ∂ x v, T Λ s v (cid:17) + (cid:15) (cid:16) [Λ s , ςv ] ∂ x v, T Λ s v (cid:17) ≡ B + B + B + B + B + B . In order to estimate B , B , B , B and B , one proceeds as for the control of B .One can check T [Λ s , T − ( (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v · )] ∂ x v = − [Λ s , T ] T − ( (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v∂ x v )+[Λ s , (cid:15)q ( (cid:15)ζ, βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v ] ∂ x v. As above, using Cauchy-Schwarz inequality, Corollary 4.13 and the commutator estimate, one obtains | B | ≤ max( (cid:15), β ) C ( (cid:13)(cid:13) U (cid:13)(cid:13) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . In the same way we control B , T [Λ s , T − ([ A ] v · )] ∂ x v = − [Λ s , T ] T − ([ A ] v∂ x v ) + [Λ s , [ A ] v ] ∂ x v. Again, using Cauchy-Schwarz inequality, Corollary 4.13 and the commutator estimate, one has µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , T ] T − ([ A ] v∂ x v ) , Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ µ max( (cid:15), β ) C ( (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) [ A ] v∂ x v (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ , and µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , [ A ] v ] ∂ x v, Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s . Thus we proved | B | ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . To control B one can check, T [Λ s , T − ([ B ] ∂ x v · )] ∂ x v = − [Λ s , T ] T − ([ B ] ∂ x v∂ x v ) + [Λ s , [ B ] ∂ x v ] ∂ x v. Again, using Cauchy-Schwarz inequality, Corollary 4.13 and the commutator estimate, one has µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , T ] T − ([ B ] ∂ x v∂ x v ) , Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ µ max( (cid:15), β ) C ( (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) [ B ] ∂ x v∂ x v (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ , and µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , [ B ] ∂ x v ] ∂ x v, Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s . Thus we proved | B | ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . In the same way we control B , T [Λ s , T − ([ C ] v∂ x · )] ∂ x v = − [Λ s , T ] T − ([ C ] v∂ x v ) + [Λ s , [ C ] v ] ∂ x v. µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , T ] T − ([ C ] v∂ x v ) , Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ µ max( (cid:15), β ) C ( (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) [ C ] v∂ x v (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ , and µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , [ C ] v ] ∂ x v, Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) v (cid:12)(cid:12) H s . Thus we proved | B | ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . To control B one can check, T [Λ s , T − ([ D ] ∂ x ( ∂ x v · ))] ∂ x v = − [Λ s , T ] T − ([ D ] ∂ x ( ∂ x v∂ x v )) + [Λ s , [ D ] ∂ x v ] ∂ x v + [Λ s , [ D ] ∂ x v ] ∂ x v = − [Λ s , T ] T − ([ D ] ∂ x ( ∂ x v∂ x v )) + [Λ s , ∂ x ([ D ] ∂ x v )] ∂ x v − [Λ s , ∂ x ([ D ]) ∂ x v ] ∂ x v + [Λ s , [ D ] ∂ x v ] ∂ x v = − [Λ s , T ] T − ([ D ] ∂ x ( ∂ x v∂ x v )) + ∂ x ([Λ s , [ D ] ∂ x v ] ∂ x v ) − [Λ s , ∂ x ([ D ]) ∂ x v ] ∂ x v. Again, using Cauchy-Schwarz inequality, Corollary 4.13 and the commutator estimate, one has µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , T ] T − ([ D ] ∂ x ( ∂ x v∂ x v )) , Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ µ max( (cid:15), β ) C ( (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) [ D ] ∂ x ( ∂ x v∂ x v ) (cid:12)(cid:12) H s − (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ , and µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , [ D ] ∂ x v ] ∂ x v, Λ s ∂ x v (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s ) (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ . and µ (cid:12)(cid:12)(cid:12)(cid:0) [Λ s , ∂ x [ D ] ∂ x v ] ∂ x v, Λ s v (cid:1)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +1 ) (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s . Thus we proved | B | ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +2 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . Finally, we turn to B = (cid:15) (cid:16) [Λ s , ςv ] ∂ x v, T Λ s v (cid:17) .From Lemma 4.4, one has | B | ≤ (cid:12)(cid:12) [Λ s , ςv ] ∂ x v (cid:12)(cid:12) H µ (cid:12)(cid:12) Λ s v (cid:12)(cid:12) H µ ≤ (cid:12)(cid:12) [Λ s , ςv ] ∂ x v (cid:12)(cid:12) L (cid:12)(cid:12) Λ s v (cid:12)(cid:12) H µ + √ µ (cid:12)(cid:12) ∂ x (cid:0) [Λ s , ςv ] ∂ x v (cid:1)(cid:12)(cid:12) L (cid:12)(cid:12) Λ s v (cid:12)(cid:12) H µ . Note the identity ∂ x (cid:0) [Λ s , ςv ] ∂ x v (cid:1) = (cid:0) [Λ s , ∂ x ( ςv )] ∂ x v (cid:1) + (cid:0) [Λ s , ςv ] ∂ x v (cid:1) . Thus we proved, | B | ≤ max( (cid:15), β ) C ( (cid:107) U (cid:107) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +1 ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . Altogether, we proved | (cid:0)(cid:2) Λ s , A [ U ] (cid:3) ∂ x U, S [ U ] (cid:3) Λ s U (cid:1) | ≤ max( (cid:15), β ) C (cid:12)(cid:12) U (cid:12)(cid:12) X s . (5.26) • Estimate of (cid:0) Λ s U, (cid:2) ∂ t , Z [ U ] (cid:3) Λ s U (cid:1) . (cid:0) Λ s U, (cid:2) ∂ t , Z [ U ] (cid:3) Λ s U (cid:1) ≡ (Λ s v, (cid:2) ∂ t , T (cid:3) Λ s v ) + (cid:16) Λ s ζ, (cid:104) ∂ t , Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) (cid:105) Λ s ζ (cid:17) = (cid:16) Λ s v, (cid:0) ∂ t q ( (cid:15)ζ, βb ) (cid:1) Λ s v (cid:17) + (cid:16) Λ s v, µ(cid:15)βκ ∂ x ∂ t ζ∂ x b Λ s v (cid:17) − µ (cid:16) Λ s v, ∂ x (cid:0) ν ( ∂ t q ( (cid:15)ζ, βb ))( ∂ x Λ s v ) (cid:1)(cid:17) + (cid:16) Λ s ζ, ∂ t (cid:16) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) (cid:17) Λ s ζ (cid:17) = (cid:15) (cid:16) Λ s v, κ ( ∂ t ζ )Λ s v (cid:17) + µ(cid:15)β (cid:16) Λ s v, κ ∂ x ∂ t ζ∂ x b Λ s v (cid:17) + µ(cid:15) (cid:16) Λ s ∂ x v, νκ ( ∂ t ζ )Λ s ∂ x v (cid:17) + µ(cid:15)β (cid:16) Λ s ∂ x v, νη ( ∂ t ζ ) b Λ s ∂ x v (cid:17) + (cid:16) Λ s ζ, ∂ t (cid:16) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) (cid:17) Λ s ζ (cid:17) = (cid:15) (cid:16) Λ s v, κ ( ∂ t ζ )Λ s v (cid:17) − µ(cid:15)β (cid:16) Λ s v, κ ∂ t ζ∂ x b Λ s ∂ x v (cid:17) − µ(cid:15)β (cid:16) Λ s v, ∂ x κ ∂ t ζ∂ x b Λ s v (cid:17) − µ(cid:15)β (cid:16) Λ s v, κ ∂ t ζ∂ x b Λ s v (cid:17) + µ(cid:15) (cid:16) Λ s ∂ x v, νκ ( ∂ t ζ )Λ s ∂ x v (cid:17) + µ(cid:15)β (cid:16) Λ s ∂ x v, νη ( ∂ t ζ ) b Λ s ∂ x v (cid:17) + (cid:16) Λ s ζ, ∂ t (cid:16) Q ( (cid:15)ζ, βb ) + (cid:15) Q ( (cid:15)ζ, βb, v ) H ( (cid:15)ζ ) (cid:17) Λ s ζ (cid:17) . From Cauchy-Schwarz inequality and since ζ and b satisfies (H1), one deduces (cid:12)(cid:12)(cid:12) (cid:0) Λ s U, (cid:2) ∂ t , Z [ U ] (cid:3) Λ s U (cid:1) (cid:12)(cid:12)(cid:12) ≤ (cid:15)C ( (cid:13)(cid:13) ∂ t U (cid:13)(cid:13) L ∞ , (cid:13)(cid:13) U (cid:13)(cid:13) L ∞ , (cid:13)(cid:13) b (cid:13)(cid:13) W , ∞ ) (cid:12)(cid:12) U (cid:12)(cid:12) X s ≤ max( (cid:15), β ) C ( (cid:13)(cid:13) ∂ t U (cid:13)(cid:13) L ∞ , (cid:13)(cid:13) U (cid:13)(cid:13) L ∞ , (cid:13)(cid:13) b (cid:13)(cid:13) W , ∞ ) (cid:12)(cid:12) U (cid:12)(cid:12) X s . and continuous Sobolev embedding yields, (cid:12)(cid:12)(cid:12) (cid:0) Λ s U, (cid:2) ∂ t , Z [ U ] (cid:3) Λ s U (cid:1) (cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C (cid:0)(cid:13)(cid:13) U (cid:13)(cid:13) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) W , ∞ (cid:1)(cid:12)(cid:12) U (cid:12)(cid:12) X s ≤ max( (cid:15), β ) C (cid:12)(cid:12) U (cid:12)(cid:12) X s . (5.27)One can now conclude the proof of the X s energy estimate. Plugging (5.23), (5.25), (5.26) and (5.27)into (5.22), and making use of the assumption of the Lemma on F.12 ddt E s ( U ) ≤ max( (cid:15), β ) C E s ( U ) + E s ( U ) (cid:0) f ( t ) + max( (cid:15), β ) C (cid:1) , with C = C ( (cid:13)(cid:13) U (cid:13)(cid:13) X sT , (cid:13)(cid:13) b (cid:13)(cid:13) H s +3 , C F ), and consequently ddt E s ( U ) ≤ max( (cid:15), β ) C E s ( U ) + (cid:0) f ( t ) + max( (cid:15), β ) C (cid:1) . Making use of the usual trick, we compute for any λ ∈ R , e max( (cid:15),β ) λt ∂ t ( e − max( (cid:15),β ) λt E s ( U )) = − max( (cid:15), β ) λE s ( U ) + ddt E s ( U ) . Thus with λ = C , one has for all t ∈ [0 , T max( (cid:15), β ) ], ddt ( e − max( (cid:15),β ) λt E s ( U )) ≤ (cid:0) f ( t ) + max( (cid:15), β ) C (cid:1) e − max( (cid:15),β ) λt . Integrating this differential inequality yields, E s ( U )( t ) ≤ e max( (cid:15),β ) λt E s ( U ) + (cid:90) t e max( (cid:15),β ) λ ( t − t (cid:48) ) (cid:0) f ( t (cid:48) ) + max( (cid:15), β ) C (cid:1) dt (cid:48) . (cid:3) .3 Well-posedness of the linearized system Proposition 5.28
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and s ≥ s +1 with s > / , and let U = ( ζ, v ) (cid:62) ∈ X sT (see Definition 5.10), b ∈ H s +3 be such that (H1) , (H2) , and (H3) are satisfied for t ∈ [0 , T / max( (cid:15), β )] ,uniformly with respect to p ∈ P CH . For any U ∈ X s , there exists a unique solution to (5.8) , U p ∈ C ([0 , T / max( (cid:15), β )]; X s ) ∩ C ([0 , T / max( (cid:15), β )]; X s − ) ⊂ X sT ,with λ T , C = C ( (cid:13)(cid:13) U (cid:13)(cid:13) X sT , T, M CH , h − , h − , h − , (cid:107) b (cid:107) H s +3 ) , independent of p ∈ P CH , such that one hasthe energy estimates ∀ ≤ t ≤ T max( (cid:15), β ) , E s ( U p )( t ) ≤ e max( (cid:15),β ) λ T t E s ( U ) + max( (cid:15), β ) C (cid:90) t e max( (cid:15),β ) λ T ( t − t (cid:48) ) dt (cid:48) and E s − ( ∂ t U p ) ≤ C e max( (cid:15),β ) λ T t E s ( U ) + max( (cid:15), β ) C (cid:90) t e max( (cid:15),β ) λ T ( t − t (cid:48) ) dt (cid:48) + max( (cid:15), β ) C . Proof . We omit here the proof of Proposition 5.28 since it can be done using the same techniques as inthe proof of [16, Proposition 2]. (cid:3)
In what follows, we consider two nonlinear systems with different initial data and right-hand sides andcontrol the difference between their solutions. This estimate is a key ingredient to the proof of thestability result.
Proposition 5.29
Let ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and s ≥ s + 1 , s > / , and assume that there exists U i for i ∈ { , } , such that U i = ( ζ i , v i ) (cid:62) ∈ X sT , U ∈ L ∞ ([0 , T / max( (cid:15), β )]; X s +1 ) , b ∈ H s +3 , U satisfy (H1) , (H2) and (H3) on [0 , T / max( (cid:15), β )] , with h , h , h > , and U i satisfy ∂ t U + A [ U ] ∂ x U + B [ U ] = F ,∂ t U + A [ U ] ∂ x U + B [ U ] = F , with F i ∈ L ([0 , T / max( (cid:15), β )]; X s ) .Then there exists constants C = C ( M CH , h − , h − , h − , max( (cid:15), β ) (cid:12)(cid:12) U (cid:12)(cid:12) X s , max( (cid:15), β ) (cid:12)(cid:12) U (cid:12)(cid:12) X s , | b | H s +3 ) and λ T = (cid:0) C × C ( | U | L ∞ ([0 ,T/ max( (cid:15),β )]; X s +1 ) ) + C (cid:1) such that for all t ∈ [0 , T max( (cid:15), β ) ] , E s ( U − U )( t ) ≤ e max( (cid:15),β ) λ T t E s ( U | t =0 − U | t =0 ) + C (cid:90) t e max( (cid:15),β ) λ T ( t − t (cid:48) ) E s ( F − F )( t (cid:48) ) dt (cid:48) . Proof .When multiplying the equations satisfied by U i on the left by Z [ U i ], one obtains Z [ U ] ∂ t U + Σ[ U ] ∂ x U + Z [ U ] B [ U ] = Z [ U ] F Z [ U ] ∂ t U + Σ[ U ] ∂ x U + Z [ U ] B [ U ] = Z [ U ] F ;with Σ[ U ] = Z [ U ] A [ U ]. Subtracting the two equations above, and defining V = U − U ≡ ( ζ, v ) (cid:62) oneobtains Z [ U ] ∂ t V + Σ[ U ] ∂ x V + ( Z [ U ] B [ U ] − Z [ U ] B [ U ]) = Z [ U ]( F − F ) − (Σ[ U ] − Σ[ U ) ∂ x U − ( Z [ U ] − Z [ U ])( ∂ t U − F ) . We then apply Z − [ U ] and deduce the following system satisfied by V : (cid:26) ∂ t V + A [ U ] ∂ x V + Z − [ U ] (cid:0) Z [ U ] B [ U ] − Z [ U ] B [ U ] (cid:1) = FV (0) = ( U − U ) | t =0 , (5.30)where, F ≡ F − F − Z − [ U ] (cid:0) Σ[ U ] − Σ[ U (cid:1) ∂ x U − Z − [ U ] (cid:0) Z [ U ] − Z [ U ] (cid:1) ( ∂ t U − F ) . (5.31)23e wish to use the energy estimate of Lemma 5.18 to the linear system (5.30).The additional term now is Z − [ U ] (cid:0) Z [ U ] B [ U ] − Z [ U ] B [ U ] (cid:1) .So we have to control, (cid:0) Λ s Z − [ U ] (cid:0) Z [ U ] B [ U ] − Z [ U ] B [ U ] (cid:1) , Z [ U ]Λ s V (cid:1) = B. One has, B = (cid:0) Λ s (cid:0) Z [ U ] B [ U ] − Z [ U ] B [ U ] (cid:1) , Λ s V (cid:1) + (cid:0)(cid:2) Λ s , Z − [ U ] (cid:3) Z [ U ] B [ U ] − Z [ U ] B [ U ] , Z [ U ]Λ s V (cid:1) = B + B Now we have to estimate the terms ( B ) and ( B ).( B ) = (cid:16) Λ s (cid:0) − Q ( (cid:15)ζ , βb ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) + Q ( (cid:15)ζ , βb ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) (cid:1) , Λ s ζ v (cid:17) + (cid:16) Λ s (cid:0) − (cid:15) Q ( (cid:15)ζ , βb, v ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) + (cid:15) Q ( (cid:15)ζ , βb, v ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) (cid:1) , Λ s ζ v (cid:17) + (cid:16) Λ s (cid:0) (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) − (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) (cid:1) , Λ s v (cid:17) + (cid:16) Λ s (cid:0) − µ [ E ] v + µ [ E ] v (cid:1) , Λ s v (cid:17) In order to control ( B ) we use the following decompositions, • (cid:16) − Q ( (cid:15)ζ , βb ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) + Q ( (cid:15)ζ , βb ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) (cid:17) = (cid:16) − Q ( (cid:15)ζ , βb ) G ( (cid:15)ζ ) H ( (cid:15)ζ ) + Q ( (cid:15)ζ , βb ) G ( (cid:15)ζ ) H ( (cid:15)ζ ) (cid:17) ( β∂ x bv ) − β ( v − v ) Q ( (cid:15)ζ , βb ) G ( (cid:15)ζ ) ∂ x bH ( (cid:15)ζ ) . • β (cid:16) − (cid:15) Q ( (cid:15)ζ , βb, v ) ∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) + (cid:15) Q ( (cid:15)ζ , βb, v ) ∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) (cid:17) = (cid:16) − (cid:15) Q ( (cid:15)ζ , βb, v ) G ( (cid:15)ζ ) H ( (cid:15)ζ ) + (cid:15) Q ( (cid:15)ζ , βb, v ) G ( (cid:15)ζ ) H ( (cid:15)ζ ) (cid:17) ( β∂ x bv ) − β ( v − v ) (cid:15) Q ( (cid:15)ζ , βb, v ) G ( (cid:15)ζ ) ∂ x bH ( (cid:15)ζ ) . • (cid:16) (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) − (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) (cid:17) = (cid:16) γq ( (cid:15)ζ , βb ) h ( h + h )( h + γh ) − γq ( (cid:15)ζ , βb ) h ( h + h )( h + γh ) (cid:17) ( (cid:15)β∂ x bv )+ (cid:16) (cid:15)γq ( (cid:15)ζ , βb ) h ( h + h ) ∂ x b ( h + γh ) (cid:17) β ( v − v ) . • µ (cid:16) − [ E ] v + [ E ] v (cid:17) = µ [ E ]( − v + v ) = µ [ E ]( v − v )( v + v ) . Using the fact that, (cid:15) Q ( (cid:15)ζ i , βb, v i ) = Q ( (cid:15)ζ i , βb, (cid:15)v i ), one deduces,24 B | ≤ C ( β | v | H s , | b | H s +2 ) (cid:15) | ζ − ζ | H s | ζ v | H s + C ( (cid:15) | ζ | H s , | b | H s +2 ) β | v − v | H s | ζ v | H s + C ( β | v | H s , | b | H s +1 ) (cid:15) | ζ − ζ | H s | ζ v | H s + C ( (cid:15) | ζ | H s , (cid:15) | v | H s , | b | H s +1 ) β | v − v | H s | ζ v | H s + C ( β | v | H s , (cid:15) | v | H s , | b | H s +1 ) (cid:15) | ζ − ζ | H s | v | H s + C ( (cid:15) | ζ | H s , (cid:15) | v | H s , (cid:15) | v | H s , | b | H s +1 ) β | v − v | H s | v | H s + C ( (cid:15) | v | H s , (cid:15) | v | H s , | b | H s +3 ) β | v − v | H s | v | H s . ≤ max( (cid:15), β ) C E s ( U − U ) E s ( V ) . ≤ max( (cid:15), β ) C E s ( V ) . with C = C ( M CH , h − , h − , max( (cid:15), β ) (cid:12)(cid:12) U (cid:12)(cid:12) X s , max( (cid:15), β ) (cid:12)(cid:12) U (cid:12)(cid:12) X s , | b | H s +3 ).With Q ( (cid:15)ζ i , βb, v i ) = Q ( (cid:15)ζ i , βb ) + (cid:15) Q ( (cid:15)ζ i , βb, v i ) for i = 1 , B ) is immediately bounded using Lemma 5.13: | B | = (cid:0) (cid:2) Λ s , Z − [ U ] (cid:3)(cid:0) Z [ U ] B [ U ] − Z [ U ] B [ U ] (cid:1) , Z [ U ]Λ s V (cid:1) ≤ C | Z [ U ] B [ U ] − Z [ U ] B [ U ] | H s − × H s − | V | X s ≤ C (cid:16)(cid:12)(cid:12)(cid:12) − Q ( (cid:15)ζ , βb, v ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) + Q ( (cid:15)ζ , βb, v ) β∂ x bG ( (cid:15)ζ ) v H ( (cid:15)ζ ) (cid:12)(cid:12)(cid:12) H s − + (cid:12)(cid:12)(cid:12) (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) − µ [ E ] v − (cid:15)γβq ( (cid:15)ζ , βb ) h ( h + h ) v ∂ x b ( h + γh ) + µ [ E ] v (cid:12)(cid:12)(cid:12) H s − (cid:17) | V | X s ≤ max( (cid:15), β ) C E s ( U − U ) E s ( V ) . ≤ max( (cid:15), β ) C E s ( V ) . So we have, | B | ≤ C max( (cid:15), β ) E s ( V ) . Now, in order to control the right hand side F we take advantage of the Lemma below. Lemma 5.32
Let ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and s ≥ s > / . Let V = ( ζ v , v ) (cid:62) , W = ( ζ w , w ) (cid:62) ∈ X s and U = ( ζ , v ) (cid:62) , U = ( ζ , v ) (cid:62) ∈ X s , b ∈ H s +2 such that there exists h > with − (cid:15)ζ ≥ h > , − (cid:15)ζ ≥ h > , δ + (cid:15)ζ − βb ≥ h > , δ + (cid:15)ζ − βb ≥ h > . Then one has (cid:12)(cid:12)(cid:12) (cid:16) Λ s (cid:0) Z [ U ] − Z [ U ] (cid:1) V , W (cid:17) (cid:12)(cid:12)(cid:12) ≤ (cid:15) C (cid:12)(cid:12) U − U (cid:12)(cid:12) X s (cid:12)(cid:12) V (cid:12)(cid:12) X s (cid:12)(cid:12) W (cid:12)(cid:12) X (5.33) (cid:16) Λ s (cid:0) Z [ U ] A [ U ] − Z [ U ] A [ U ] (cid:1) V , W (cid:17) ≤ (cid:15) C (cid:12)(cid:12) U − U (cid:12)(cid:12) X s (cid:12)(cid:12) V (cid:12)(cid:12) X s (cid:12)(cid:12) W (cid:12)(cid:12) X (5.34) with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β | b | H s +2 ) .Proof .Let V = ( ζ v , v ) (cid:62) , W = ( ζ w , w ) (cid:62) ∈ X and U = ( ζ , v ) (cid:62) , U = ( ζ , v ) (cid:62) ∈ X s . By definition of Z [ · ](see (5.9)), one has (cid:16) Λ s (cid:0) Z [ U ] − Z [ U ] (cid:1) V, W (cid:17) = (cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb ) + (cid:15) Q ( (cid:15)ζ , βb, v ) H ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb ) + (cid:15) Q ( (cid:15)ζ , βb, v ) H ( (cid:15)ζ ) (cid:1) ζ v , ζ w (cid:17) + (cid:16) Λ s (cid:0) T [ (cid:15)ζ , βb ] − T [ (cid:15)ζ , βb ] (cid:1) v, w (cid:17) . Now, one can check that T [ (cid:15)ζ , βb ] v − T [ (cid:15)ζ , βb ] v = (cid:0) q ( (cid:15)ζ , βb ) − q ( (cid:15)ζ , βb ) (cid:1) v + µ(cid:15)βκ ∂ x b∂ x (cid:0) ζ − ζ (cid:1) v − µ∂ x (cid:8)(cid:0) νq ( (cid:15)ζ , βb ) − νq ( (cid:15)ζ , βb ) (cid:1) ∂ x v (cid:9) = (cid:15)κ ( ζ − ζ ) v + µ(cid:15)βκ ∂ x b∂ x ( ζ − ζ ) v − µ(cid:15)∂ x (cid:8) νκ ( ζ − ζ ) ∂ x v (cid:9) − µ(cid:15)β∂ x (cid:8) νη ( ζ − ζ ) b∂ x v (cid:9) , κ ∂ x ( ζ − ζ ) ∂ x bv = ∂ x ( κ v ( ζ − ζ ) ∂ x b ) − ∂ x ( κ v )( ζ − ζ ) ∂ x b − κ v ( ζ − ζ ) ∂ x b ,so that, after one integration by part, and using Cauchy-Schwarz inequality and the product estimatein Sobolev spaces (see [1, 13, 14]), one has (cid:12)(cid:12)(cid:12) (cid:16) Λ s (cid:0) T [ (cid:15)ζ , βb ] − T [ (cid:15)ζ , βb ] (cid:1) v , w (cid:17) (cid:12)(cid:12)(cid:12) ≤ (cid:15) C ( β | b | H s +2 ) (cid:12)(cid:12) ζ − ζ (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) w (cid:12)(cid:12) H µ . (5.35)Now, applying Cauchy-Schwarz inequality, one has (again thanks to continuous Sobolev embeddingfor s ≥ s > / (cid:12)(cid:12)(cid:12)(cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) (cid:1) ζ v , ζ w (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:15) C (cid:12)(cid:12) ζ − ζ (cid:12)(cid:12) H s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L , (5.36)and since, (cid:15) Q ( (cid:15)ζ i , βb, v i ) = Q ( (cid:15)ζ i , βb, (cid:15)v i ) for ( i = 1 , (cid:12)(cid:12)(cid:12)(cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) (cid:1) ζ v , ζ w (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:15) C (cid:12)(cid:12) U − U (cid:12)(cid:12) X s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L , (5.37)with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ). Estimates (5.35), (5.36) and (5.37) yield (5.33).Let us now turn to (5.34). One has (cid:16) Λ s (cid:0) Z [ U ] A [ U ] − Z [ U ] A [ U ] (cid:1) V , W (cid:17) (5.38)= (cid:15) (cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v − Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v (cid:1) ζ v , ζ w (cid:17) + (cid:15) (cid:16) Λ s (cid:16) Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v − Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v (cid:17) ζ v , ζ w (cid:17) + (cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb ) − Q ( (cid:15)ζ , βb ) (cid:1) v , ζ w (cid:17) + (cid:16) Λ s (cid:0) (cid:15) Q ( (cid:15)ζ , βb, v ) − (cid:15) Q ( (cid:15)ζ , βb, v ) (cid:1) v , ζ w (cid:17) + (cid:16) Λ s (cid:0) Q ( (cid:15)ζ , βb ) − Q ( (cid:15)ζ , βb ) (cid:1) ζ v , w (cid:17) + (cid:16) Λ s (cid:0) (cid:15) Q ( (cid:15)ζ , βb, v ) − (cid:15) Q ( (cid:15)ζ , βb, v ) (cid:1) ζ v , w (cid:17) + (cid:16) Λ s (cid:0) Q [ (cid:15)ζ , βb, v ] − Q [ (cid:15)ζ , βb, v ] (cid:1) v , w (cid:17) + (cid:15) (cid:16) Λ s (cid:0) T [ (cid:15)ζ , βb ]( ςv v ) − T [ (cid:15)ζ , βb ]( ςv v ) (cid:1) , w (cid:17) = ( I ) + ( II ) + ( III ) + ( IV ) + ( V ) + ( V I ) + (
V II ) + (
V III ) . (5.39)( III ) and ( V ) may be estimated exactly as in (5.36), the ( IV ) and ( V I ) terms may be estimated exactlyas in (5.37) , and we do not detail the precise calculations. The ( I ) and ( II ) terms follow in the sameway, using the decompositions below, (cid:15) (cid:16) Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v − Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v (cid:17) = (cid:16) Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) (cid:17) ( (cid:15)v )+ (cid:15) ( v − v ) Q ( (cid:15)ζ , βb ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) , and, (cid:15) (cid:16) Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v − Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) v (cid:17) = (cid:16) Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) (cid:17) ( (cid:15)v ) + (cid:15) ( v − v ) Q ( (cid:15)ζ , βb, (cid:15)v ) H ( (cid:15)ζ ) H (cid:48) ( (cid:15)ζ ) ,
26o that one has (cid:12)(cid:12) ( I ) (cid:12)(cid:12) ≤ C ( (cid:15) (cid:12)(cid:12) v (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:15) (cid:12)(cid:12) ζ − ζ (cid:12)(cid:12) H s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L + C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:15) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L , and, (cid:12)(cid:12) ( II ) (cid:12)(cid:12) ≤ C ( (cid:15) (cid:12)(cid:12) v (cid:12)(cid:12) H s ) (cid:15) (cid:12)(cid:12) U − U (cid:12)(cid:12) X s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L + C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , (cid:15) (cid:12)(cid:12) v (cid:12)(cid:12) H s +2 ) (cid:15) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ v (cid:12)(cid:12) H s (cid:12)(cid:12) ζ w (cid:12)(cid:12) L . Let us detail now (
V II ). One has (cid:0) Q [ (cid:15)ζ , βb, v ] − Q [ (cid:15)ζ , βb, v ] (cid:1) v = (cid:15) (cid:16) q ( (cid:15)ζ , βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v − q ( (cid:15)ζ , βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v (cid:17) v − µ [ A ]( v − v ) v − µ [ B ] ∂ x ( v − v ) v − µ [ C ]( v − v ) ∂ x v − µ [ D ] ∂ x (cid:0) ∂ x ( v − v ) v (cid:1) = (cid:15) (cid:16) q ( (cid:15)ζ , βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v − q ( (cid:15)ζ , βb )( H (cid:48) ( (cid:15)ζ ) − ς ) v (cid:17) v − µ [ A ]( v − v ) v − µ [ B ] ∂ x ( v − v ) v − µ [ C ]( v − v ) ∂ x v − µ∂ x (cid:0) [ D ]( ∂ x ( v − v ) v ) (cid:1) + µ∂ x (cid:0) [ D ] (cid:1) ( ∂ x ( v − v ) v ) . (5.40)Again, the contribution of the first term in (5.40) is estimated as above (recalling that this term ismultiplied by a (cid:15) - factor), and the contributions of the other terms in (5.40) are easily estimated, so onehas, (cid:12)(cid:12) ( V II ) (cid:12)(cid:12) ≤ C ( (cid:15) (cid:12)(cid:12) v (cid:12)(cid:12) H s ) (cid:15) (cid:12)(cid:12) ζ − ζ (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) L + C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s ) (cid:15) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) L + C ( β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:15) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) w (cid:12)(cid:12) H µ . We conclude by estimating (
V III ). One has T [ (cid:15)ζ , βb ]( ςv v ) − T [ (cid:15)ζ , βb ]( ςv v ) = (cid:0) q ( (cid:15)ζ , βb ) v − q ( (cid:15)ζ , βb ) v (cid:1) ςv + µ(cid:15)βκ ∂ x b ( ∂ x ζ v − ∂ x ζ v ) ςv − µ∂ x (cid:8) ν (cid:0) q ( (cid:15)ζ , βb ) ∂ x ( ςv v ) − q ( (cid:15)ζ , βb ) ∂ x ( ςv v ) (cid:9) = (cid:0) T [ (cid:15)ζ , βb ] − T [ (cid:15)ζ , βb ] (cid:1) ( ςv v ) + ( v − v ) (cid:0) q ( (cid:15)ζ , βb ) ςv (cid:1) + ( v − v ) µ(cid:15)βκ ∂ x b∂ x ζ ςv − µ∂ x (cid:8) νq ( (cid:15)ζ , βb ) ∂ x (cid:0) ς ( v − v ) v (cid:1)(cid:9) = (cid:0) T [ (cid:15)ζ , βb ] − T [ (cid:15)ζ , βb ] (cid:1) ( ςv v ) + ( v − v ) (cid:0) q ( (cid:15)ζ , βb ) ςv (cid:1) + µ(cid:15)β∂ x (( v − v ) κ ∂ x bζ ςv ) − µ(cid:15)β∂ x ( v − v ) κ ∂ x bζ ςv − µ(cid:15)β ( v − v ) ∂ x bκ ζ ςv − µ(cid:15)β ( v − v ) ∂ x bζ ∂ x ( κ ςv ) − µ∂ x (cid:8) νq ( (cid:15)ζ , βb ) ∂ x (cid:0) ς ( v − v ) v (cid:1)(cid:9) . One finally uses Cauchy-Schwarz inequality, the product estimate in Sobolev spaces, as well as (5.35),and obtain (cid:12)(cid:12) ( V III ) (cid:12)(cid:12) ≤ (cid:15) C ( β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) ζ − ζ (cid:12)(cid:12) H s (cid:12)(cid:12) ςv v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) w (cid:12)(cid:12) H µ + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) ςv (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) L + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +1 ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) ςv (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) H µ + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +1 ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) ςv (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) L + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s (cid:12)(cid:12) w (cid:12)(cid:12) L + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +1 ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) w (cid:12)(cid:12) L + (cid:15) C ( (cid:15) (cid:12)(cid:12) ζ (cid:12)(cid:12) H s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s ) (cid:12)(cid:12) v − v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) v (cid:12)(cid:12) H s +1 µ (cid:12)(cid:12) w (cid:12)(cid:12) H µ . Altogether, we obtain (5.34), and the Lemma is proved. (cid:3)
Let us continue the proof of Proposition 5.29, by estimating F defined in (5.31).27ore precisely we want to estimate (cid:0) Λ s F , Z [ U ]Λ s V (cid:1) = (cid:0) Λ s F − Λ s F , Z [ U ]Λ s V (cid:1) − (cid:0) Λ s (Σ[ U ] − Σ[ U ]) ∂ x U , Λ s V (cid:1) − (cid:0) (cid:2) Λ s , Z − [ U ] (cid:3) (Σ[ U ]] − Σ[ U ]) ∂ x U , Z [ U ]Λ s V (cid:1) − (cid:0) Λ s (cid:0) Z [ U ] − Z [ U ] (cid:1) ( ∂ t U − F ) , Λ s V (cid:1) − (cid:0) (cid:2) Λ s , Z − [ U ] (cid:3) ( Z [ U ] − Z [ U ])( ∂ t U − F ) , Z [ U ]Λ s V (cid:1) . Let us estimate each of these terms. The first term is immediately bounded using Lemma 5.13: (cid:12)(cid:12) (cid:0) Λ s F − Λ s F , Z [ U ]Λ s V (cid:1) (cid:12)(cid:12) ≤ C | F − F | X s | V | X s , (5.41)with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) W , ∞ , β (cid:12)(cid:12) b (cid:12)(cid:12) W , ∞ ) ..The contributions of the second and fourth terms follow from Lemma 5.32. Indeed, recalling that V ≡ U − U , (5.33) yields immediately (cid:12)(cid:12) (cid:0) Λ s (cid:0) Z [ U ] − Z [ U ] (cid:1) ( ∂ t U − F ) , Λ s V (cid:1) (cid:12)(cid:12) ≤ C(cid:15) | ∂ t U − F | X s | V | X s , (5.42)and (5.34) yields (cid:12)(cid:12) (cid:0) Λ s (Σ[ U ] − Σ[ U ]) ∂ x U , Λ s V (cid:1) (cid:12)(cid:12) ≤ C(cid:15) | ∂ x U | X s | V | X s , (5.43)with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β | b | H s +2 ).Finally, we control the third and fifth terms using Lemma 5.13, (5.16): (cid:12)(cid:12)(cid:0) (cid:2) Λ s , Z − [ U ] (cid:3) U , Z [ U ]Λ s V (cid:1)(cid:12)(cid:12) ≤ C | U | H s − × H s − | V | X s , with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ).Thus it remains to estimate | U | H s − × H s − , where U ≡ U ( i ) ≡ (Σ[ U ] − Σ[ U ]) ∂ x U or U ≡ U ( ii ) ≡ ( Z [ U ] − Z [ U ])( ∂ t U − F ).We proceed as in Lemma 5.32, helped by the fact that one is allowed lose one derivative in our es-timates.Let W ≡ ∂ t U − F ≡ ( ζ w , w ) (cid:62) . One has U ( ii ) ≡ ( Z [ U ] − Z [ U ]) W ≡ (cid:16) Q ( (cid:15)ζ , βb ) + (cid:15) Q ( (cid:15)ζ , βb, v ) H ( (cid:15)ζ ) − Q ( (cid:15)ζ , βb ) + (cid:15) Q ( (cid:15)ζ , βb, v ) H ( (cid:15)ζ ) (cid:17) ζ w (cid:16) T [ (cid:15)ζ , βb ] − T [ (cid:15)ζ , βb ] (cid:17) w ≡ (cid:18) ζ ( ii ) u ( ii ) (cid:19) . Recall that T [ (cid:15)ζ , βb ] w − T [ (cid:15)ζ , βb ] w = (cid:15) (cid:16) κ ( ζ − ζ ) w + µβκ ∂ x b∂ x ( ζ − ζ ) w − µ∂ x (cid:8) νκ ( ζ − ζ ) ∂ x w (cid:9)(cid:17) = (cid:15) (cid:16) κ ( ζ − ζ ) w + µβ∂ x ( κ ∂ x b ( ζ − ζ ) w ) − µβ∂ x ( κ ∂ x bw )( ζ − ζ ) − µ∂ x (cid:8) νκ ( ζ − ζ ) ∂ x w (cid:9) − µ∂ x (cid:8) νη ( ζ − ζ ) b∂ x w (cid:9)(cid:17) , so that one has straightforwardly | u ( ii ) | H s − ≤ (cid:15)C ( β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ) | ζ − ζ | H s | w | H s +1 µ . As for the first component, we apply (5.36), (5.37) and deduce | ζ ( ii ) | H s − = (cid:0) Λ s − ζ ( ii ) , Λ s − ζ ( ii ) (cid:1) ≤ (cid:15)C | U − U | X s − | ζ w | H s − | ζ ( ii ) | H s − .
28t follows (cid:12)(cid:12) (cid:0) (cid:2) Λ s , Z − [ U ] (cid:3) ( Z [ U ] − Z [ U ])( ∂ t U − F ) , Z [ U ]Λ s V (cid:1) (cid:12)(cid:12) ≤ C(cid:15) | ∂ t U − F | X s | V | X s , (5.44)with C = C ( M CH , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β | b | H s +2 ).Now, recall U ( i ) ≡ (Σ[ U ] − Σ[ U ]) ∂ x U . Proceeding as above, one obtains (cid:12)(cid:12) U ( i ) (cid:12)(cid:12) H s − × H s − ≤ (cid:15)C | ∂ x U | X s | V | X s , and thus (cid:12)(cid:12) (cid:0) (cid:2) Λ s , Z − [ U ] (cid:3) (Σ[ U ] − Σ[ U ]) ∂ x U , Z [ U ]Λ s V (cid:1) (cid:12)(cid:12) ≤ C(cid:15) | ∂ x U | X s | V | X s . (5.45)Altogether, we proved (using Lemma 5.12) that F , as defined in (5.31), satisfies (cid:12)(cid:12)(cid:12)(cid:16) Λ s F, Z [ U ]Λ s V (cid:17)(cid:12)(cid:12)(cid:12) ≤ C ( | ∂ x U | X s + | ∂ t U − F | X s ) (cid:15)E s ( V ) + CE s ( V ) E s ( F − F ) . (5.46)with with C = C ( M CH , h − , h − , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , (cid:15) (cid:12)(cid:12) U (cid:12)(cid:12) X s , β (cid:12)(cid:12) b (cid:12)(cid:12) H s +2 ).Notice also that by the system satisfied by U , one has (see detailed calculations in the proof of [16,Proposition 2]) (cid:12)(cid:12) ∂ t U − F (cid:12)(cid:12) X s ≡ − (cid:12)(cid:12) ( A [ U ] + A [ U ]) ∂ x U + B [ U ] (cid:12)(cid:12) X s ≤ C ( | U | X s +1 ) + C (cid:12)(cid:12)(cid:12)(cid:16) Λ s F, Z [ U ]Λ s V (cid:17)(cid:12)(cid:12)(cid:12) ≤ max( (cid:15), β ) C × ( | ∂ x U | X s + | ∂ t U − F | X s ) E s ( V ) + C E s ( V ) E s ( F − F ) . We can now conclude by Lemma 5.18, and the proof of Proposition 5.29 is complete. (cid:3)
We start by defining the full justification terminology (initiated in [15]). An asymptotic model is fullyjustified as an approximation of the full Euler system if the asymptotic model is consistent with the fullEuler system, if both exact and approximate models are well-posed for the same class of initial data, andif their solutions remain close uniformly with respect to time. In what follows, we will fully justify thenew Green-Naghdi model (3.19) derived in a previous section. To this end, we will state the three mainingredients of full justification. For the sake of simplicity, we do not state the proofs of these results. Infact, following the same techniques as in [16] adjusted to our new model, the following results can beproved without any difficulties.
Proposition 6.1 (Consistency)
For p = ( µ, (cid:15), δ, γ, β, bo) ∈ P SW , let U p = ( ζ p , ψ p ) (cid:62) be a family of so-lutions of the full Euler system (2.2) such that there exists T > , s ≥ s + 1 , s > / for which ( ζ p , ∂ x ψ p ) (cid:62) is bounded in L ∞ ([0 , T ); H s + N ) with sufficiently large N , and uniformly with respect to p ∈ P SW . Moreover assume that b ∈ H s + N satisfy (H0) and there exists h > such that h = 1 − (cid:15)ζ p ≥ h > , h = 1 δ + (cid:15)ζ p − βb ≥ h > . (H1) Define v p as in (3.1) . Then ( ζ p , v p ) (cid:62) satisfies (3.19) up to a remainder term, R = (0 , r ) (cid:62) , boundedby (cid:107) r (cid:107) L ∞ ([0 ,T ); H s ) ≤ ( µ + µ(cid:15) ) C, with C = C ( h − , M SW , | b | H s + N , (cid:107) ( ζ p , ∂ x ψ p ) (cid:62) (cid:107) L ∞ ([0 ,T ); H s + N ) ) . Theorem 6.2 (Existence and uniqueness)
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and s ≥ s + 1 , s > / ,and assume U = ( ζ , v ) (cid:62) ∈ X s , b ∈ H s +3 satisfies (H0) , (H1) , (H2) , and (H3) . Then there exists amaximal time T max > , uniformly bounded from below with respect to p ∈ P CH , such that the system of quations (3.19) admits a unique strong solution U = ( ζ, v ) (cid:62) ∈ C ([0 , T max ); X s ) ∩ C ([0 , T max ); X s − ) with the initial value ( ζ, v ) | t =0 = ( ζ , v ) , and preserving the conditions (H1) , (H2) and (H3) (with dif-ferent lower bounds) for any t ∈ [0 , T max ) .Moreover, there exists λ, C = C ( h − , h − , h − , M CH , T, (cid:12)(cid:12) U (cid:12)(cid:12) X s , | b | H s +3 ) , independent of p ∈ P CH , suchthat T max ≥ T / max( (cid:15), β ) , and one has the energy estimates ∀ ≤ t ≤ T max( (cid:15), β ) , (cid:12)(cid:12) U ( t, · ) (cid:12)(cid:12) X s + (cid:12)(cid:12) ∂ t U ( t, · ) (cid:12)(cid:12) X s − ≤ C e max( (cid:15),β ) λt + max( (cid:15), β ) C (cid:90) t e max( (cid:15),β ) λ ( t − t (cid:48) ) dt (cid:48) + max( (cid:15), β ) C If T max < ∞ , one has | U ( t, · ) | X s −→ ∞ as t −→ T max , or one of the conditions (H1) , (H2) , (H3) ceases to be true as t −→ T max . Theorem 6.3 (Stability)
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH and s ≥ s + 1 with s > / , and assume U , = ( ζ , , v , ) (cid:62) ∈ X s , U , = ( ζ , , v , ) (cid:62) ∈ X s +1 , and b ∈ H s +3 satisfies (H0) , (H1) , (H2) , and (H3) . Denote U j the solution to (3.19) with U j | t =0 = U ,j .Then there exists T, λ, C = C ( M CH , h − , h − , h − , (cid:12)(cid:12) U , (cid:12)(cid:12) X s , | U , | X s +1 , | b | H s +3 ) such that ∀ t ∈ [0 , T max( (cid:15), β ) ] , (cid:12)(cid:12) ( U − U )( t, · ) (cid:12)(cid:12) X s ≤ C e max( (cid:15),β ) λt (cid:12)(cid:12) U , − U , (cid:12)(cid:12) X s . Theorem 6.4 (Convergence)
Let p = ( µ, (cid:15), δ, γ, β, bo) ∈ P CH (see (2.8) ) and s ≥ s + 1 with s > / ,and let U ≡ ( ζ , ψ ) (cid:62) ∈ H s + N ( R ) , b ∈ H s + N satisfy the hypotheses (H0) , (H1) , (H2) , and (H3) , with N sufficiently large. We suppose U ≡ ( ζ, ψ ) (cid:62) a unique solution to the full Euler system (2.2) with initialdata ( ζ , ψ ) (cid:62) , defined on [0 , T ] for T > ,and we suppose that U ≡ ( ζ, ψ ) (cid:62) satisfies the assumptionsof our consistency result, Proposition (6.1) . Then there exists C, T > , independent of p , such that • There exists a unique solution U a ≡ ( ζ a , v a ) (cid:62) to our new model (3.19) , defined on [0 , T ] and withinitial data ( ζ , v ) (cid:62) (provided by Theorem 6.2); • With v , defined as in (3.1) ,one has for any t ∈ [0 , T ] , (cid:12)(cid:12) ( ζ, v ) − ( ζ a , v a ) (cid:12)(cid:12) L ∞ ([0 ,t ]; X s ) ≤ C µ t. We conclude with a detailed statement of the functions A , B , C , D , E , and F given in Section (3.1). To our knowledge, the local well-posedness of the full Euler system in the two-fluid configuration over a variabletopography seems to be an open problem. Functions A = 2 (cid:15)β (1 + βw b ) ∂ x bν (cid:48) ( b ) ∂ x ( f ( b ) − γg ( b ) ) + 2 (cid:15)β ν∂ x bw (cid:48) b∂ x ( f ( b ) − γg ( b ) )+ 2 (cid:15)βw ν∂ x b∂ x ( f ( b ) − γg ( b ) ) + 3 (cid:15)βw νb∂ x ( f ( b ) − γg ( b ) ) − (cid:15) bo ∂ x ( f ( b ) − γg ( b ) ) − (cid:15)βw b [ λ ( b )] ∂ x ( f ( b ) − γg ( b ) )+ (cid:15) (1 + βw b ) θ [2 β∂ x ( ∂ x bg (cid:48) ( b )) + ∂ x g ( b )] + (cid:15) (1 + βw b )[2 θ + ( γ − g ( b )] ∂ x ( β∂ x bg (cid:48) ( b ))+ (cid:15)β (1 + βw b ) α ( f ( b ) − γg ( b ) ) ∂ x b + 4 (cid:15)β (1 + βw b ) α∂ x ( f ( b ) − γg ( b ) ) ∂ x b + 3 (cid:15)β (1 + βw b )[ γ f ( b ) + 23 δ − f ( b )] ∂ x ( f ( b ) − γg ( b ) ) b + (cid:15)β (1 + βw b )( θ − α ) g ( b ) ∂ x b + (cid:15)β (1 + βw b )(2 θ − α ) β ( ∂ x b ) g (cid:48) ( b )+ (cid:15)β (1 + βw b )[2 θ − α + 13 ( δ − − βb ) f (cid:48) ( b ) − γ g (cid:48) ( b )]( β∂ x bg (cid:48) ( b ) + ∂ x ( g ( b ))) ∂ x b + (cid:15)β (1 + βw b ) η ( b )( ∂ x b ) ( f ( b ) − γg ( b ) ) + (cid:15)β (1 + βw b ) γ f (cid:48) ( b ) b∂ x b ( f ( b ) − γg ( b ) )+ 2 (cid:15)β (1 + βw b )[ 2 γ f (cid:48) ( b )] b∂ x b∂ x ( f ( b ) − γg ( b ) ) − (cid:15)β (1 + βw b ) f ( b ) ∂ x ( f ( b ) − γg ( b ) )+ (cid:15)β (1 + βw b ) η ( b )( ∂ x b ) g ( b ) + (cid:15)β (1 + βw b )[ γ f (cid:48)(cid:48) ( b )]( ∂ x b ) b ( f ( b ) − γg ( b ) ) − β∂ x bν (cid:48) ( b )(1 + βw b ) (cid:15)∂ x ( ς ) − (cid:15)ν (1 + βw b ) ∂ x ( ς ) − νβ [ ∂ x ( w ) b + w ∂ x b ] (cid:15)∂ x ( ς ) + (cid:15) (1 + βw b )[2 s ( b ) + ∂ x ( t ( b ))] . (A.1) B = (cid:15)β∂ x bν (cid:48) ( b )(1 + βw b )( f ( b ) − γg ( b ) ) + (cid:15)β ν∂ x bw (cid:48) ( b ) b ( f ( b ) − γg ( b ) )+ (cid:15)βνw ∂ x b ( f ( b ) − γg ( b ) ) + 3 (cid:15)βw bν∂ x ( f ( b ) − γg ( b ) ) − (cid:15) bo ∂ x ( f ( b ) − γg ( b ) )) − (cid:15)βw b [ λ ( b )] ∂ x ( f ( b ) − γg ( b ) )+ (cid:15) (1 + βw b )[2 θ + ( γ − g ( b )]( β∂ x bg (cid:48) ( b ) + ∂ x ( g ( b )))+ (cid:15)β (1 + βw b )[2 α ]( f ( b ) − γg ( b ) ) ∂ x b + 3 (cid:15)β (1 + βw b )[ γ f ( b ) + 23 δ − f ( b )] ∂ x ( f ( b ) − γg ( b ) ) b + (cid:15)β (1 + βw b )(2 θ − α ) g ( b ) ∂ x b + (cid:15)β (1 + βw b )[ 2 γ f (cid:48) ( b )] b∂ x b ( f ( b ) − γg ( b ) ) − (cid:15)β (1 + βw b ) f ( b ) b ∂ x ( f ( b ) − γg ( b ) ) − (cid:15)β (1 + βw b ) ∂ x bν (cid:48) ( b ) ς − (cid:15)ν (1 + βw b ) ∂ x ( ς ) − (cid:15)βν [ ∂ x ( w ) b + w ∂ x b ] ς + (cid:15) (1 + βw b )[ 12 ∂ x ((1 − γ ) g ( b ) ) + t ( b )] . (A.2) C = (cid:15)β∂ x bν (cid:48) ( b )(1 + βw b )( f ( b ) − γg ( b ) ) + (cid:15)β ν∂ x bw (cid:48) ( b ) b ( f ( b ) − γg ( b ) )+ (cid:15)βνw ∂ x b ( f ( b ) − γg ( b ) ) + 3 (cid:15)βw bν∂ x ( f ( b ) − γg ( b ) ) − (cid:15) bo ∂ x ( f ( b ) − γg ( b ) )) − (cid:15)βw b [ λ ( b )] ∂ x ( f ( b ) − γg ( b ) )+ (cid:15) (1 + βw b )[ θ ]( β∂ x bg (cid:48) ( b ) + 2 ∂ x ( g ( b ))) + (cid:15) (1 + βw b )[ θ + 23 ( γ − g ( b )] β∂ x bg (cid:48) ( b )+ (cid:15)β (1 + βw b )[2 α ]( f ( b ) − γg ( b ) ) ∂ x b + 3 (cid:15)β (1 + βw b )[ γ f ( b ) + 23 δ − f ( b )] ∂ x ( f ( b ) − γg ( b ) ) b + (cid:15)β (1 + βω b )[2 θ − α + 13 ( δ − − βb ) f (cid:48) ( b ) − γ g (cid:48) ( b )] g ( b ) ∂ x b + (cid:15)β (1 + βw b )[ 2 γ f (cid:48) ( b )] b∂ x b ( f ( b ) − γg ( b ) ) − (cid:15)β (1 + βw b ) f ( b ) b ∂ x ( f ( b ) − γg ( b ) ) − (cid:15)β∂ x bν (cid:48) ( b )(1 + βw b ) ς − (cid:15)ν (1 + βw b ) ∂ x ( ς ) − (cid:15)βν [ ∂ x ( w ) b + w ∂ x b ] ς + (cid:15) (1 + βw b )[ 13 ∂ x ((1 − γ ) g ( b ) )) + t ( b )] . (A.3)31 = 3 (cid:15) νβw b ( f ( b ) − γg ( b ) ) − (cid:15) f ( b ) − γg ( b ) )) − (cid:15) βw b [ λ ( b )]( f ( b ) − γg ( b ) )+ (cid:15) βw b )[2 θ + ( γ − g ( b )] g ( b )+ (cid:15) βw b )[ θ + 23 ( γ − g ( b )] g ( b ) + 3 (cid:15)β βw b )[ γ f ( b ) + 23 δ − f ( b )]( f ( b ) − γg ( b ) ) b − (cid:15)β βw b )[ 13 f ( b )] b ( f ( b ) − γg ( b ) )) − ν(cid:15)ς βw b ) + 2 (cid:15) (1 − γ ) g ( b ) βw b )= 2 (cid:15) βω b )( γ − g ( b ) . (A.4) E = (cid:15) β∂ x bν (cid:48) ( b )(1 + βw b ) ∂ x ( f ( b ) − γg ( b ) ) + (cid:15) β ν∂ x bw (cid:48) b∂ x ( f ( b ) − γg ( b ) )+ (cid:15) νβw ∂ x b∂ x ( f ( b ) − γg ( b ) )) + (cid:15) νβw b∂ x ( f ( b ) − γg ( b ) ) − (cid:15) ∂ x ( f ( b ) − γg ( b ) ) − (cid:15) βw bλ ( b ) ∂ x ( f ( b ) − γg ( b ) )+ (cid:15)β (1 + βw b )[ θ ] ∂ x ( ∂ x bg (cid:48) ( b )) + (cid:15) β (1 + βw b )[ α ] ∂ x ( f ( b ) − γg ( b ) ) ∂ x b + (cid:15) β (1 + βw b )[2 α ] ∂ x ( f ( b ) − γg ( b ) ) ∂ x b + (cid:15) β (1 + βw b )[ γ f ( b ) + 23 δ − f ( b )] ∂ x ( f ( b ) − γg ( b ) ) b + (cid:15)β (1 + βw b )[ θ − α ] ∂ x bg (cid:48) ( b ) ∂ x b + (cid:15)β (1 + βw b )[2 θ − α + 13 ( δ − − βb ) f (cid:48) ( b ) − γ g (cid:48) ( b )] ∂ x ( β∂ x bg (cid:48) ( b )) ∂ x b + (cid:15) β (1 + βw b )[ η ]( ∂ x b ) ∂ x ( f ( b ) − γg ( b ) ) + (cid:15) β (1 + βw b )[ γ f (cid:48) ( b )] b∂ x b∂ x ( f ( b ) − γg ( b ) )+ (cid:15) β (1 + βw b )[ 2 γ f (cid:48) ( b )] b∂ x b∂ x ( f ( b ) − γg ( b ) ) − (cid:15) β (1 + βw b )[ 13 f ( b )] b ∂ x ( f ( b ) − γg ( b ) )+ (cid:15)β (1 + βw b )[ η ]( ∂ x b ) β∂ x bg (cid:48) ( b ) + (cid:15) β (1 + βw b )[ γ f (cid:48)(cid:48) ( b )]( ∂ x b ) b∂ x ( f ( b ) − γg ( b ) )+ (cid:15) (1 + βw b ) ∂ x ( s ( b )) . 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