The linear and non linear Rayleigh-Taylor instability for the quasi isobaric profile
aa r X i v : . [ m a t h . A P ] J un The linear and non linear Rayleigh-Taylorinstability for the quasi-isobaric profile
Olivier Lafitte ∗† June 29, 2007
Abstract
We study the stability of the system of the Euler equations in theneighborhood of the stationary solution associated with the quasi isobaricprofile in a gravity field. This situation corresponds to a Rayleigh-Taylortype problem with a smooth base density profile which goes from 0 to ρ a (of Atwood number A = 1) given by the ablation front model with athermal conductivity exponent ν >
1. This linear analysis leads to thestudy of the Rayleigh equation for the perturbation of the velocity at thefrequency k : − ddx ( ρ ( x ) dudx ) + k [ ρ ( x ) − gγ ρ ′ ( x )] u = 0 . We denote by the terms ’eigenmode and eigenvalue’ a L solution of theRayleigh equation associated with a value of γ . Let L > ρ ( x ) = ρ a ξ ( xL ), where ˙ ξ = ξ ν +1 (1 − ξ ). We provethat there exists L m ( k ), such that, for all 0 < L ≤ L m , there exists aneigenmode u such that the unique associated eigenvalue γ is in [ α , α ], α >
0. Its limit when L goes to zero is √ gk . We obtain an expansionof γ in terms of L as follows: γ = √ gk q ν )) − ( kL ν ) ν + O (( kL )min (1 , ν ) ) . We identify in this paper the expression of the next term of the expansionof γ in powers of L ν .Using the existence of a maximum growth rate Λ and the existence of atleast one eigenvalue belonging to ] Λ2 , Λ[ (thanks to a semiclassical analy-sis), we perform the nonlinear analysis of the incompressible Euler systemof equations using the method introduced by Grenier. This generalizes theresult of Guo and Hwang (which was obtained in the case ρ ( x ) ≥ ρ l > ρ → x → −∞ and k ( x ) = ρ ′ ( x ) ρ ( x ) satisfy k regular enough, bounded, and k ρ − bounded, which is the case in themodel associated with the quasi-isobaric profile, according to ν > . ∗ Universit´e de Paris XIII, LAGA, 93 430 Villetaneuse † CEA/DM2S, Centre d’Etudes de Saclay, 91191 Gif sur Yvette Cedex Statement of the problem and main result
In this paper, we study a theoretical system of equations deduced of the fluiddynamics analysis of an ablation front model. Such models have been studiedfrom a physical point of view by many authors (see H.J. Kull and S.I. Anisimov[12], V. Goncharov, [6], P. Clavin and L. Masse [17]). They can be considered asa generalization in the ablation case of the Rayleigh-Taylor instability, studiedin the pioneering works of J.W. Strutt (Lord Rayleigh) [19] and G. Taylor [20].The Rayleigh equation models the Rayleigh-Taylor instability. It is obtained byconsidering the linearization of the incompressible 2d Euler equations around thesolution ( ρ ( x ) , , , p ( x )) (density, velocity, pressure) with dp dx + ρ ( x ) g = 0.The system of equations write ∂ t ρ + ∂ x ( ρU ) + ∂ z ( ρV ) = 0 ∂ t ( ρU ) + ∂ x ( ρU + P ) + ∂ z ( ρU V ) = − ρg∂ t ( ρV ) + ∂ x ( ρU V ) + ∂ z ( ρV + P ) = 0 ∂ x U + ∂ z V = 0 (1)Write ρ = ρ + σ , U = v , V = v , P = p + p , the linearized system is ∂ t σ + dρ dx v = 0 ρ ( x ) ∂ t v + ∂ x p = − σgρ ( x ) ∂ t v + ∂ z p = 0 ∂ x v + ∂ z v = 0 . (2)from which one deduces, using v = ˜ ue ikz , the partial differential equation − ∂∂x ( ρ ( x ) ∂∂x ∂ t ˜ u ) + k ρ ( x ) ∂ t ˜ u = gk ρ ′ ( x )˜ u. Introduce T ( x, z, t ) = ρ ( x ) ρ ( x, z, t ) , Q ( x, z, t ) = p ( x, z, t ) − p ( x ) ρ ( x )the system (1) is equivalent to ∂ t T + ~U . ∇ T = k ( x ) uT∂ t ~U + ( ~u. ∇ ) ~U + T ∇ Q + T Qk ( x ) ~e = (1 − T ) ~g div ~U = 0 . (3)It is a consequence of the equality T ρ − ∇ p = T ∇ Q + k T Q~e + T ~g and of ∂ t T + ~U . ∇ T = − ρ − ( ∂ t ρ + ~U ∇ ρ ).The associated linearized system in the neighborhood of ~U = 0, T = 1, Q = 0is ∂ t T = k ( x )˜ u div˜ ~u = 0 ∂ t ˜ ~u + ρ − ∇ ( ρ ˜ Q ) + ˜ T~g = 0 . e γt e ikz u ( x, kL ) , where k is the wavelength of the transversal perturbation and γ is the growthrate in time of this perturbation. We obtain the Rayleigh equation (4) (see C.Cherfils, P.A. Raviart and O.L. [3]): − ddx ( ρ ( x ) dudx ) + ( k ρ ( x ) − gk γ ρ ′ ( x )) u ( x ) = 0 . (4)We consider a family of density profiles ρ ( x ) such that ρ ( x ) = ρ ( xL ), where L is a characteristic length of the base solution. In one of the physical applica-tions, namely the case of the ICF, its magnitude is 10 − meters, hence allowingus to consider the limit L → ρ is given by ρ ( x ) = ρ a ξ ( xL ) , (5)where the function ξ is a non constant solution of˙ ξ = ξ ν +1 (1 − ξ ) , (6) ν is called the thermal conduction index.Note that this equation on the density is NOT obtained from the incompress-ible Euler equations but from a compressible model with thermal conductionintroduced by Kull and Anisimov [12] and used for example in [11] or in [14].The Kull-Anisimov profile satisfies lim x → + ∞ ρ ( x ) = ρ a , where ρ a denotes thedensity of the ablated fluid, and the convergence is exponential, whereaslim x →−∞ ρ ( x ) = 0and the convergence is rational (( − x ) ν ρ ( x ) → C > x → −∞ ). Theassociated Atwood number is thus 1. Remark also that all non constant solu-tions of (6) differ from a translation.This case may be related to the case of the water waves (the density of airbeing much smaller than the density of water). It is thus a limit case in all thetheoretical set-up used for the study of Euler equations for fluids of differentdensities.Note that, in this case, the self adjoint operator associated with the equation(4) is not coercive in H ( R ). The methods of [3], [10] cannot be used directly.Moreover, the properties of ρ do not allow us to apply [8], because it relies on ρ ( x ) ≥ ρ l > k ( x ) = ρ ′ ( x ) ρ ( x ) introduced in the abstract. In our case, it isequal to L − ξ ν (1 − ξ ), hence it is a continuous bounded function which admitsa maximum L − eff , and, for ν > , k ρ − is bounded. These properties are (fora more general profile) what is needed to obtain the nonlinear result.3 emarks Define the function r ( t, ε ) through:1 ε ( ξ ( − tε )) ν (1 − ξ ( − tε )) = 1 νt + ε ν t − − ν r ( t, ε ) . (7)There exists t > ε > r ( t, ε ) is bounded for t ≥ t , ≤ ε ≤ ε ,and has a C ∞ expansion in ε, ε ν . Define S through ε ν S ′ ( t, ε ) = ε − ξ ′ ( − tε ) ξ ( − tε ) − νt , lim t → + ∞ S ( t, ε ) = 0 . We have the identity ξ ( − tε )( νtε ) ν exp( ε ν S ( t, ε )) = 1 (8)which implies that there exists a function r bounded for t ≥ t and ε ≤ ε suchthat exp( − νε ν S ( t, ε )) = 1 + ε ν t − ν r ( t, ε ) . Let u ( y ) = u ( L y ). The Rayleigh equation rewrites − ddy ( ξ ( y ) dudy ) + ( ε ξ ( y ) − λεξ ′ ( y )) u ( y ) = 0 , (9)where ε = kL and λ = gkγ . We will consider this equation from now on.We shall introduce two equivalent versions of this equation, which are:1. the system on ( U + , V + ) such that U + ( y, ε ) = u ( y, ε ) e εy and V + (given bythe first equation of the system below), v ( y, ε ) = V + ( yε ) e − εy : ( dU + dy = ε (1 − λ ) U + + εξ ( y ) V + dV + dy = ε ( λ + 1) V + + ε (1 − λ ) ξ ( y ) U + , (10)2. if we introduce w = vξ ( y ) , the system on ( u, w ) is (cid:26) dudt = λu − w dwdt = ( λ − u − λw + ( νt + ε ν S ′ ( t, ε )) w. (11)The first part of the main result of this paper was presented in [13], and thecase where ξ ( y ) = ξ (1)( y + 1) − ν for y ≥ L ( R ) of (9), then λ satisfiesthe inequality (see [10]) λ ≥ max(1 , ε ( ν + 1) ν +1 ν ν ) . (12)The main result of the first part of this paper is It is a consequence of max( ˙ ξξ ) = ν ν ( ν +1) ν +1 heorem 1
1. There exists ε > , and C > such that, for all ε ∈ ]0 , ε [ there exists λ ( ε ) ∈ [ , ] such that the Rayleigh equation (9) admits abounded solution u for λ = λ ( ε ) , which corresponds to the eigenmode u and the eigenvalue γ ( k, ε ) = q gkλ ( ε ) , and λ ( ε ) satisfies | λ ( ε ) − | ≤ C ε ν .
2. We have the estimate λ ( ε ) = 1 + 2( εν ) ν (Γ(1 + ν )) − + o ( ε ν )= 1 + 2( εν ) ν (Γ(1 + ν )) − + O ( ε α ) with α = min (1 , ν ) .3. We have the expansion − λ ( ε ) = 2( εν ) ν ( B (0)) − [1 + 2( εν ) ν C (1 , B (0)) + o ( ε ν )] , where B (0) = − R ∞ s ν e − s ds = − − ν Γ(1+ ν ) and C (1 , is calculatedbelow in Proposition 4. This result is a result, for k fixed, in the limit L →
0. It writes also, for k fixedand for L < ε k as γ = s gk εν ) ν (Γ(1 + ν )) − + o ( ε ν ) . (13)Note that, in this case, the order of magnitude of γ − √ gk is not in kL as in [3],but the result of [10], based on ρ − ρ a x> ∈ L ν + θ ′ for all θ ′ > k going to infinity, which can be stated as Proposition 1 a) Any value λ ( ε ) such that (9) has a L non zero solutionsatisfies kg ( λ ( ε )) ≤ Λ , where Λ = gL eff .b) Any sequence k → λ ( k ) k satisfies the followinglim k → + ∞ λ ( ε ) k = L eff = min y ξ ( y ) ξ ′ ( y ) L . It is proven in [10].Remark that formula (13) and Proposition 1 are not in contradiction. Theylead to two different stabilizing mechanisms induced by the transition region:one is a low frequency stabilization when L → k → + ∞ . It is important to notice thatPropositions 2 and 3 below allow us to construct an (exact) solution u ( y, λ ( ε ) , ε )of the Rayleigh equation hence giving an unstable mode˜ u ( x, z, t ) = e ikz u ( xL , kL , λ ( ε )) e √ gk √ λ ( ε ) t k ≥ λ ( ε ) and u ( y ) such that ε = kL , u solution of (9), γ ( k, ε ) = q gkλ ( ε ) , Λ2 < γ ( k, ε ) < Λ, || u ( y ) || L = 1, u (0) > γ ( k ) the eigenvalue γ ( k, L ).From u , one deduces a solution U = ℜ [( u , v , Q , T ) e ikz + γ ( k ) t ] = ℜ [( u ( x ) , − ik u ′ ( x ) , − γ ( k ) k u ′ ( x ) , k ( x ) γ ( k ) u ( x )) e ikz + γ ( k ) t ]of the linearized system. We thus consider a function V N = (0 , , p ρ , L x ) + P Nj =1 δ j V j ( x, y, t ) satisfying ( Emod )( V N ) = δ N +1 R N +1 , V N ( x, z, − (0 , , p ρ , L x ) = δU ( x, z, V ( x, y, t ) of the Euler system suchthat Emod ( V ) = 0 and V ( x, z,
0) = (0 , , p ρ , L x ) + δU ( x, y, V d ( x, y, t ) = V ( x, y, t ) − V N ( x, y, t ). This procedure constructs a solu-tion of the nonlinear system.We have the Theorem 2
1. There exists two constants A and C , depending only onthe properties of the Euler system, on the stationary solution and on thesolution ˆ u ( x ) , such that, for all θ < , for all t ∈ ]0 , γ ( k ) ln θδC A [ , one hasthe control of the approximate solution V N in H s , namely || T N − || H s + || ~u N || H s + || Q N − q || H s ≤ C δAC e γ ( k ) t − δAC e γ ( k ) t and the leading order term of the approximate solution is the solution ofthe linear system as follows || T N − || L ≥ δ || T (0) || L e γ ( k ) t − AC C e γ ( k ) t − δAC e γ ( k ) t
2. There exists N such that for any N ≥ N , the function V d is well definedfor t < γ ( k ) ln δ and satisfies the inequality || V d || ≤ δ N +1 e ( N +1) γ ( k ) t , ∀ t ∈ [0 , γ ( k ) ln 1 δ [ .
3. We have the inequality, for ǫ < C A || ~u ( 1 γ ( k ) ln ε C Aδ ) || L ≥ ε || ~u (0) || L . This paper is organized as follow. The sections 1, 2, 3 study the linear systemand identify the behavior of the growth rate γ ( k ) when L → y → + ∞ and we extend such solutions, for ( ε, λ ) in a compact B , on[ ξ − ( εR ) , + ∞ [, where R is a constant depending only on B (Proposition 2). Inthe second section, for all t >
0, we calculate a solution of (9) which is boundedon ] − ∞ , − t ε ] (Proposition 3).A solution u of (9) which is in L ( R ) goes to zero when y → + ∞ as well aswhen y → −∞ . Moreover, as ρ ( x ) is a C ∞ function on R , any solution u of(4) is also in C ∞ .Notice that lim ε → ( − εξ − (( εR ) ν )) = νR , from which one deduces that thereexists t such that 0 < t < lim ε → ( − εξ − (( εR ) ν )).The regions ] − ∞ , − t ε ] and [ ξ − (( εR ) ν ) , + ∞ [ overlap and[ ξ − (( εR ) ν ) , − t ε ] ⊂ [ − ενR , − ενR ] . Hence the solution u belongs to the family of solutions described in proposition2 (of the form C ∗ u + ( y, ε )) and belongs to the family of solutions described inproposition 3 (of the form C ∗∗ U ( − εy, ε )), that is (cid:26) u ( y ) = C ∗ u + ( y, ε ) , y ≥ ξ − (( εR ) ν ) u ( y ) = C ∗∗ U ( − εy, ε ) , y < − t ε From the continuity of u and of u ′ , one deduces that, for all y ⊥ ∈ [ − ενR , − ενR ](corresponding to t ⊥ = − εy ⊥ ∈ [ νR , νR ]), we have C ∗ u + ( y ⊥ , ε ) = C ∗∗ U ( t ⊥ , ε ), C ∗ ddy u + ( y ⊥ , ε ) = − C ∗∗ εU ′ ( t ⊥ , ε ).Introduce the Wronskian (where ε − has been added for normalization purposes) W ( y ) = ε − ( u + ( y, ε ) ddy ( U ( − εy, ε )) − ddy ( u + ( y, ε )) U ( − εy, ε )) . It is zero at y ⊥ = − εt ⊥ . Conversely, if λ and ε are chosen such that theWronskian is zero (in particular at a point y ⊥ = − t ⊥ ε ), the function˜ u ( y ) = ( C ∗∗ U ( − εy, ε ) , y ≤ y ⊥ C ∗∗ U ( − εy ⊥ ,ε ) u + ( y ⊥ ,ε ) u + ( y, ε ) , y ≥ y ⊥ (14)is, thanks to the Cauchy-Lipschitz theorem, a solution of (4). Moreover, itbelongs to L ( R ) owing to the properties of u + and of U .In Section 3, we compute the function W . As U and u + are solutions of theRayleigh equation, which rewrites d dy ( u + ( y, ε )) = − ξ ′ ( y ) ξ ( y ) du + dy + ( ε − ελ ξ ′ ( y ) ξ ( y ) ) u + ( y, ε )the function W is solution of ddy W = − ξ ′ ( y ) ξ ( y ) W , which implies the equality ξ ( y ) W ( y ) = ξ ( y ) W ( y ) for all y, y (15)7his Wronskian can be computed for y ⊥ ∈ [ − ενR , − ενR ] using the expressionsobtained for U and u + . We prove that it admits a unique root for 0 < ε < ε and λ in a fixed compact, and we identify the expansion of this root in ε , henceproving Theorem 1. Precise estimates of this solution are given in Section 3.In Section 4, after proving a H s result on a general solution of the linear system(taking into account a mixing of modes), we calculate all the terms V j of theexpansion of the approximate solution, the perturbation of order δ being aneigenmode with a growth rate γ ∈ ] Λ2 , Λ[, where Λ = max k ( x ) gL . The system (10) writes ddy ~U + = εM ( ξ ( y ) , λ ) ~U + . When y → + ∞ , the matrixconverges exponentially towards M (1 , λ ), which eigenvalues are 0 and 2, ofassociated eigenvectors (1 , λ −
1) and (1 , λ + 1).It is classical that
Lemma 1
There exists a unique solution ( U + , V + ) of (10) which limit at y → + ∞ is (1 , λ − . Moreover, there exists ξ > such that this solution admitsan analytic expansion in ε for ξ ( y ) ∈ [ ξ , . The proof of this result is for example a consequence of Levinson [16].The aim of this section is to express precisely the coefficients of this expansionwhen ξ ( y ) → ξ ∈ [ ξ ( εR ) , ξ ].We consider, in what follows, the change of variable ζ = εξ ( y ) ν . (16)We prove in this section the Proposition 2
Let K be a compact set and λ ∈ K . There exists ε > and R > such that, for < ε < ε , the family of solutions of (10) which is boundedwhen y → + ∞ is characterized , for y such that ξ ( y ) ≥ ( εR ) ν , by ( U + ( y, ε ) = 1 + (1 − ξ ( y ))(1 − λ ) ξ ( y ) ζA ( ζ, ε ) V + ( y, ε ) = λ − − λ )(1 − ξ ) ζB ( ζ, ε ) . The associated solution of (9) is u + ( y, ε ) = U + ( y, ε ) e − εy . It can also be shown that there exists a unique solution ( ˜ U, ˜ V ) such that ( ˜ U, ˜ V ) e − εy → (1 , λ + 1) a general solution is K + ( U, V ) where K + is a constant roof of Proposition 2 We write the analytic expansion in ε : U = 1 + X j ≥ ε j u j , V = λ − X j ≥ ε j v j . We deduce, in particular, ( du dy = λ − ξ ( y ) (1 − ξ ( y )) dv dy = ( λ − − ξ ( y ))hence assuming u , v → ξ → ξ →
1) we get ( u = − λν +1 1 − ξ ν +1 ξ ν +1 v = − λ ν − ξ ν ξ ν . The following recurrence system for j ≥ ( du j +1 dy = ξ ( v j − ( λ − ξu j ) dv j +1 dy = ( λ + 1)( v j − ( λ − ξu j ) . (17)Usual methods for asymptotic expansions lead to the estimates (which are notsufficient for the proof of Proposition 2) | u j ( y ) | + | v j ( y ) | ≤ M A j ξ ( ν +1) j . However, using the relation 1 − ξ = ˙ ξξ ν +1 , we obtain the following estimates: Lemma 2
Let ξ > given. For all j ≥ , introduce a j and b j , such that u j ( y ) = (1 − ξ ( y ))(1 − λ ) ξ νj +1 a j ( ξ ( y )) , v j ( y ) = (1 − ξ ( y ))(1 − λ ) ξ νj b j ( ξ ( y )) . The functions a j and b j are bounded, analytic functions of ξ , for ξ ∈ [ ξ , .They satisfy | a j ( ξ ) | ≤ AR j , | b j ( ξ ) | ≤ AR j , (18) where R depends only on λ . We prove Lemma 2 by recurrence. Assume that this relation is true for j . Wehave the relations ( du j +1 dy = (1 − λ )( b j − ( λ − a j ) ˙ ξξ ν ( j +1)+2 dv j +1 dy = (1 − λ )( λ + 1)( b j − ( λ − a j ) ˙ ξξ ν ( j +1)+1 from which we deduce, using the limit 0 at ξ → u j +1 ( y ) = (1 − λ ) Z ξ ( y )1 b j ( η ) − ( λ − a j ( η ) η ν ( j +1)+2 dη v j +1 ( y ) = (1 − λ )( λ + 1) Z ξ ( y )1 b j ( η ) − ( λ − a j ( η ) η ν ( j +1)+1 dη We thus deduce that ξ ν ( j +1) v j +1 ( y ) and ξ ν ( j +1)+1 u j +1 ( y ) are bounded functionswhen ξ ∈ ]0 , | b j | ≤ AR j and | a j | ≤ AR j , then | u j +1 | ≤ AR j | − λ | ( | λ − | + 1) R ξ dηη ν ( j +1)+2 | v j +1 | ≤ AR j | − λ || λ + 1 | ( | λ − | + 1) R ξ dηη ν ( j +1)+1 . We end up with | u j +1 | ≤ | λ − | AR j ( | λ − | +1) ξ ν ( j +1)+1 − ξ ν ( j +1)+1 ν ( j +1)+1 , | v j +1 | ≤ | λ − | AR j | λ + 1 | ( | λ − | +1) ξ ν ( j +1) − ξ ν ( j +1) ν ( j +1) . As − ξ a a ≤ − ξ, ξ ∈ [0 , | u j +1 | ≤ | λ − | AR j ( | λ − | +1)(1 − ξ ( y )) ξ ν ( j +1)+1 , | v j +1 | ≤ AR j | λ − || λ + 1 | ( | λ − | +1)(1 − ξ ( y )) ξ ν ( j +1) . Consider R λ = ( | λ − | + 1)max(1 , | λ + 1 | ) . (19)The previous inequalities become | u j +1 | ≤ AR j +1 λ (1 − ξ ( y )) | λ − | ξ ν ( j +1)+1 , | v j +1 | ≤ AR j +1 λ (1 − ξ ( y )) | λ − | ξ ν ( j +1) , hence we proved the inequality for j + 1.The inequality is true for j = 1, hence the end of the proof of Lemma 2, wherewe may choose the value of R for λ ∈ [ , ] as R = . Finally we have theequalities, for all y such that ξ ( y ) ≥ ξ : U + ( y, ε ) = 1 + (1 − ξ ( y ))(1 − λ ) ξ ( y ) P j ≥ a j ( ξ ( y ))( ε ( ξ ( y )) ν ) j = 1 + (1 − ξ ( y ))(1 − λ ) ξ ( y ) ( ε ( ξ ( y )) ν ) P j ≥ a j +1 ( ξ ( y ))( ε ( ξ ( y )) ν ) j V + ( y, ε ) = λ − − λ )(1 − ξ ( y )) P j ≥ b j ( ξ ( y ))( ε ( ξ ( y )) ν ) j = λ − − λ )(1 − ξ ( y ))( ε ( ξ ( y )) ν ) P j ≥ b j +1 ( ξ ( y ))( ε ( ξ ( y )) ν ) j . Using the estimates (18) and the change of variable (16), for ζ < R − the series P a j ( ε ν ζ ν ) ζ j is normally convergent and the following functions are well defined ( ˜ U ( y, ε ) = 1 + (1 − λ )(1 − ξ ( y )) ξ ( y ) ζA ( ζ, ε )˜ V ( y, ε ) = λ − − λ )(1 − ξ ( y )) ζB ( ζ, ε ) . It is straightforward to check that ˜ U and ˜ V solve system (10) and that wehave, for ξ ( y ) ≥ ξ , ζ ( ξ ) ≤ εξ ν , hence for ε < ε = ξ ν R and ξ ( y ) ≥ ξ wehave ˜ U ( y, ε ) = U + ( y, ε ) and ˜ V ( y, ε ) = V + ( y, ε ). We extended the solutionconstructed for ξ ( y ) ∈ [ ξ ,
1[ to the region ζ < R . This proves Proposition 2. Note that these inequalities depend on a given arbitrary ξ > The solution in the low density region
In this section, we obtain the family of solutions of (9) bounded by | y | A e εy when y → −∞ , that is in the low density region ξ →
0. Introduce the new variable t = − εy . Commonly, I call this solution the hypergeometric solution, because ithas been observed that, in the model case ρ ( x ) = ( − x − − ν studied in [3] aswell as in [6], the Rayleigh equation rewrites as the hypergeometric equation.Introduce τ ( s, ε ) = − dds ( ξ ( − sε ))( ξ ( − sε )) − = ξ ν ε (1 − ξ ) = 1 νs + ε ν S ′ ( s, ε ) . We define the operators R ε , K ε and ˜ K λε through R ε ( g )( s ) = [ Z ∞ s τ ( y, ε ) e − y ( ξ ( − yε )) − λ g ( y, ε ) dy ] e s ( ξ ( − sε )) λ , (20) K ε ( g )( t ) = (1 − λ ) ˜ K λε ( g )( t ) = 1 − λ Z + ∞ t τ ( s, ε ) R ε ( g )( s, ε ) ds. (21)These operators rewrite R ε ( g )( s, ε ) = Z + ∞ s ( 1 νy + ε ν S ′ ( y, ε )) e − y − s ) s − λν y λν exp( ε ν λ ( S ( y ) − S ( s ))) g ( y, ε ) dy.K ε ( g )( t, ε ) = 1 − λ Z + ∞ t ( 1 νs + ε ν S ′ ( s, ε )) R ε ( g )( s, ε ) ds. We have the inequalities, for g uniformly bounded, (and λ < ν , which implies ξ ( − sε ) ν − λ ≤ ξ ( − tε ) ν − λ for t ≥ s ) | R ε ( g )( s ) | ≤ || g || ∞ [ Z + ∞ s ε ξ ν − λ (1 − ξ ) e − y dy ] e s ξ λ ≤ || g || ∞ ξ ν ε (22) | K ε ( g )( t ) | ≤ | λ − | || g || ∞ Z ∞ t τ ( s, ε ) ξ ν ε ds ≤ | λ − | ν || g || ∞ ξ ν ε . (23)Moreover, the following inequality is true: | g ( s, ε ) | ≤ C p ( ξ ν ε ) p ⇒ | K ε ( g )( t, ε ) | ≤ | λ − | ν ( p + 1) C p ( ξ ν ε ) p +1 . (24)In a similar way, we introduce K λ ( g )( t ) = − λ R + ∞ t νs R λ ( g )( s ) dsR λ ( g )( s ) = R + ∞ s νy e − y − s ) s − λν y λν g ( y ) dy. ε > < ε < ε . Under suitable assumptions on g (we canfor example consider g in C ∞ ([ t , + ∞ [) such that | ∂ p g | ≤ C p y α − p for all p ), theoperators K ε , R ε , K , R are well defined. Moreover, one proves that g ( t, λ, ε ) = X n ≥ K ( n ) ε (1)( t, ε ) (25) g ( t, λ ) = X n ≥ K ( n )0 (1)( t ) (26)are normally converging series on [ t , + ∞ [, and that we have: g = 1 + K ε ( g ) , g = 1 + K ( g ) . (27)Moreover, we know that g is defined on R , because the series P ( | λ − | A ) p p ! ( ξ ν ε ) p converges and is majorated by exp( | λ − | A ξ ν ε ), from the inequality (24). Weobtain the inequalities | g ( t, λ ) | ≤ exp( | λ − | ν t ) , | g ( t, λ, ε ) | ≤ exp( | λ − | ν ζ − ) . (28)We cannot thus consider the limit ζ → g as (28).We shall assume that λ belongs to a compact set and that λ ≥ . We prove Proposition 3
Let g be defined through (25). The family of solutions of thesystem (11) on ( u, w ) which is bounded by | y | A e εy when y → −∞ is given by u ( y, ε ) = C ( F ( t, λ, ε )+ G ( t, λ, ε )) , ξ ( y ) w ( y, ε ) = v ( y, ε ) = Cξ ( y )[( λ − F ( t, λ, ε )+( λ +1) G ( t, λ, ε )] where C is a constant, t ∈ [ t , + ∞ [ , t = − εy and F and G are given by equalities(30) and (31) below.We have the estimates, for t ∈ [ t , ε [ | g ( t, λ, ε ) − g ( t, λ ) | ≤ C ε ν | g ( t, λ ) || u ( − tε , ε ) − u ( − tε , ε ) | ≤ C ε ν | u ( − tε , ε ) || v ( − tε , ε ) − v ( − tε , ε ) | ≤ C ε ν | v ( − tε , ε ) | proof The system (11) rewrites on F and G given by Proposition 3: (cid:26) F ′ ( t, λ, ε ) = F ( t, λ, ε ) − ( νt + ε ν S ′ ( t, ε ))[( λ − F ( t, λ, ε ) + ( λ + 1) G ( t, λ, ε )] G ′ ( t, λ, ε ) = − G ( t, λ, ε ) + ( νt + ε ν S ′ ( t, ε ))[( λ − F ( t, λ, ε ) + ( λ + 1) G ( t, λ, ε )] . (29)A non exponentially growing solution of the system (29) is obtained throughthe following procedure. We denote by g ( t, λ, ξ ) the function g ( t, λ, ε ) = G ( t, λ, ε ) e t ( ξ ( − tε )) λ +12 ( εν ) − λ +12 ν = G ( t, λ, ε ) e t t − λ ν exp( − ε ν λ S ( t, ε )) . (30)12e first get, from the fact that F is bounded when t → + ∞ , that F ( t, λ, ε ) e − t t λ − ν e ε ν λ − S ( t,ε ) = F ( t, λ, ε ) e − t ( ξ ( − tε )) − λ ( εν ) λ − ν = λ +12 R + ∞ t ( νs + ε ν S ′ ( s, ε )) s λ − ν e ε ν λ − S ( s,ε ) e − s G ( s, λ, ε ) ds = − λ +12 R + ∞ t ξ − dds ( ξ ) g ( s, λ, ε ) e − s ξ − λ ( εν ) λν ds = − λ +12 R + ∞ t ξ − dds ( ξ ) ξ − λ ( εν ) λ − ν ds. (31)We deduce from the system (29) the equality ddt ( G ( t, λ, ε ) e t t − λ ν exp( − ε ν λ S ( t, ε ))) = λ −
12 ( 1 νt + ε ν S ′ ) e t t − λ ν exp( − ε ν λ S ( t, ε )) F ( t, λ, ε ) . Under the assumptions g bounded and satisfies the conditionlim t →∞ g ( t, λ, ε ) = 1 (32)one gets the equality g ( t, λ, ε ) − K ε ( g )( t, ε ) . (33)Using the usual Volterra method and inequalities (23), (24) and (28), we deducethat the only solution of (33) satisfying assumptions (32) is given through (25).One gets G through (30) then F thanks to F ( t, λ, ε ) e − t ξ − λ ( εν ) λ − ν = ( εν ) λν λ + 12 Z ∞ t τ ( s, ε ) e − s ξ − λ g ( s, λ, ε ) ds. (34)The first part of Proposition 3 is proven.Denote by ( u , w ) the leading order term in ε of ( u, w ) when t and λ are fixed.Introduce F ( t, λ ) and G ( t, λ ) through the equalities u ( t, λ ) = F ( t, λ ) + G ( t, λ ) , w ( t, λ ) = ( λ − F ( t, λ ) + ( λ + 1) G ( t, λ ) . The functions ( F ( t, λ ) , G ( t, λ )) are solution of (cid:26) dF dt ( t, λ ) = F ( t, λ ) − λ − νt F ( t, λ ) − λ +12 νt G ( t, λ ) dG dt ( t, λ ) = − G ( t, λ ) + λ − νt F ( t, λ ) + λ +12 νt G ( t, λ ) . The second part of Proposition 3 comes from the following estimates on theoperators R ε and K ε , valid for ε ≤ ε and t ≥ t > | R ε ( f ) − R λ ( f ) | ≤ C ε ν | R λ ( f ) | , | K ε ( g ) − K λ ( g ) | ≤ C ε ν | K λ ( g ) | , (35)from which we deduce the uniform estimates for g given by (30) solution of (33) | g ( t, λ, ε ) − g ( t, λ ) | ≤ C ε ν | g ( t, λ ) | , t ≥ t , ε ≤ ε (36)because the Volterra series associated with K is normally convergent in [ t , + ∞ [.This ends the proof of Proposition 3. 13 ote that the previous estimates, as well as the behavior of the solution and theoperator R , are valid only for t > , because, for example, R λ (1)( s ) ≃ when s → . The integral defining K λ is nevertheless convergent at + ∞ , because for t ≥ t we have the equality νsR λ (1)( s ) = 1 − Z ∞ s y ( ys ) ν e − y − s ) dy. ε = 0 We prove in this Section
Lemma 3
The solution ( F ( t, λ ) , G ( t, λ )) constructed through (30), (33), (34)for ε = 0 is given by (cid:26) F ( t, λ ) = e − t ( U ( t, λ ) + dU dt ( t, λ )) G ( t, λ ) = e − t ( U ( t, λ ) − dU dt ( t, λ )) where U ( t ) = 2 − λ +12 ν U ( − λ ν , − ν , t ) the function U ( a, b, T ) being the Loga-rithmic Kummer’s solution of the confluent hypergeometric equation (see [1]). This allows to obtain the limit of the ( F ( t, λ ) , G ( t, λ )) for t → U ( t, λ ) = u ( t ) e t is tU ′′ − (2 t + 1 ν ) U ′ + λ + 1 ν U = 0 . (37)Introducing T = 2 t , we recognize (see [1]) the equation for hypergeometricconfluent functions for b = − ν and a = − λ ν : T d U dT − ( 1 ν + T ) dU dT + 1 + λ ν U = 0 . The family of solutions of this Kummer’s equation is generated by two functions M ( a, b, T ) and U ( a, b, T ). Note that T − b M (1 + a − b, − b, T ) is also a solu-tion of (37), independant of M ( a, b, T ), hence U ( a, b, T ) can be expressed using M ( a, b, T ) and T − b M (1 + a − b, − b, T ). The family of solutions of (37) whichgo to zero when T → + ∞ is generated by U ( a, b, T ), called the logarithmicsolution. It is the subdominant solution of the hypergeometric equation.The expression of the subdominant solution U ( a, b, T ) is the following: U ( a, b, T ) = π sin πb [ M ( a, b, T )Γ(1 + a − b )Γ( b ) − T − b M (1 + a − b, − b, T )Γ( a )Γ(2 − b ) ]where Γ is the usual Gamma function (Γ( s ) = R ∞ t s e − t dt ). The relation be-tween U ( a, b,
0) and U ′ ( a, b,
0) characterize the subdominant solution of the ordi-nary differential equation, and this particular solution has been chosen through14he limit when z → + ∞ : U ( a, b,
0) = Γ(1 − b )Γ(1 + a − b ) , lim z → + ∞ z a U ( a, b, z ) = 1 . (38)As we imposed that g ( t, ε ) → t → + ∞ , we get that G ( t, λ ) e t t − λ +12 ν → t → + ∞ and that there exists a constant ˜ C such that F ( t, λ ) e t t − λ +12 ν → ˜ C when t → + ∞ . Hence( F ( t, λ ) + G ( t, λ )) e t t − λ +12 ν → . As T a U ( a, b, T ) →
1, we get that t − λ ν U ( − λ ν , − ν , t ) → λ +12 ν . We thusobtain the equality t − λ ν U ( t, λ ) = t − λ ν e t ( F ( t, λ ) + G ( t, λ )) = 2 − λ +12 ν t − λ ν U ( − λ ν , − ν , t ) , hence U ( t, λ ) = 2 − λ +12 ν U ( − λ ν , − ν , t ) . (39)Introduce C ( λ ) = U ( − λ ν , − ν ,
0) = − π sin πν Γ( − ν )Γ(1 + λ − ν ) = Γ(1 + ν )Γ(1 + λ − ν ) . (40)We get that u ( t ) = 2 − λ +12 ν U ( − λ ν , − ν , t ) e − t . As w = λu − du dt = (( λ +1) U − dU dt ) e − t one deduces G ( t, λ ) = ( U ( t, λ ) − dU dt ( t, λ )) e − t , F ( t, λ ) = ( U ( t, λ ) + 12 dU dt ( t, λ )) e − t . (41)Using [1] and (37), we finally obtain G ( t, λ ) → − λ +12 ν C ( λ ) , F ( t, λ ) → − λ +12 ν C ( λ ) when t → . (42)We deduce the equality U (0 , λ ) = 2 − λ +12 ν Γ(1 + ν )Γ(1 + − λ ν ) , lim t → + ∞ t − ν − λ − ν U ( t, λ ) = 2 ν + λ − ν . Note that we can deduce the expressions of F + G and of G . We thuscheck that( F + G )( t, λ ) e t = C ( M ( − λ , − ν , t ) − C ∗ (2 t ) ν +1 M (1 + 1 − λ ν , ν , t ))(43) − b )Γ(1+ a − b ) = π sin πb Γ( b )Γ(1+ a − b ) t G ( t, λ ) = C (( M − M ′ )( − λ , − ν , t ) − C ∗ (2 t ) ν +1 ( M − M ′ )(1 + − λ ν , ν , t )) − C C ∗ ν (1 + ν ) t ν M (1 + − λ ν , ν , t )) . (44)We note that ( M − M ′ )( − λ , − ν ,
0) = − λ . We deduce that e t ( F + G )(0 , λ ) =2 − λ ν C ( λ ) and e t G (0 , λ ) = 2 − λ ν C ( λ )(1 − λ ), hence ( λ − e t ( F + G )(0 , λ )+2 e t G (0 , λ ) = 0. In the next Section, we combine the results of Section 1 andof Section 2. The Wronskian is related to a function independant of the variable t , calledthe Evans function, introduced below in (45) and denoted by Ev ( λ, ε ). In thepresent Section, we shall identify the leading order term in ε of the Evansfunction, and all the terms of the form ε ν ( λ −
1) of the Evans function. Weshall finish by the calculation of the term of the form ε ν . More precisely, weprove Lemma 4
The function Ev ( λ, ε ) = ξ ( y ) W ( y ) (45) is independant of y . It is analytic in λ and in ε ν , ε . Moreover, one has Ev (1 , ε ) = 2( εν ) ν and ∂ λ Ev (1 ,
0) = 2 − ν Γ(1 + ν ) . This function is called theEvans function of the equation (9). Using the expressions of ddy ( U ( − εy, ε )) and ddy u + , we have ε W ( y, ε ) = u + ( y, ε )( − ελU ( − εy, ε ) + εW ( − εy, ε )) − U ( − εy, ε )( − ελu + ( y, ε ) + εξ ( y ) v + ( y, ε ))= εξ ( y ) ( ξ ( y ) u + ( y, ε ) W ( − εy, ε ) − U ( − εy, ε ) v + ( y, ε )) . Hence we have the following constant function to study, which depends only on λ, ε : Ev ( λ, ε ) = ξ ( y ) W ( y ) = [ ξ ( y ) u + ( y, ε ) V ( − εy, ε ) − ξ ( y ) v + ( y, ε ) U ( − εy, ε )] . We shall use the equalities, valid for all y (and t ) such that both solutions aredefined (which means y ∈ [ − ενR , − ενR ]) Ev ( λ, ε ) = ξ ( y ) W ( y ) = ( ξ W )( − t ε ) . We begin with the
Lemma 5
The Evans function has an analytic expansion in λ , which coeffi-cients depend analytically on ε and ε ν . Ev ( λ, ε ), we introduce ξ = ξ ( − tε ) , ζ = εξ ν , ζ = νt = ζ ( t, , for t ≥ t > . We check that the function Ev ( λ, ε ) is analytic in λ and has an analytic expan-sion in ε ν and ε thanks to the equality [ ν ] X p =0 ξ ν +1 − p + 11 − ξ + 1 − ξ ν − [ ν ] ξ ν − [ ν ] (1 − ξ ) = 1 ξ ν +1 (1 − ξ )which implies that the relation between t and ζ is analytic in ε and ε ν .Assume from now on λ ≥ and ν > ξ ( y ) by ε ν ζ − ν . Using thisLemma, there exists two functions B ( ε ) and C ( λ, ε ) such that Ev ( λ, ε ) = Ev (1 , ε ) + B ( ε )( λ −
1) + C ( λ, ε )( λ − . (46) Direct relations
Considering the limit in (52) for ε = 0, we obtain Ev ( λ,
0) = ( λ − e t ( F + G )( t, λ )[ − − λ ) νtA ( νt,
0) + νtB ( νt, . As this quantity is independant of t , we consider the limit when t →
0, hencewe deduce that Ev ( λ,
0) = − ( λ − − λ +12 ν C ( λ ) . (47)Remark that this implies the identity ( λ − e t ( F + G )( t, λ )[ − − λ ) νtA ( νt, νtB ( νt, − − λ +12 ν C ( λ )( λ − e t ( F + G )( t, λ )[ − − λ ) νtA ( νt,
0) + νtB ( νt, − − λ +12 ν C ( λ ) . In a similar way, we check that, for λ = 1, U + = 1 and V + = 0, and g ( t, , ε ) = 1,which implies G ( t, , ε ) = e − t ( νε ) − ν ξ − = e − t ( ζν ) ν (49)from which one deduces Ev (1 , ε ) = 2 e t G ( t, , ε ) ξ = 2( εν ) ν . (50)From (46), the unique root λ ( ε ) of Ev ( λ, ε ) in the neighborhood of λ = 1 satisfies λ ( ε ) − − Ev (1 , ε ) B ( ε ) + C ( λ ( ε ) , ε )( λ ( ε ) − . The two first terms of the expansion of λ ( ε ) − ε ν under theassumption ν > ( ε ) − − Ev (1 , ε ) B ( ε ) + C (1 , λ ( ε ) −
1) + o ( ε ν ) . As λ ( ε ) − − Ev (1 ,ε ) B (0) + o ( ε ν ), we write λ ( ε ) − − Ev (1 ,ε ) B ( ε ) − C (1 , B (0)) − Ev (1 ,ε ) + o ( ε ν )= − Ev (1 ,ε ) B ( ε ) − C (1 , B (0)) − ( Ev (1 , ε )) + o ( ε ν ) . (51)One is thus left with the calculus of C (1 ,
0) and of B ( ε ) up to the order1. For the computation of B ( ε ), we need the behavior of the solutions of theoverdense system for λ = 1.As in Section 1, we introduce a j ( ξ ) = a j + ξa j + O ( ξ ) and b j ( ξ ) = b j + ξb j + O ( ξ ). We recall that ζA ( ζ, ε ) = P ∞ j =1 a j ( ξ ) ζ j and ζB ( ζ, ε ) = P ∞ j =1 b j ( ξ ) ζ j .Introduce u ( ζ ) = X j ≥ ζ j − b j , v ( ζ ) = X j ≥ ζ j − jb j , w ( ζ ) = X j ≥ ζ j − ja j , k ( ζ ) = X j ≥ ζ j − ja j . Lemma 6
The following relations are true e t ( F + G )( t, , ε ) − e t ( F + G )( t,
1) = − ε ν ν − + O ( ε ν )2 e t G ( t, , ε ) − e t G ( t,
1) = − ε ν ν − + O ( ε ν ) ζ ( t, ε ) − ζ ( t ) = − ξ νν − ζζA ( ζ, ε )(1 − ξ ( y )) − ζ A ( ζ ,
0) = ξζ [ k ( ζ ) − w ( ζ ) − νν − ( ζw ′ ( ζ ) + w ( ζ ))] + O ( ε ν ) ζB ( ζ, ε )(1 − ξ ( y )) − ζ B ( ζ ,
0) = ξζ [ v ( ζ ) − u ( ζ ) − νν − ( ζu ′ ( ζ ) + u ( ζ ))] + O ( ε ν ) For the computation of C (1 , , one has C (1 ,
0) = − lim λ → ,t → e t ( F + G )( t, λ ) − e t ( F + G )( t, λ − . From these two results, one obtains the following
Proposition 4
Introduce the function R ( t ) = ν ln t − ˜ K (1)( t ) − B (0) t − ν ,where ˜ K (1) has been introduced in (21)and note that the terms B and C which have been introduced in (46) are calculated through B (0) = − Z + ∞ s ν e − s ds = − − ν Γ(1 + 1 ν ) We have B ( ε ) = B (0) + ε ν ν − εν ) ν lim t → R ( t ) , and C (1 ,
0) = R + ∞ s ν e − s ds − ν R + ∞ ln se − s ds + ν R + ∞ s ν − e − s ˜ K (1)( s ) ds + B (0)2 ν R
10 1 − e − s s ds + ν R s ν − e − s [ ν ln s − R ( s )] ds Ev ( λ, ε ) = [( λ − e t ( F + G )( t, λ, ε ) + 2 e t G ( t, λ, ε )]( ξ ( y ) + (1 − λ )(1 − ξ ) ζA ( ζ, ε )) − e t ( F + G )( t, λ, ε )( λ − − λ )(1 − ξ ( y )) ζB ( ζ, ε )) . (52)Remember that we have( λ − B ( ε ) + C ( λ, ε )( λ − Ev ( λ, ε ) − Ev (1 , ε ) . We thus deduce the equality B ( ε ) + C ( λ, ε )( λ −
1) = ξ ( y ) e t G ( t,λ,ε ) − e t G ( t, ,ε ) λ − − (1 − ξ ( y ))[ e t ( F + G )( t, λ, ε )(1 − ζB ( ζ, ε )) + 2 e t G ( t, λ, ε ) ζA ( ζ, ε )]+(1 − λ )(1 − ξ ( y )) ζA ( ζ, ε ) e t ( F + G )( t, λ, ε )Recall that G ( t, λ, ε ) e t = ( ζν ) λ +12 ν g ( t, λ, ε ) and use g ( t, , ε ) = 1. We use also therelation (34) to get( F + λ +12 λ G )( t, λ, ε ) e − t ( ζν ) λ − ν = 2 R ∞ t ξ − λ ( εν ) λν e − s g ( s, λ, ε ) ds − λ +12 λ R ∞ t ξ − λ ( εν ) λν e − s [ dgds − g − ds (53)Note that we need two terms of G and of F + G , and that we use dgds = (1 − λ ) dds ( ˜ K λε ( g )) , g − − λ ) ˜ K λε ( g ) . This will contribute to the term in C . Rewrite the first term of (52) as ξ ( y ) 2 e t G ( t, λ, ε ) − e t G ( t, , ε ) λ − εν ) ν [ ( ζν ) λ − ν − λ − − ( ζν ) λ − ν ˜ K λε ( g )] . Its limit when λ goes to 1 is 2( εν ) ν [ ν ln( ζν ) − ˜ K ε (1)]. Hence we get the identity B ( ε ) = − (1 − ε ν ζ − ν )[ e t ( F + G )( t, , ε )(1 − ζB ( ζ, ε )) + 2 e t G ( t, , ε ) ζA ( ζ, ε )]+2( εν ) ν [ ν ln( ζν ) − ˜ K ε (1)] (54)and the right hand side is independant on t . Using Lemma 6, we obtain B ( ε ) = 2( εν ) ν [ ν ln( ζν ) − ˜ K ε (1)( t )]+(1 − ε ν ζ − ν )( − ε ν ν − )(1 − ζB ( ζ,
0) + 2 ζA ( ζ,
0) + o ( ε ν ))+(1 − ε ν ζ − ν )[ e t ( F + G )( t, − ζB ( ζ, ε )) + 2 e t G ( t, ζA ( ζ, ε )](55)from which one deduces B ( ε ) = 2( εν ) ν [ ν ln( ζν ) − ˜ K ε (1)] − ε ν ν − (1 − ζB ( ζ,
0) + 2 ζA ( ζ,
0) + o ( ε ν )) − (1 − ε ν ζ − ν ) B (0) − [ e t ( F + G )( t, ζ ( B ( ζ, − B ( ζ, ε )) + 2 e t G ( t, ζ ( A ( ζ, ε ) − ζA ( ζ, . G ( t,
1) = ( ζ ν ) ν e − t and F ( t,
1) = 2 e t R + ∞ t s ν e − s ds , onededuces that G ( t, F ( t,
1) goes to a constant when t → B ( ε ) = B (0) + ( εν ) ν lim t → [ 1 ν ln t − K (1)( t ) − t − ν B (0))] . (56)The second part consists in the calculus of C (1 , ε = 0 in (52), one obtains the two identities B (0) = − e t ( F + G )( t, , − ζ B ( ζ , − e t G ( t, , ζ A ( ζ , , ζ = νt.B (0) + C ( λ, λ −
1) = − e t ( F + G )( t, λ )(1 − ζ B ( ζ , − e t G ( t, λ ) ζ A ( ζ , − ( λ − ζ A ( ζ , e t ( F + G )( t, λ ) . Hence C ( λ, λ −
1) = − ( λ − ζ A ( ζ , e t ( F + G )( t, λ )+ e t ( F + G )( t, ζB ( ζ, − ζB ( ζ, − ζ B ( ζ , e t ( F + G )( t, − e t ( F + G )( t, λ ))+2 e t G ( t, ζA ( ζ, − ζA ( ζ, ζ A ( ζ , e t G ( t, − e t G ( t, λ )) . We get (as we work for ε = 0, we should write ζ but we drop this notation andwe use ζ = νt ) C ( λ,
0) = − ζA ( ζ, e t ( F + G )( t, λ ) − e t ( F + G )( t, ζ B ( ζ, − B ( ζ, λ − − (1 − ζB ( ζ, e t ( F + G )( t,λ ) − e t ( F + G )( t, λ − − e t G ( t, λ ) ζ A ( ζ, − ζA ( ζ, λ − − ζA ( ζ, e t G ( t,λ ) − e t G ( t, λ − . In this equality, one only needs the value for λ →
1, and it is independant of ζ .We thus consider the limit when λ → ζ →
0, hence one obtains C (1 ,
0) = − lim ζ → ,λ → e t ( F + G )( t, λ ) − e t ( F + G )( t, λ − . Equality (53) rewrites ( F + λ +12 λ G )( t, λ, ε ) e − t ( ζν ) λ − ν = − λ +12 λ R ∞ t ( ε λν ν − λν ξ − λ ) dds [ e − s g ( s, λ, ε )] ds +( λ − λ λ R ∞ t ( ε λν ν − λν ξ − λ ) dds [ ˜ K λε ( g ) e − s ] ds Hence, considering the limit ε →
0, one obtains( F + λ +12 λ G )( t, λ ) e − t ( ζν ) λ − ν = − λ +12 λ R ∞ t ( ζν ) λν dds [ e − s ] ds +( λ − λ λ R ∞ t ( ζν ) λν dds [ ˜ K λ ( g ) e − s ] ds λ = 1 is thus ( F + G )( t, e − t = 2 R + ∞ t ( ζν ) ν e − s ds . Hence e − t [( − λ λ G )( t, λ )( ζν ) λ − ν + ( F + G )( t, λ )( ζν ) λ − ν − ( F + G )( t, λ +1 λ R ∞ t ( ζν ) ν (( ζν ) λ − ν − e − s ds +( λ − λ λ R ∞ t ζ λν dds [ ˜ K λ ( g ) e − s ] ds Dividing by λ −
1, one deduces e − t [ − λ G ( t, λ )( ζν ) λ − ν + ( F + G )( t,λ ) − ( F + G )( t, λ − ( ζν ) λ − ν + ( F + G )( t, ( ζν ) λ − ν − λ − ]= λ +1 λ R ∞ t ( ζν ) λν − λ − e − s ds + λ λ R ∞ t ( ζν ) λν dds [ ˜ K λ ( g ) e − s ] ds We consider the limit when λ →
1, and recalling that for ε = 0 one has ζν = s ,denoting by H ( t, λ ) = ( F + G )( t,λ ) − ( F + G )( t, λ − , we obtain e − t [ − G ( t,
1) + H ( t,
1) + ( F + G )( t, ν ln ζν ]=2 R ∞ t ln se − s ds + R ∞ t s ν dds [ ˜ K (1) e − s ] ds Using again the integration by parts on the last term hence one gets e − t [ − G ( t,
1) + H ( t,
1) + ( F + G )( t, ν ln ζν ]=2 R ∞ t ln se − s ds − t ν ˜ K (1)( t ) e − t − R ∞ t dds ( s ν ) ˜ K (1) e − s ds We notice that the function R ( t ) = − ν ln t + 2 ˜ K (1)( t ) + t − ν B (0) has a finitelimit when t goes to zero, according to (56). We have the equality ˜ K ( t ) = R ( t ) + ν ln t − B (0) t − ν . We deduce that R t dds ( s ν ) ˜ K (1) e − s ds = R t dds ( ζ ν )[ R ( s ) + ν ln s − B (0) s − ν ] e − s ds = ν ν − R t s ν − [ R ( s ) + ν ln s − B (0) s − ν ] e − s ds In this last term, the only term which matters when t → − B (0) ν ν − Z t s − e − s ds = − B (0) ν ν − [ Z t e − s − s ds − ln t ] . Note that B (0) = − R + ∞ s ν e − s ds . One obtains B (0) = − Z ∞ ( a ν e − a da = − − ν Γ(1 + 1 ν ) . .1 Reduction of the Evans function Lower order terms
Recall that the operator ˜ K ε (1) is defined through (21).We prove the following lemma of reduction: Proof of Lemma 6
It is enough to prove that the relation giving ζ is − tε = C − νξ ν − ν − ξ ν − − ξ − ν R ( ξ )hence we deduce t = − Cε + ζν + ξ ζν − − R ( ξ ) ξ ζ. We thus obtain t = ζ ν , hence ζ − ζν = ξ ζν − O ( ξ ) ζ. We deduce that w ( ζ ) − w ( ζ ) = ( ζ − ζ ) w ′ ( ζ ) + O (( ζ − ζ ) ), hence w ( ζ ) − w ( ζ ) − ( ζ − ζ ) w ′ ( ζ ) = 0( ε ν ) and w ( ζ ) − w ( ζ ) − ( ζ − ζ ) w ′ ( ζ ) = 0( ε ν ).We use e t G ( t, λ, ε ) = ( εν ) λ +12 ν ξ − λ +12 g ( t, λ, ε ), hence for λ = 1 we obtain e t G ( t, , ε ) = η − . The equality giving F ( t, , ε ) being e − t F ( t, , ε ) = Z + ∞ t τ ( s, ε ) e − s η ( s, ε ) − ds = − e − t η ( t, ε ) − +2 Z + ∞ t e − s η ( s, ε ) − ds, one obtains e t ( F + G )( t, , ε ) − e t ( F + G )( t,
1) = 2 e t Z ∞ t e − s ( 1 η ( s, ε ) − η ( s,
0) ) ds.
Similarily e t (2 G ( t, , ε ) − G ( t, η ( s, ε ) [1 − η ( s, ε ) η ( s,
0) ] . Using the relation 1 η ( t, ε ) ν (1 + νν − ε ν η ( t, ε ) + O ( ε α )) = 1 η ( t, ν one obtains η ( t, ε ) η ( t, − ν − ε ν η ( t,
0) + O ( ε α ) . This gives directly the two equalities of Lemma 6.22 .2 Limit for large k of the growth rate Recall that was proven in [10] the following estimate on any value of γ such thatthere exists a solution of (9) associated with λ = gkγ and ε = kL : γ → Λ = r gL r ν ν ( ν + 1) ν +1 when k → + ∞ . If we compare with (13), one may see the difference between the result for L → k is fixed and the result for L > k → + ∞ . Note forexample that the limit of √ gk q kL ν ) ν Γ(1 + ν )when k → + ∞ is + ∞ because ν >
1. This is not surprising because we did notget the lower order terms up to the order ε of the expansion of λ . Remark thatthe term in ε comes from the terms in ε in the functions A ( ζ, ε ) and B ( ζ, ε ).We have the following result (according to [10]) Lemma 7
There exists k ∗ > such that, for all k ≥ k ∗ , there exists a real γ ( k ) and a non zero solution u ( x ) e iky + γ ( k ) t of the Rayleigh equation (4) such that Λ2 < γ ( k ) < Λ . We have the following behavior of the eigenmode || ρ u || + || ρ u ′ || + || u || + || u ′ || + || u ′′ || < + ∞ As the result of this Lemma is important for the nonlinear analysis, we rewritean idea of the proof, based on Remark 8.1 of [10]. We denote by L ρ the spaceof functions u such that ρ u ∈ L (IR). Finding γ is equivalent to finding 0 as an eigenvalue (in L (IR) ) of − k ρ − ddx ( ρ ddx ρ − ) + 1 − gγ k ( x ) . This operator rewrites − k d dx + 1 − gγ k ( x ) + k − W ( x ) where W ( x ) = k ′ ( x ) + ( k ( x )) , which is bounded when ρ ′′ ρ is bounded (or equivalently when k ′ is bounded). We introduce the operator Q = − k ρ − ddx ( ρ ddx ρ − )+1, whichis coercive, thanks to the Poincare estimates, for k large enough. The eigenvalueproblem rewrites γ g ∈ σ p ( Q − k Q − ) . Under the (natural) hypothesis that k has a nondegenerate minimum L , onededuces that for k large enough one has at least a value of γ ( k ) such that23 < g ( γ ( k )) < L using usual results on semiclassical Schrodinger operatorswhich potential has a well.We thus constructed v ∈ L (IR) and γ ( k ) such that v is the eigenvector of Q − k Q − associated with the eigenvalue ( γ ( k )) g .To v is associated a solution of (4) which is u = ρ − Q − v , u ′ ∈ L ρ , u ∈ L ρ .Remembering that u solves − u ′′ + k u − k ( x ) u ′ − gk ( γ ( k )) u = 0 , multiplying this equation by u and integrating, one gets Z ( k u + ( u ′ ) ) dx = Z k ( x ) ρ − u. [ gk ( γ ( k )) ρ u + ρ u ′ ] dx hence, using the hypothesis ρ ′ ρ − ≤ M one obtains (the norm on the Sobolev space H is || u || = R ( u ′ ) + k u dx ) || u || ≤ M [ gk ( γ ( k )) || ρ u || + || ρ u ′ || ]hence a control on the H norm of u (instead of having the weight ρ ).Moreover, as u ′′ = gk ( γ ( k )) k ( x ) u + k ( x ) u ′ − k u , one deduces that u ′′ ∈ L , andwe have iteratively the control of u in H s ( s ≤ s max , according to the numberof derivatives of k that we consider). 24 Towards a non linear analysis
We show in this Section that the result of Guo and Hwang [8] can be extendedin our set-up, even if the density profile ρ ( x ) does not satisfy the coercivityassumption (3) of [8]. The quantity k ( x ) = ρ ′ ( x ) ρ ( x ) plays a crucial role. It has aphysical interpretation, being the inverse of a length: it is called the inverse ofthe density gradient scalelength. We need the assumptions( H ) k ( x ) bounded , k ( x ) ρ − bounded . Note that k bounded is fulfilled in the case studied by Guo and Hwang (where ρ is bounded below), and in the case of the striation model (studied by R.Poncet [18]) but is not automatically fulfilled by a profile such that ρ ( x ) → x → −∞ . However, for the particular case of the ablation front profile,we have k ( x ) = L − ξ ( xL ) ν (1 − ξ ( xL )), hence it is bounded and belongs to[0 , L − ν ν ( ν +1) ν +1 ].Before starting the proof of Theorem 2, which is rather technical, let us de-scribe our procedure.Firstly, we prove that the linear system reduces to an elliptic equation on thepressure, from which we obtain a general solution. We identify a normal modesolution of this system using the first part of the paper.Once this normal mode solution U is constructed, with suitable assumptions onthe growth rate, one introduces a perturbation solution of the nonlinear system,which initial condition is δU | t =0 and an approximate solution V N of the nonlinear system which admits an expansion in δ N up to the order N with the sameinitial condition.Using the Duhamel principle for the construction of the j − th term of the ex-pansion in δ of V N , one obtains a control of all the terms of V N .The natural energy inequalities are on the quantities ρ u j , ρ v j , ρ − p j , ρ − ρ j .We verify that the properties of ρ ( x ) imply that we can deduce inequalitieson u j , v j , ρ − p j and T j .Note that we have, as a consequence of the method that we chose, a control in t s e Λ t of the H s norm of all solutions of the homogeneous linear system (withany initial condition U ( x, y, e jγ ( k ) t (with no additionalpower in t ) of the H s norm of the j − th term of the expansion. Remark 1
When an initial value mixes eigenmodes, the H s norm of the solu-tion behaves as t s e Λ t . If one starts from a pure eigenmode with Λ2 < γ ( k ) < Λ the exponential behavior comes at most from the growth of the pure eigenmode. .1 Obtention of a solution of the linear system Consider the system ∂ t σ + ρ ′ v = f ρ ∂ t v + ∂ x p = σg + f ρ ∂ t v + ∂ y p = f ∂ x v + ∂ y v = 0We know that the relevant quantities are ρ v , , ρ − σ , and we denote thesethree quantities by X, Y, τ . To have the same behavior when ρ →
0, consider ψ such that, once ψ is obtained, we revert to v and v using v = − ∂ y ( ρ − ψ ), v = ∂ x ( ρ − ψ ). Introduce b = ρ − [ ∂ y ( ρ v ) − ∂ x ( ρ v )] . (57)The system on v , v , σ, p implies the two equations ( ∂ t b = g∂ y τ + ρ − ( ∂ y f − ∂ x f ) ∂ t τ + k ( x ) X = ρ − f . (58)We obtain ψ from b through the elliptic equation∆ ψ − ( 12 k ′ + 14 k ) ψ = − b. (59)We then revert to X through the equality X = − ∂ y ψ . Finally, the pressure p is obtained through the elliptic equation ρ ∂ x ( ρ − ∂ x p ) + ∂ y p = ρ [ ρ ∂ x ( ρ − τ ) g + ρ ∂ x ( ρ − f ) + ρ − f ]which rewrites∆ p − k ∂ x p = ρ [ ∂ x τ g − k τ g ) + ρ − (div ~f − k f )] (60)Hence we solve the system ∂ t τ = k ∂ y ψ ( b ) + ρ − f ∂ t b = g∂ y τ + ρ − ( ∂ y f − ∂ x f ) τ (0) = τ ( x, y ) , b (0) = b ( x, y )∆ ψ − ( k ′ + k ) ψ = − b ∆ p − k ∂ x p = ρ [ ∂ x τ g − k τ g ) + ρ − (div ~f − k f )] (61)which has the same properties as the system (13) of [8], the Poincare estimatebeing still valid.From b and τ , one reverts to X and Y , hence a solution of the system. Moreover,one checks that ( X, Y ) ∈ L (IR) (according to the energy equality), hence X ∈ H (IR) under the assumption k bounded.26 roposition 5 Under the hypotheseses (H), and under the hypothesis h j ∈ L , j = 0 , , , the functions u , v , T , p solution of ∂ t T − k u = h ρ ∂ t u + ∂ x p + ρ gT = h ρ ∂ t v + ∂ y p = h ∂ x u + ∂ y v = 0 satisfies u ( t ) , v ( t ) , T ( t ) ∈ L when it is true for t = 0 . Moreover, one has ρ − p ( t ) ∈ L (IR ) . Proof
The proof of this result follows two steps: first of all the assumption k bounded implies that ρ u , ρ v , ρ − ∇ p , ρ T belong to L . We thus multiplythe equality ∂ t ~u + ρ − ∇ p + T ~g = ~h by ∇ ( ρ − p ). We get, integrating in x, y : Z ( ∇ q ) + k ( x ) q ∇ q .~e + T ~g ∇ q = Z ~h ∇ q from which one deduces ||∇ q || ≤ max( k ρ − ) || ρ q || + g || T || ∞ + || ~h || . It is then enough to use the Poincare estimate between ρ q and ρ − ∇ p toobtain the estimate on ∇ q , from which one deduces the estimate on q .Finally, from the estimate on q and on ∇ q , multiplying the equation on thevelocity by ~u and integrating, we get the Gronwall type inequality ddt || ~u || ≤ C || q || H + || ~h || + g || T || ∞ hence a control on || ~u || on [0 , T ] for all t as soon as it is true for t = 0.The system writes ∂ t T + ~u. ∇ T = uT k ( x ) ∂ t ~u + ( ~u. ∇ ) ~u + T ∇ Q + T Qk ( x ) ~e = (1 − T ) ~g div ~u = 0 (62)In the system (62), appear only quadratic terms. When one wants to deducethe term of order N in the system, plugging in the expansions T N , u N , v N and Q N one obtains source terms of the form S N = P N − j =2 u j T N − j k ( x ) − u j ∂ x T N − j − v j ∂ y T N − j R ,N = − P N − j =2 u j ∂ x u N − j + v j ∂ y u N − j + T j ∂ x Q N − j + T j Q N − j k ( x ) R ,N = − P N − j =2 u j ∂ x v N − j + v j ∂ y v N − j + T j ∂ y Q N − j and the system rewrites ∂ t T N − u N k ( x ) = S N ∂ t u N + ∂ x Q N + Q N k ( x ) + gT N = R ,N ∂ t v N + ∂ y Q N = R ,N ∂ x u N + ∂ y v N = 0 . (63)27 igher order Sobolev regularity (preparatory equality) One of themain tools that we have to use is the divergence free condition, in order to getrid of the pressure p or of the reduced pressure Q when obtaining the energyinequality. Recall that the system (63) rewrites ∂ t T N − u N k ( x ) = S N ρ ∂ t ~u N + ∇ ( ρ Q N ) + gρ T N ~e = ρ ~R N div ~u N = 0where ~R N = ( R ,N , R ,N ).Denote by ~G N = ρ ∂ t ~R N − gρ S N ~e . Applying the operator ∂ t ∂ nx n to equationon the velocity and using the equation on the specific volume, one obtains ∂ nx n ( ρ ∂ t ~u N ) + ∇ ∂ t ∂ nx n ( ρ Q N ) + g∂ nx n ( ρ k u N ) = ∂ nx n ( ~G N ) . (64)One deduces the Lemma 8
For all n , one has the estimate || ρ ∂ t ∂ nx n ~u N || ≤ C n ( X p ≤ n || ρ ∂ px p u N || + || ρ ∂ px p G N || ) . Moreover, as the coefficients of the system depend only on x , this inequality isalso true with the same constants when ∂ nx n is replaced by ∂ nx n ∂ qy q for all q ≥ . Proof
One notices that (64) writes ρ ∂ t ∂ nx n ~u N + ∇ ( ∂ t ∂ nx n ( ρ Q ))+ ~gk ( x ) ρ ∂ nx n u N = ~G N − n − X p =0 C pn ρ ( n − p )0 ∂ px p ∂ t ~u N − ~g n − X p =0 C pn ρ ( n − p +1)0 ∂ px p u N . Multiplying by ∂ t ∂ nx n ~u N and integrating, using the recurrence hypothesis that || ρ ∂ t ∂ px p ~u N || ≤ C p ( X m ≤ p − || ρ ∂ t ∂ mx m ~u N || )+2 g Λ ( X m ≤ p − || ρ ∂ mx m u N || )+ || G Nn − || as well as the inequalities | k ( x ) g | ≤ Λ , | ρ − ρ ( p )0 | ≤ Λ p (which are true as soon as k is a C ∞ function which derivatives are bounded,because ρ ′ = k ρ ) one obtains the inequality || ρ ∂ t ∂ nx n ~u N || ≤ C n ( X m ≤ n || ρ ∂ t ∂ mx m ~u N || ) . Lemma 8 is proven. 28 .2 The energy equalities
Note that the system for the leading term of the perturbation is the system (63)with a null source term. Owing to this remark, we shall treat the general caseand apply the equality to the particular cases.Multiplying (64) by ∂ t ∂ nx n ~u N and integrating, using the divergence free relation,one obtains R ∂ nx n ( ρ ∂ t ~u N ) .∂ nx n ∂ t ~u N dxdy + R g∂ nx n ( ρ k u N + ρ S N ) ~e .∂ t ∂ nx n ~u N dxdy = R ∂ nx n ( ρ ∂ t ~S N ) .∂ t ∂ nx n ~u N dxdy. In this equality, we can consider (for Sobolev inequalities) the term containingthe largest number of derivatives of ~u N . We obtain, denoting by ~R Nn = ∂ nx n ( ρ ∂ t ~u N ) − ρ ∂ nx n ∂ t ~u N B Nn = ∂ nx n ( ρ k u N ) − ρ k ∂ nx n u N the equality R ρ ∂ nx n ∂ t ~u N .∂ nx n ~u N dxdy + R gρ k ∂ nx n u N .∂ t ∂ nx n u N dxdy + R ~R Nn .∂ nx n ∂ t ~u N dxdy + R gB Nn .∂ t ∂ nx n u N dxdy = R ∂ nx n ( ρ ∂ t ~R N ) .∂ t ∂ nx n ~u N dxdy − R g∂ nx n ( ρ S N ) ~e .∂ t ∂ nx n ~u N dxdy. The terms ~R Nn and B Nn contain only derivatives of order less than n −
1, henceit will appear as a source term in the application of the Duhamel principle lateron. The two first terms of the previous equality are the exact derivative in timeof E Nn ( t ) = 12 [ Z ρ ( ∂ nx n ∂ t ~u N ) dxdy + Z gρ k ( ∂ nx n u N ) dxdy ] . The energy equality is thus E Nn ( t ) = E Nn (0) + Z t g Nn ( s ) ds where g Nn ( t ) = R ∂ nx n ( ρ ∂ t ~R N ) .∂ t ∂ nx n ~u N dxdy − R g∂ nx n ( ρ S N ) ~e .∂ t ∂ nx n ~u N dxdy = − ( R ~R Nn .∂ nx n ∂ t ~u N dxdy + R gB Nn .∂ t ∂ nx n u N dxdy ) . Note that this source term satisfies | g Nn ( t ) | ≤ || ρ ∂ nx n ∂ t ~u N || L K Nn ( t ) (65)where one has K Nn ( t ) ≤ || ρ − ~R Nn || + || ρ − ∂ nx n ( ρ ∂ t ~S N ) || + | g | [ || ρ − B Nn || + || ∂ nx n ( ρ S N ) || ] . (66)We are ready to prove the Duhamel inequality associated with this problem,using gk ( x ) ≤ Λ . 29 .3 The Duhamel principle Two versions of the behavior of the semi group will be deduced. The first onecorresponds to the general case for the terms in δ at least.We consider the (general) system ρ ( x ) ∂ t ~w + ∇ ( ρ ∂ t Q ) + gk ρ w~e = ~M , div ~w = 0 . (67)with the initial conditions ~w | t =0 = 0 , ∂ t ~w | t =0 = 0 . (68)Note that this system is easily deduced from the system obtained for the N thterm of the expansion in δ of the solution. Proposition 6
Assume that there exists two constants K and L , with L > Λ ,sich that || ρ − ~M || ≤ Ke Lt . (69) The unique solution of the linear system (67) with initial Cauchy conditions(68) satisfies the estimate || ρ ~w || ≤ KL ( L − Λ) (1 + Λ ( L − Λ) ) e Lt ≤ K ( L − Λ) e Lt || ρ ∂ t ~w || ≤ KL − Λ (1 + Λ ( L − Λ) ) e Lt || ρ ∂ t ~w || ≤ K (1 + ( L − Λ) ) e Lt Proof
We begin by multyplying the equation (67) by ∂ t ~w and integrate inspace. One deduces that || ρ ∂ t ~w || ≤ Λ || ρ w || + Ke Lt . We will make use of this equality later.Let us multiply the equation (67) by ∂ t ~w . We obtain the identity ddt ( 12 Z ρ ( ∂ t ~w ) dxdy + 12 Z k ρ k w dxdy = Z M ( x, y, t ) ∂ t ~wdxdy. Integrating in time and using the initial condition (68) as well as the estimate(69), we obtain the inequality Z ρ ( ∂ t ~w ) dxdy ≤ Λ Z k ρ w dxdy + 2 K Z t e Ls || ρ ∂ t ~w || ( s ) ds. Let us introduce now u ( t ) = R t || ρ ∂ t ~w || ( s ) ds . We obtain, considering ddt R ρ ~w dxdy ,that || ρ ~w || ( t ) ≤ Z t || ρ ∂ t ~w || ( s ) ds || ρ ~w || ( t ) ≤ u ( t ) . Hence the inequality( u ′ ( t )) ≤ Λ ( u ( t )) + Z t Ke Ls u ′ ( s ) ds. From this inequality, we deduce that( u ′ ( t )) ≤ Λ ( u ( t )) + 2 Ke Lt u ( t )hence u ′ ( t ) ≤ Λ u ( t ) + q Ke Lt u ( t ) . Introduce h such that u ( t ) = ( h ( t )) e Λ t . We obtain the inequality2 hh ′ e Λ t ≤ √ Khe L +Λ2 t hence h ′ ( t ) ≤ √ Ke L − Λ2 t that is h ( t ) ≤ √ KL − Λ e L − Λ2 t which leads to u ( t ) ≤ K ( L − Λ) e Lt The estimate on u ′ ( t ) follows, using ( u ′ ) ≤ Λ u + 2 Ke Lt u . We thus, byintegration, deduce another estimate on u . The estimate on ρ ∂ t ~w is the con-sequence of (8).If one wants a general formulation of the Duhamel principle (taking into ac-count non zero initial values), one states the following proposition, which willlead to the result of proposition 8, hence allowing a mixing of modes and a weaknonlinear result. The mixing of modes is not our purpose here, but we shallnot speak of weak nonlinear results. See Cherfils, Garnier, Holstein [4] for moredetails. Proposition 7
The solution of ddt ( Z ( ρ ( ∂ t ~u N ) − g ρ ′ ρ ρ ( u N ) ) dxdy ) = g ( t, x, ∂ t ~u N ) with initial condition ∂ t ~u N (0) , ~u N (0) , with the assumption | g ( t, x, ∂ t ~u N ) | ≤ K ( t ) || ρ ∂ t ~u n || L where K is a positive increasing function for t ≥ satisfies the inequalities || ρ ~u N || ≤ [ C + R t p K ( s ) e − Λ s ds ] e Λ2 t || ρ ∂ t ~u N || ≤ [ C + R t p K ( s ) e − Λ s ds ] e Λ t where C depends on the initial data. roof We deduce from the energy equality the following inequality: Z ρ ( x )( ∂ t ~u N ) dxdy − g Z k ( x ) ρ ( x ) u N dxdy ≤ C , + +2 Z t K ( s ) || ρ ∂ t ~u N || L ( s ) ds where C = R ρ ( x )( ∂ t ~u N ) (0) dxdy − g R k ( x ) ρ ( x ) u N (0) dxdy and C , + =max( C , u ( t ) = || ρ ~u N (0) || + R t || ρ ∂ t ~u N ( s ) || ds = || ρ ~u N (0) || + R t || ρ ∂ t ~u N || ( s ) ds . We notice that u ′ ( t ) = || ρ ∂ t ~u N || ( t ) hence u ′ ( t ) ≥
0. Recall that gk ( x ) ≤ Λ . The inequality implies( u ′ ( t )) ≤ Λ ( u ( t )) + C , + + 2 R t K ( s ) u ′ ( s ) ds ≤ Λ ( u ( t )) + C , + + 2 K ( t ) u ( t ) ≤ (Λ u + K ( t )Λ ) + C , + − K ( t ) Λ . Use now the inequality ( a + b + c ) ≤ a + b + c for positive numbers a, b, c to obtain u ′ ( t ) ≤ Λ u ( t ) + p C , + + p K ( t ) u ( t ) . Introducing v ( t ) = u ( t ) e − Λ t which satisfies v ( t ) ≥ u (0) e − Λ t , we deduce v ′ ( t ) ≤ p C , + e − Λ t + q K ( t ) e − Λ t v ( t ) . • Assume u (0) >
0. We obtain, denoting by h ( t ) = p v ( t )2 hh ′ ≤ p C , + e − Λ t + q K ( t ) e − Λ t h ( t )hence 2 h ′ ≤ ( C , + u (0) ) e − Λ t + q K ( t ) e − Λ t . We deduce the inequality h ( t ) ≤ h (0) + Λ − ( C , + u (0) ) (1 − e − Λ t ) + 1 √ Z t q K ( s ) e − Λ s ds. which imply that there exists A and B such that u ( t ) ≤ ( A e Λ t + B e Λ t ( Z t q K ( s ) e − Λ s ds ) ) . • Assume u (0) = u ′ (0) = 0. As C , + = 0, we have the inequality u ′ ( t ) ≤ Λ u ( t ) + p K ( t ) u ( t )from which one deduces, with the same notations as above, that h ′ ( t ) ≤ r K ( t ) e − Λ t h (0) = 0 one obtains h ( t ) ≤ Z t r K ( s ) e − Λ s ds. • Assume finally u (0) = 0 and u ′ (0) >
0. We obtain v ′ ( t ) ≤ p C , + e − Λ t + q K ( t ) e − Λ t v ( t ) . Introduce ˜ v ( t ) = v ( t ) − p C , + 1 − e − Λ t Λ . We have˜ v ′ ( t ) ≤ r K ( t ) e − Λ t (˜ v ( t ) + p C , + − e − Λ t Λ ) ≤ s K ( t ) e − Λ t (˜ v ( t ) + p C , + Λ )from which one deduces the inequality2 s ˜ v + p C , + Λ ≤ s p C , + Λ + Z t q K ( s ) e − Λ s ds. In all the previous cases, we deduced the inequality u ( t ) ≤ [ C + R t p K ( s ) e − Λ s ds ] e Λ t .Using finally the relation ddt || ρ ~u N || ≤ || ρ ∂ t ~u N || = u ′ ( t )we get || ρ ~u N || ≤ u ( t ) − u (0) . These are the two estimates of Proposition 7.Of course, the proof is much simpler in the case we are interested in, that is ∂ t ~u N = 0, ~u N = 0, where (using the notations of this paragraph, C = C , + = u (0) = u ′ (0) = 0), where one deduces easily q u ( t ) e − Λ t ≤ Z t − q K ( s ) e − Λ s ds. H s estimates for a general solution of the linearizedsystem The H s inequalities for the solution of the homogeneous system Weconsider the system satisfied by the leading order term of the perturbation ofthe Euler system (which is the system (62), particular case of (63) for N = 1.We prove in this section the analogous of the Proposition 1 of [8], with a slightlybetter estimate which shows essentially that the relevant growth rate is, up to polynomial terms , Λ: 33 roposition 8 Let T ( t ) , ~u ( t ) be the solution of the modified linearized Eulersystem (62). There exists a constant C s depending only on the characteristicsof the system, that is of k and g , such that || ρ T ( t ) || H s + || ρ ~u ( t ) || H s ≤ C s (1+ t ) s exp(Λ t )( || ρ T (0) || H s + || ρ ~u (0) || H s ) . Note that in these inequalities (which are general) a power of t appears in thebound for the norm H s . This is the general case. Note that similar estimateswere obtained independantly by R. Poncet [18].An important feature of this result takes in consideration an initial conditionwhich is not an eigenmode of the Rayleigh equation, and which is a combinationof different eigenmodes. As we shall see in what follows, the interaction of thesedifferent eigenmodes lead to a linear growth of the form (1 + t ) s e Λ t for the H s norm of the solution. Proof
We prove in a first stage the H s inequality result for the system satisfiedby ( T , u , v , Q ). We use the pressure p in the analysis. The system implythe equation ρ ( x ) ∂ t ~u + ∇ ∂ t p = ρ ~gk ′ u . We apply the operator D m,p to this equation. The energy inequality deducedfrom (65) and from the inequality (66) is(( u n ) ′ ) ≤ Λ ( u n ) + C + K n ( t ) u n ( t )where we have the estimate K n ( t ) ≤ || ρ − ~R n || + | g ||| ρ − B n || .
1. principal termThe inequation on || ρ ~u || writes( ddt || ρ ~u || ) ≤ Λ || ρ ~u || + C hence one obtains the inequality || ρ ~u || ≤ || ρ ~u (0) || cosh Λ t + r C Λ + || ρ ~u (0) || sinh Λ t ≤ D e Λ t .
2. derivative of the principal termIn the inequality obtained for D ,p ~u , the source term g is bounded by M D e Λ t || D ,p ∂ t ~u || because it contains only derivatives of order n − u ) ′ ) ≤ Λ ( u ) + C + 2 M D e Λ t u ( t )34rom which one deduces(( u ) ′ ( t )) ≤ (Λ u + M D Λ e Λ t ) + C − ( M D Λ ) e t hence ( u ) ′ ( t ) ≤ Λ u ( t ) + M D Λ e Λ t + p C that is ddt ( u e − Λ t ) ≤ M D Λ + p C e − Λ t from which one deduces u ( t ) ≤ Λ − ( p C + M D t + Λ u (0)) e Λ t .
3. Greater order term:We prove thus by recurrence that there exists A n and B n such that u n ( t ) ≤ ( A n + B n t ) n e Λ t , according to the inequality ddt ( u n ( t ) e − Λ t ) ≤ p C n e − Λ t + ( A n − + tB n − ) n − . One deduces the same inequality for ddt u n ( t ).4. In the derivative D n,p , the only term which matters for the order of thepower of t is n , hence one deduces that X n + p = s ( || ρ D n,p ~u N || + || ρ D n,p ∂ t ~u N || ) ≤ ( C s + tD s ) s e Λ t Proposition 8 is proven. Note that this improvement does not change the be-havior of the approximate solution we intend to construct, because for a normalmode solution u ( x, y, t ) = ˆ u ( x ) e iky + γ ( k ) t , where γ ( k ) has been calculated and where ˆ u ( x ) is solution of the Rayleighequation, one has the following equalities: || ρ D m,p ~u ( t ) || = || ρ D m,p ~u (0) || e γ ( k ) t || D m,p T ( t ) || = || T (0) || e γ ( k ) t || ρ D m,p Q ( t ) || = || ρ D m,p Q (0) || e γ ( k ) t . (70)Remark that, according to Lemma 7, and to the equality ikQ ( x, y, t ) = γ ( k ) ik ∂ x u ( x, y, t ),we have also the relations || D m,p ~u ( t ) || = || D m,p ~u (0) || e γ ( k ) t || D m,p Q ( t ) || = || D m,p Q (0) || e γ ( k ) t . (71)35 .5 The H s inequalities for the linearized system We consider the system (63). We apply the operator D m,p = ∂ mx ∂ py . This systembecomes ∂ t D m,p T N − D m,p u N k ( x ) = D m,p S N + P p − q =0 C qp D m,q u N k ( q − p )0 ( x ) ∂ t D m,p u N + ρ − ∂ x ( ρ D m,p Q N ) + gD m,p T N = D m,p S N − P p − q =0 C qp D m,q Q N k ( q − p )0 ( x ) ∂ t D m,p v N + ∂ y D m,p Q N = D m,p S N ∂ x D m,p u N + ∂ y D m,p v N = 0 . (72)We notice that this system writes as the system (63) with a source term involvingderivatives of the solution at a lesser order of derivatives in x .We introduce u Nn ( t ) = || ρ ∂ nx n ~u N (0) || L + Z t || ρ ∂ t ∂ nx n ~u N ( s ) || L ds and v Nn ( t ) = u Nn ( t ) e − Λ t .We are now ready to study the behavior of the lower order terms of theexpansion, assuming that we found a γ ( k ) such that Λ2 < γ ( k ) < Λ.We have to deal in a second part with terms of the form u Nn , where N ≥
2. Inthis set-up one has to use Proposition 7, because we cannot obtain the sharpestinequality using the estimate u ′ ≤ Λ u + K + √ C . Recall that from Lemma 7 (proven in [10]), there exists a normal mode solutionof the linearized system of the form ˆ u ( x, k ) e iky + γ ( k ) t where Λ2 ≤ γ ( k ) < Λ.With this normal mode solution one constructs an approximate solution of thenonlinear system, of the form T N ( x, y, t ) = 1 + P Nj =1 δ j T j ( x, y, t ) u N ( x, y, t ) = P Nj =1 δ j u j ( x, y, t ) v N ( x, y, t ) = P Nj =1 δ j v j ( x, y, t ) Q N ( x, y, t ) = Q ( x ) + P Nj =1 δ j Q j ( x, y, t ) . There is an important Lemma, which depends on Hypothesis (H):
Lemma 9
The functions u j , v j , Q j , T j belong to L . The proof of this Lemma is a consequence of Proposition 5, which will lead tothe control of the source term of the linear system on T N , u N , v N , Q N .We shall use the estimates of Cordier, Grenier and Guo [5], and the method ofGuo and Hwang [8] to give an H s estimate of T N , u N , v N , Q N and a L estimateof T N − T − δT , u N − δu , v N − δv to obtain a lower bound on T N , u N , v N .36e prove in this section the H s estimate ~u N in the weighted norm || ρ . || . Usingthe assumption k ρ − bounded, we deduce estimates in H s for ~u N . The firstresult reads as Proposition 9
There exists constants C p and A p , depending only on the char-acteristics of the system (namely g , k ( x ) and its derivatives) and on the H p norm of the initial data such that u Np ( t ) ≤ ( C p ) N ( A p ) N − e Nγ ( k ) t . Remark 2
This estimate relies heavily, as in [7], on the quadratic structure ofthe nonlinearity, and that we give the precise estimate on the constant C j whichappears in (13) of [7]. This estimate could not be obtained in the set-up of Guoand Hwang [8] because the nonlinearity was written using ρ~u. ∇ ~u , hence a cubicnonlinearity. A second comment is the following: the inequality 2 γ ( k ) > Λ allows us to forgetthe coefficient (1 + t ) s in the H s estimate for a general solution of the linearsystem (obtained in Proposition 8). This is a consequence, as we shall see below,of the relation e Λ t Z t e ( Nγ ( k ) − Λ) s ds ≤ N γ ( k ) − Λ e Nγ ( k ) t (to be compared with the relation e Λ t R t e (Λ − Λ) s ds ≤ te Λ t ). Case N = 2Recall that we have the following system ∂ t T − k ( x ) u = − u ∂ x T − v ∂ y T − T u ρ ( x ) ∂ t ~u + ∇ ( ρ Q ) + gT = − ρ ( x )[ ~u . ∇ ~u ] − ρ T ∂ x Q − ρ ′ Q T div ~u = 0We have thus the estimates || ρ ∂ t S j || + || ρ S j || ≤ C j e γ ( k ) t . This means that K ( t ) ≤ D e γ ( k ) t , hence || ρ ∂ t ~u || ≤ ( C + Z t p D e ( γ ( k ) − Λ2 ) s ds ) e Λ t hence the inequality || ρ ∂ t ~u || + || ρ ~u || ≤ M e γ ( k ) t . We need to derive estimates for the terms T and Q . For the term T , one has ddt Z ρ T dxdy = Z k ( x ) ρ u T + Z S ρ T dxdy ddt || ρ T || ≤ M || ρ u || + || ρ S || ≤ ( M M + C ) e γ ( k ) t hence the estimate || ρ T ( t ) || ≤ || ρ T (0) || + C + M M γ ( k ) ( e γ ( k ) t − . As for the estimate on Q , one deduces ∂ x ( ρ − ∂ x ( ρ Q )) + ∂ y Q + g∂ x T = div ~S which imply estimates on Q . Case N ≥ . We start with the induction hypothesis that, for j ≤ N −
1, there exists C and A such that || ρ ~u j || + || ρ ∂ x ~u j || + || ρ ∂ y ~u j || + || ρ ∂ x T j || + || ρ ∂ y T j || + || ρ T j || ≤ A j − C j e jγ ( k ) t and that the derivative in time of all quantities is bounded by jγ ( k ) A j − C j e jγ ( k ) t .Thus there exists M (independant on the number of terms which appear in thesource term and which depends only on the coefficients of the system) such thatthe source term of (66) for n = 0 is bounded by: K N ( t ) ≤ M A N − C N N γ ( k ) e Nγ ( k ) t . (73)Note that in this estimate the N term comes, one from the number of the termsin the expansion P N − j =0 A j B N − j and a second one from the derivative in timewhich appears in the source term ∂ t ~S N . We thus obtain, using h N ( t ) ≤ Z t q K N ( s ) e − Λ s ds the inequality h N ( t ) ≤ q M A N − C N N γ ( k ) Z t p e ( Nγ ( k ) − Λ) s ds which yields h n ( t ) ≤ A N − C N e ( Nγ ( k ) − Λ) t M N γ ( k )( N γ ( k ) − Λ) A .
The choice of A is thus induced by MN γ ( k )( Nγ ( k ) − Λ) A ≤ N (forgetting thatwe have to be more precise to obtain estimates not only on ~u N but also on T N ) Note also that if we consider a cubic model, the number of terms in the source term is N ( N − N in the estimate. As we can see in thefollowing lines, this gives a less efficient estimate A = Mγ ( k ) γ ( k ) − Λ2 . The value of C is thus given by thenorm of the leading term ( T , u , v , Q ).The final estimate is || ρ ~u N || ≤ C N A N − e Nγ ( k ) t We proved the assumption (73).We use this result and the estimates for a normal mode solution (on which nopowers of t appear for the norms of the derivatives). We obtain h ( t ) ≤ h (0) + Λ − ( C , + u (0) ) + N C N γ ( k ) Z t e Nγ ( k ) − Λ2 s ds hence as N γ ( k ) > Λ one gets h ( t ) ≤ h (0) + Λ − ( C , + u (0) ) + NN γ ( k ) − Λ C N γ ( k ) e Nγ ( k ) − Λ2 t . We deduce the inequality (using ( a + b ) ≤ a + b )) u ( t ) ≤ h (0) + u (0)) + Λ − ( C , + u (0) ) ) e Λ t + 2( NN γ ( k ) − Λ ) C N γ ( k ) e Nγ ( k ) t . Remark
If the system has a cubic source term, at each stage of the construc-tion one gets N M N − C N as estimate, hence the convergence of the infiniteseries is not ensured by these estimates. In this paragraph, we derive estimates on the global approximate solution. Weshall use throughout what follows the Moser estimates, that we recall here || D α ( f g ) || L ≤ C ( || f || ∞ || g || s + || g || ∞ || f || s ) (74)and || D α ( f g ) − f D α g || L ≤ C ( || Df || ∞ || g || s − + || g || ∞ || f || s ) (75)and the Sobolev embedding || f || ∞ ≤ C || f || s for s > d and ||∇ f || ∞ ≤ C || f || s for s > d + 1. More precisely, we prove that Proposition 10
For all θ < and for all t < γ ( k ) ln θδC A , we have || T N − || H s + || ~u N || H s + || Q N − q || H s ≤ C δAC e γ ( k ) t − δAC e γ ( k ) t || T N − || L ≥ || T (0) || L δe γ ( k ) t − AC C δ e γ ( k ) t − δAC e γ ( k ) t || u N || L ≥ || u (0) || L δe γ ( k ) t − AC C δ e γ ( k ) t − δAC e γ ( k ) t | v N || L ≥ || v (0) || L δe γ ( k ) t − AC C δ e γ ( k ) t − δAC e γ ( k ) t . We have also the following estimates for the remainder terms || ~R N || H s + || S N || H s ≤ M δ N +1 ( N + 1) A N − C N +20 δ N +1 e ( N +1) γ ( k ) t . Proof
We have proven the H s estimates for all the terms of the expansion u j , v j , T j , Q j . It is this easy to deduce, using (73), the estimate for the remainderterms. This comes from the inequality (1 ≤ j ≤ N − || D α ( u j ∂ u N − j ) || ≤ C ( || u j || ∞ || u N − j || H | α | +1 + || u j || | α | || ∂ u N − j || ∞ )(and subsequent inequalities), the Solobev embedding || f || ∞ ≤ || f || and the H s estimate for s = 2 , H s of the terms of the expansion in δ j of order less than N is boundedby C j A j − e jγ ( k ) t . We thus deduce that || N X j =1 T j || H s ≤ N X j =1 CA j − C j δ j e jγ ( k ) t = CC δe γ ( k ) t − ( C Aδ ) N − e ( N − γ ( k ) t − C Aδe γ ( k ) t . When t < T θδ = γ ( k ) ln θδC A , we obtain 1 − C Aδe γ ( k ) t ≥ − θ , hence we deducethe estimate || T N − || H s = || N X j =1 T j || H s ≤ CC − θ δe γ ( k ) t . Moreover, one has || T N − || L ≥ δ || T || L − N X j =2 δ j || T j || hence using N X j =2 δ j || T j || L ≤ N X j =2 δ j || T j || L ≤ C AC − θ δ e γ ( k ) t one obtains || T N − || L ≥ δ || T (0) || L e γ ( k ) t − C AC − θ δ e γ ( k ) t . One may thus consider C = C − θ We thus deduce that, for t < γ ( k ) ln || T (0) || L (1 − θ ) CAC ,we obtain || T N − || L ≥ δ || T (0) || L e γ ( k ) t . Similar estimates hold for || ~u N || L .Note that this proves that the first term of the expansion is the leading term of40he approximate total solution.For all what follows, we introduce I ( t ) = AC e γ ( k ) t − δAC e γ ( k ) t (76) I N +1 ( t ) = N A N − C N +10 e ( N +1) γ ( k ) t . (77) We constructed in the previous section a solution T N , ~u N , Q N such that ∂ t T N + ~u N ∇ T N − k ( x ) u N T N = S N ∂ t ~u N + ~u N . ∇ ~u N + T N ρ − ∇ ( ρ Q N ) = ~g + ~R N div ~u N = 0 (78)with the following properties for the remainder terms: || ρ ∂ nx n R Nj || + || ρ ∂ nx n S N || ≤ C n δ N +1 I N +1 ( t ) (79) || ∂ nx n R Nj || + || ∂ nx n S N || ≤ C n δ N +1 I N +1 ( t ) (80)the constant C n depending on the Sobolev norm with weight ρ of the initialvalue of the normal mode solution and of the characteristic constants of theproblem.We deduced from this equality and the additional assumption k ρ − boundedthat we have identical estimates on ~R N and S N : || ∂ nx n R Nj || + || ∂ nx n S N || ≤ C n δ N +1 I N +1 ( t ) (81)We study in this Section the global solution of the Euler system (62) to obtainSobolev estimates on the difference between the approximate solution and thefull solution. Let T d = T − T N , ~u d = ~u − ~u N , Q d = Q − Q N . We have thefollowing system of equations: ∂ t T d + ~u N ∇ T d + ~u d ∇ T N = k ( uT d + u d T N ) − S N ρ ( ∂ t ~u d + ~u d ∇ ~u + ~u N ∇ ~u d ) + T ∇ ( ρ Q ) − T N ∇ ( ρ Q N ) = − ρ ~R N div ~u d = 0 . (82)Before stating the results on the difference quantities according to the sys-tem, we use the properties of T N − , ~u N , Q N : Lemma 10
Let t ∈ [0 , T θδ ] .For all α , there exists a constant C ( | α | ) such that || D α ( ~u d . ∇ ~u N ) || ≤ C ( | α | ) || ~u d || | α | CC δ − θ e γ ( k ) t || D α ( T d ( ∇ Q N + k Q N )) || ≤ C ( | α | ) || T d || | α | CC δ − θ e γ ( k ) t || D α ( k u d ( T N − || ≤ C ( | α | ) || ~u d || | α | CC δ − θ e γ ( k ) t || D α ( T N − ∇ Q d + k Q d ~e ) || ≤ C ( | α | ) || Q d || | α | +1 CC δ − θ e γ ( k ) t D α ( f g N ) = X C βα D β f D α − β g N and we use the estimate || D α − β g N || ∞ ≤ C || g N || | α |−| β | , as well as the H s result on any term of the form g N = P Nj =1 δ j g j , where g j = u j , v j , T j , Q j toconclude for any term studied in the Lemma. Moreover, we use the Moserestimates to obtain || D α ( ~u. ∇ f ) − ~u. ∇ D α f || ≤ C ( ||∇ ~u || ∞ ||∇ f || | α |− + ||∇ f || ∞ || ~u || | α | )hence, using ~u = ~u N + ~u d , one deduces || D α ( ~u. ∇ f ) − ~u. ∇ D α f || ≤ C ( ||∇ ~u d || ∞ || f || | α | + δI ( t ) || f || | α | + ||∇ f || ∞ || ~u || | α | + ||∇ f || ∞ δI ( t ))and, similarily || D α ( ~u. ∇ f ) || ≤ C ( δI ( t )( ||∇ f || ∞ + || f || | α | +1 ) + || ~u d || ∞ || f || | α | +1 + || ~u d || α ||∇ f || ∞ , || D α ( T ∇ Q d ) − T ∇ D α Q d || ≤ C ( δI ( t ) || Q d || | α | + ||∇ T d || ∞ ||∇ Q d || | α |− + ||∇ Q d || ∞ || T d || | α | according to the equality D α ( T ∇ Q d ) − T ∇ D α Q d = D α (( T − ∇ Q d ) − ( T − ∇ D α Q d .We shall also use the following estimates || D α ( ~u d . ∇ ~u d ) || ≤ C ( | α | ) || ~u d || || ~u d || | α | +1 (83) || D α ( ~u d . ∇ ~u d ) − ~u d . ∇ D α ~u d || ≤ C ( | α | ) || ~u d || || ~u d || | α | . (84)These equalities come respectively from (74) and (75).Introduce in what follows ~V = ~u d . ∇ ~u N + ~u N . ∇ ~u d , ~W = ~V + T d ρ − ∇ ( ρ Q N ).We have the estimates || D α ~V || ≤ M | α | I ( t ) δ || ~u d || | α | +1 || D α ~W || ≤ M | α | I ( t ) δ ( || ~u d || | α | +1 + || T d || | α | ) , ∀ α || D α ( T d ∇ Q d ) − T d ∇ D α Q d || ≤ C ( || T d || || Q d || + || T d || || Q d || ) for | α | = 2 || T d ρ − ∇ ( ρ Q N ) || ≤ δI ( t ) || T d || The equation on the density yield ∂ t T d + ~u. ∇ T d − k uT d = k u d T N − ~u d . ∇ T N − S N . Apply the operator D α and denote by W α = D α ( ~u. ∇ T d ) − ~u. ∇ D α T d . Thisequation rewrites ∂ t D α T d + ~u. ∇ D α T d + W α − D α ( k uT d ) + D α ( ~u d . ∇ T N ) − D α ( k u d T N ) = 0 .
42e can decompose W α − D α ( k uT d ) into two parts, the one with ~u N , the otherone with ~u d , denoted respectively by W α and W Nα . It is clear that || W Nα − D α ( k uT d ) + D α ( ~u d . ∇ T N ) − D α k u d T N ) || ≤ Cδ ( || ~u d || | α | + || T d || | α | ) . it is also clear that, using Moser estimates, || W α || ≤ C ( ||∇ ~u d || ∞ || T d || | α | + || ~u d || | α | || T d || ∞ ). One is thus left with the inequality ddt || D α T d || ≤ || W α || + || D α S N || + CδI ( t )( || ~u d || | α | + || T d || | α | ) . We have thus the estimate ddt || D α T d || ≤ C ( ||∇ ~u d || ∞ || T d || | α | + || ~u d || | α | || T d || ∞ ) + δ N +1 I N +1 M + CδI ( t )( || ~u d || | α | + || T d || | α | ) . We obtained the relations ||∇ Q d || ≤ M ( || ~u d . ∇ ~u d || + δI ( t )( || ~u d || + || T d || ) + δ N +1 I N +1 ( t )) . P | α =1 ||∇ D α Q d || ≤ M ( P | α | =1 || D α ( ~u d . ∇ ~u d ) || + δI ( t )[ || ~u d || + || T d || ]+ δ N +1 I N +1 ( t ) + (1 + δI ( t ) + || T d || )( || ~u d . ∇ ~u d || + δI ( t )( || T d || + || ~u d || ) + δ N +1 I N +1 ( t ))Using the fact that t ≤ T δ , one obtains ||∇ Q d || ≤ M ( || ~u d . ∇ ~u d || + || ~u d || + || T d || + δ N +1 I N +1 ( t )) . P | α =1 ||∇ D α Q d || ≤ M ( P | α | =1 || D α ( ~u d . ∇ ~u d ) || + || ~u d || + || T d || +(1 + || T d || )( || ~u d . ∇ ~u d || + || T d || + || ~u d || ) + δ N +1 I N +1 ( t ))In what follows, we introduce ~G αN = D α ( T N ∇ Q d ) − T N ∇ D α Q d + D α ( T N Q d k ~e )+ D α ( T d ∇ Q N + k T d Q N ~e )+ D α ~R N ,~G α = D α ( T d ∇ Q d ) − T d ∇ D α Q d + D α ( T d Q d k ~e ) . The equation on D α ~u d is ∂ t D α ~u d + D α ( ~u d . ∇ ~u d ) + D α ~R N + ~G α + ~G αN + T ∇ D α Q d = 0 . When one multiplies by ∇ D α Q d , one uses the divergence free condition on D α ~u d to get the estimate23 ||∇ D α Q d || ≤ || D α ( ~u d . ∇ ~u d ) || + || D α ~R N || + || ~G α || + || ~G αN || . We use || ~G αN || ≤ C (1 + t ) | α | +3 ( || Q d || | α | + || T d || | α | + || ~u d || | α | +1 ) δI ( t )43nd || ~G α || ≤ C ( ||∇ T d || ∞ || Q d || | α | + || T d || | α | ||∇ Q d || ∞ + || T d || ∞ || Q d || | α | ) . Hence we obtain (and it is pertinent for | α | > ||∇ D α Q d || ≤ C ′ ( || D α ( ~u d . ∇ ~u d ) || + || D α ~R N || ) + CδI ( t )( || Q d || | α | + || T d || | α | + || ~u d || | α | +1 )+ C ( || T d || || Q d || | α | + || T d || | α | || Q d || ) . For | α | = 2, we will obtain || Q d || , which is important.We use the equality, for | α | = 2 || ~G α || = || D α T d ∇ Q d + X <β<α D β T d ∇ D α − β Q d C βα || which leads to the inequality || ~G α || ≤ D ( || T d || || Q d || + || T d || || Q d || ) . Replacing this estimate in the inequality for α such that | α | = 2, one gets || D α ∇ Q d || ≤ C ( || D α ( ~u d . ∇ ~u d ) || + || T d || + || ~u d || +(1+ || T d || ) || Q d || + || T d || || Q d || + δ N +1 I N +1 ( t ) . Using the inequalities on || Q d || and || Q d || , one gets || Q d || ≤ M ( || ~u d . ∇ ~u d || + || ~u d || + || T d || + δ N +1 I N +1 ( t )) || Q d || ≤ M ( || ~u d . ∇ ~u d || + || ~u d || + || T d || + δ N +1 I N +1 ( t )(1+ || T d || )+(1+ || T d || )(1+ || T d || + || ~u d . ∇ ~u d || )) || Q d || ≤ M ( || ~u d . ∇ ~u d || + || ~u d || + || T d || + (1 + || T d || + || T d || ) || ~u d . ∇ ~u d || + (1 + || T d || ) || ~u d || + || T d || || ~u d || + δ N +1 I N +1 ( t )(1 + || T d || + (1 + || T d || ) ))We use then the inequalities ||∇ D α Q d || ≤ C ( || D α ( ~u d . ∇ ~u d ) || + || D α ~R N || + || Q d || | α | + || T d || | α | (1 + || Q d || ) + || ~u d || | α | +1 + (1 + || T d || ) || Q d || | α | )from which one obtains || Q d || | α | +1 ≤ M | α | +1 ( || ~u d . ∇ ~u d || | α | + || ~u d || | α | +1 + δ N +1 I N +1 ( t )+ || T d || | α | (1 + || Q d || ) + || Q d || | α | (1 + || T d || )) . Note that we have the estimate || ~u d . ∇ ~u d || | α | ≤ C || ~u d || || ~u d || | α | +1 . (85)hence || Q d || | α | +1 ≤ M | α | +1 ((1 + || ~u d || ) || ~u d || | α | +1 + δ N +1 I N +1 ( t )+ || T d || | α | (1 + || Q d || ) + (1 + || T d || ) || Q d || | α | (86)44t is then enough to use a recurrence argument to control the norm of Q d in H s +1 using the control of the norm of Q d in H s .For the control on ~u d , let us rewrite the equation on D α ~u d . We introduce ~V α = D α ( ~u. ∇ ~u d ) − ~u. ∇ D α ~u d , ~W α = D α ( T. ∇ Q d ) − T. ∇ D α Q d . We have the estimates || ~V α || ≤ C (1 + || ~u d || ) || ~u d || | α | , || ~W α || ≤ C (1 + || T d || ) || Q d || | α | . Using the relation Z T ∇ D α Q d D α ~u d dxdy = − Z D α Q d ( ∇ ( T N −
1) + ∇ T d ) D α ~u d dxdy thanks to the divergence free condition, as well as Z ~u. ∇ D α ~u d .D α ~u d dxdy = 0one obtains the estimate ddt || D α ~u d || ≤ || ~V α || + || ~W α || + || D α ~R N || + || D α ( k T Q d ) || + || D α Q d || (1 + || T d || ) , hence the inequality ddt || D α ~u d || ≤ C [(1 + || T d || ) || Q d || α + (1 + || ~u d || ) || ~u d || | α | + δ N +1 I N +1 ( t )] (87)For | α | ≥
3, this inequality is an a priori inequality. We have to state theidentical inequalities for | α | = 0 , , ddt || ~u d || ≤ C ((1 + || T d || ) || Q d || + δ N +1 I N +1 ( t )) (88)because ~V α = ~W α = 0, ddt ||∇ ~u d || ≤ C ((1+ || T d || ) || Q d || +(1+ || ~u d || ) || ~u d || +(1+ || T d || ) || Q d || + δ N +1 I N +1 ( t ))(89)and ddt || ~u d || ≤ C ((1+ || T d || ) || Q d || +(1+ || T d || ) || Q d || +(1+ || ~u d || ) || ~u d || + δ N +1 I N +1 ( t ))(90)We thus deduce an estimate of the form ddt ( || T d || + || ~u d || ) ≤ C (1+ || T d || ) + || ~u d || )( || T d || + || ~u d || )+ δ N +1 I N +1 ( t )( || T d || + || ~u d || ) from which one deduces an estimate of the form ddt ( || T d || + || ~u d || ) ≤ C (1+ || T d || ) + || ~u d || )( || T d || + || ~u d || ) + Cδ N +1 I N +1 ( t )45 nd of the proof We thus know that, for t ≤ T δ , we have δ N +1 I N +1 ( t ) ≤ ddt H ( t ) ≤ C ((1 + ( H ( t )) ) H ( t ) + 1)where H ( t ) = ( || T d || + || ~u d || ) .As we have H (0) = 0, one deduces that Z H ( t )0 ds (1 + s ) s + 1 ≤ Ct.
The function H → R H ds (1+ s ) s +1 is a bijection from [0 , + ∞ [ onto [0 , R + ∞ ds (1+ s ) s +1 [.For H ( t ) ≥
1, one deduces Ct ≥ R ds (1+ s ) s +1 , hence for t < C R ds (1+ s ) s +1 = T , one obtains H ( t ) ≤
1. The set of points t such that t > H ( t ) ≤ T ), we obtain Lemma 11
Let h be a function such that dhdt ≤ C (1 + h ( t )) h ( t ) + Cδ N +1 e ( N +1) γ ( k ) t , h (0) = 0 . For δ < and ( N + 1) γ ( k ) > C , denoting by T δ = γ ( k ) ln δ , one has ∀ t ∈ [0 , T δ ] , h ( t ) ≤ δ N +1 e ( N +1) γ ( k ) t . Proof
The inequality we start with is ddt h ( t ) ≤ C (1 + h ( t )) h ( t ) + Cδ N +1 e ( N +1) γ ( k ) t . We consider N such that ( N + 1) γ ( k ) > C . We study the interval where h ( t ) ∈ [0 , h (0) = 0. Consider t the first time (if it exists)where h ( t ) = 1. If it does not exist, then h ( t ) ≤ t ∈ [0 , T δ ] and we have,for all t ∈ [0 , T δ ] the inequality h ′ ( t ) ≤ Ch ( t ) + Cδ N +1 e ( N +1) γ ( k ) t . from which one deduces h ( t ) ≤ Cδ N +1 ( N + 1) λ − C e ( N +1) γ ( k ) t < δ N +1 e ( N +1) γ ( k ) t hence h ( T δ ) < t exists, we have, for all t ∈ [0 , t ], the inequality ddt ( h ( t ) e − Ct ) ≤ − C (1 − h ( t )) h ( t ) R ( h ( t )) e − Ct + Cδ N +1 e ( N +1) γ ( k ) t − Ct R ( x ) = (1 + x ) + 2(1 + x ) + 4(1 + x ) + 8, from which one deduces that h ( t ) e − Ct ≤ C ( N + 1) γ ( k ) − C δ N +1 e ( N +1) γ ( k ) t − Ct < δ N +1 e ( N +1) γ ( k ) t − Ct hence h ( t ) <
1, contradiction.We thus deduce that h ( t ) ≤ t ∈ [0 , T δ ], hence h ( t ) ≤ δ N +1 e ( N +1) γ ( k ) t , t ∈ [0 , T δ ] . Lemma 11 is proven.We have thus the inequalities || ~u || ≥ || ~u N || − || ~u d || ≥ δ || ~u (0) || − C Aδ e γ ( k ) t − C δAe γ ( k ) t − δ N +1 e ( N +1) γ ( k ) t . Choose t = T δ = γ ( k ) ln θC Aδ . We have || ~u || ≥ δe γ ( k ) t [ || ~u (0) || − C θ − θ − θ N ] . We thus check that there exists ε ≤ such that θ < ε implies [ || ~u (0) || − C θ − θ − θ N ] ≥ || ~u (0) || . Hence for t ≤ γ ( k ) ln ε C Aδ , one has || ~u ( t ) || ≥ || ~u (0) || δe γ ( k ) t . (91)In particular || ~u ( 1 γ ( k ) ln ε C Aδ ) || ≥ ε || ~u (0) || . We proved Theorem 2.It is then clear that, for 0 ≤ t ≤ T δ , this term is smaller than θ ,as small as onewants, hence the inequality on T d , ~u d .As T = T N + T d , ~u = ~u N + ~u d , one obtains || T − || ∞ ≥ || T N || − || T d || which imply the result. References [1]
M. Abramovitz and I.A. Stegun : Handbook of mathematical func-tions with formulas, graphs, and mathematical tables. 10th printing, withcorr.
National Bureau of Standards. A Wiley-Interscience Publication.New York etc.: John Wiley & Sons. XIV (1972). [2]
S. Chandrasekhar
Hydrodynamic and hydromagnetic stability
OxfordUniversity press, 1961
C. Cherfils, O. Lafitte, P.A. Raviart : Asymptotic results for theRayleigh-Taylor instability
In Advances in Mathematical Fluid Mechanics(Birkhauser) (2001), 47-71 [4]
J. Garnier, C. Cherfils, P.A. Holstein
Statisticial analysis of mul-timode weakly nonlinear Rayleigh-Taylor instability in the presence ofsurface tension
Physical review E 68 036401 (2003) [5]
S. Cordier, E. Grenier and Y. Guo : Two stream instabilites inplasmas
Methods Appl. Anal. 7 (2), 391-406 [6]
V. Goncharov
Self consistent stability analysis of ablation fronts ininertial confinement fusion
Ph D Thesis, Rochester, 1998 [7]
B. Desjardins and E. Grenier
On Nonlinear Rayleigh-Taylor Insta-bilities
Acta Math. Sinica, English Series, 22 (4) 1007-1016 (2006) [8]
Y. Guo and H.J. Hwang
On the dynamical Rayleigh-Tayor instability
Arch. Ration. Mech. Anal.167 (3) 235-253, 2003 [9]
E. Grenier : On the nonlinear instability of Euler and Prandtl Equations
Commun. Pure Appl. Math. 13 (2000) 1067-1091 [10]
B. Helffer and O. Lafitte
Asymptotic growth rate for the linearizedRayleigh equation for the Rayleigh-Taylor instability
Asympt. An.33 (3-4)189-235, 2003 [11]
B. Helffer and O. Lafitte
The Semiclassical Regime for AblationFront Models
Arch. Ration. Mech. Anal. 183 (3) 371-409, 2007 [12]
H.J. Kull and S.I. Anisimov
Ablative stabilization in the incompress-ible Rayleigh-Taylor instability
Phys. Fluids 29 (7) 1986 2067-2075 [13]
O. Lafitte
Sur la phase lin´eaire de l’instabilit´e de Rayleigh-Taylor
S´eminaire `a l’Ecole Polytechnique, CMAT, Avril 2000 and Preprint 2000-21, CMAT[14]
O. Lafitte
Study of the linear ablation growth rate for the quasi-isobaricmodel of Euler equations with thermal conductivity
Pr´epublication 2005-29 du LAGA, Universit´e de Paris-Nord, November 2005, accepted forpublication in Indiana Univ. Math. J, 2007 [15] Study of the linear ablation growth rate for the quasi-isobaric model ofEuler equations with thermal conductivity accepted for publication in In-diana Univ. Math. J, 2007 [16]
N. Levinson
The asymptotic nature of solutions of linear systems ofdifferential equations
Duke Math. J. 15, (1948). 738
P. Clavin and L. Masse
Instabilities of ablation fronts in inertial con-finement fusion: a comparison with flames
Phys. Plasmas, 11(2): 690-705,2004 [18]
R. Poncet
Nonlinear instability of the two-dimensional striation modelabout smooth steady states accepted for publication in Comm. P. D. E.,2007 [19]
J.W. Strutt (Lord Rayleigh)
Investigation of the character of theequilibrium of an Incompressible Heavy Fluid of Variable Density
Proc.London Math. Society 14, 170-177, 1883 [20]
G. Taylor
The instability of liquid surfaces when accelerated in a direc-tion perpendicular to their planes