A geometric proof of the Quasi-linearity of the water-waves system
aa r X i v : . [ m a t h . A P ] F e b A GEOMETRIC PROOF OF THE QUASI-LINEARITY OF THEWATER-WAVES SYSTEM
AYMAN RIMAH SAID
Abstract.
In the first part of this paper we prove that the flow associated tothe Burgers equation with a non local term of the form H h D i α u fails to be uni-formly continuous from bounded sets of H s ( D ) to C ([0 , T ] , H s ( D )) for T > s > + 2, 0 ≤ α < D = R or T and H is the Hilbert transform. Fur-thermore we show that the flow cannot be C from bounded sets of H s ( D ) to C ([0 , T ] , H s − α − + + ǫ ( D )) for ǫ >
0. We generalize this result to a large classof nonlinear transport-dispersive equations in any dimension, that in particularcontains the Whitham equation and the paralinearization of the water waves sys-tem with and without surface tension. The current result is optimal in the sensethat for α = 2 and D = T the flow associated to the Benjamin-Ono equation isLipschitz on function with 0 mean value H s .In the second part of this paper we apply this method to deduce the quasi-linearity of the water waves system, which is the main result of this paper. Keywords—
Flow map, Regularity, Quasi-linear, nonlinear Burgerstype hyperbolic equations, nonlinear Burgers type dispersive equa-tions, Water Waves system.
Contents
1. Introduction 12. Study of the model equation 113. A technical generalization 184. Quasi-linearity of the Water-Waves system with surface tension 285. Quasi-Linearity of the Gravity Water Waves 32Appendix A. Pseudodifferential and Paradifferential operators 35Appendix B. Energy estimates and well-posedness of some pulled backhyperbolic equations 41References 441.
Introduction
A commonly found definition is that a partial differential equation is said to bequasi-linear if it is linear with respect to all the highest order derivatives of theunknown function, for example equations of the form: ∂ t u + X A j ( u ) ∂ j u = F ( u ) . Which we compare to the definition of semi-linearity as a partial differential equationwhose highest order terms are linear, for example equations of the form: ∂ t u + X A j ∂ j u = F ( u ) . This distinction is supposed to classify the equations in accordance to how one solvestheir respective Cauchy problems. For example, semi-linear equations are expectedto be solved locally by a Picard iteration scheme and thus the associated flow map isexpected to depend regularly on the data. On the other hand quasi-linear equationsare expected to be solved by a compactness method and no more information than
PhD student at CMLA, Batiment Laplace 61, Avenue du President Wilson 94235 Cachan Cedex.email: [email protected] . ontinuity can be recovered on the flow map. The problem of those broad definitionswith the count of derivatives is that they fail to classify the equations according tothis simple criteria on their Cauchy problem. Indeed by those definitions the (KPI)and (KPII) equations are semi-linear by the count of the derivatives, indeed theyare given by: ( u t + uu x + u xxx ) x + u yy = 0 , (KPI)( u t + uu x + u xxx ) x − u yy = 0 . (KPII)Bourgain showed in [9] that (KPII) can be solved by an iteration scheme andthat the flow map is regular. But Moulinet, Saut and Tzvetkov showed in [18] thatthe flow map associated to (KPI) cannot be C and that it cannot be solved by aPicard iteration scheme. They introduce the following definitions to quasi-linearityand semi-linearity [18], that we will use here: • A partial differential equation is said to be semi-linear if its flow map isregular (at least C ). • A partial differential equation is said to be quasi-linear if its flow map is not C .It is well known that the flow map associated to the Burgers equation: ∂ t u + u∂ x u = 0 on R , fails to be uniformly continuous, giving the equation its quasi-linear nature, as forexample shown in [22]. An important class of equations that arises in the study ofasymptotic models of the water waves equations is Burgers type equation with adispersive term, for example the Benjamin-Ono equation: ∂ t u + u∂ x u + H∂ x u = 0 on R , (BO)and Korteweg-de Vries equation: ∂ t u + u∂ x u + ∂ x u = 0 on R . (KdV)It was also shown in [16], that the flow map associated to the Benjamin-Onoequation on H s ( R ) , s > fails to be uniformly continuous. The proof relies heavilyon the dimension, the structure of the equation and on some interactions betweensmall and high frequencies thus it does not generalize to the case of T . More generallyin [22], it is shown that the flow map fails to be C (thus the equations are unsolvableby a Picard fixed point scheme) for equations of the form: ∂ t u + u∂ x u + ω ( D ) ∂ x u = 0 , with | ω ( ξ ) | ≤ | ξ | γ , γ < . Here the proof relies heavily on the Duhamel formula, on the explicit solvability ofthe linear part using the Fourier transform and again on some interactions betweensmall and high frequencies thus it does not generalize to the case of T .In [22], for the KdV equation, using Strichartz type dispersive estimates theCauchy problem is solved by a Picard fixed point scheme and thus the flow mapis regular, showing a change in nature for the problem. This shows that an interest-ing phenomena happening where the dispersive term can dominate the nonlinearity.On R , the previous examples show that this change of regime happens for a disper-sive term of order 3. Thus the result obtained in [22] is optimal in d = 1.In this paper we improve these results in several directions: • we prove the result for a generic dispersive perturbation of order α < • we prove the strongest result possible by proving that the flow is not uni-formly continuous, for ǫ > C from H s ( D ) to C ([0 , T ] , H s − α − + + ǫ ( D ))) . • we prove the result in any dimension.For the sake of clarity we begin by stating a result in dimension 1. Theorem 1.1.
Consider three real numbers α ∈ [0 , , s ∈ ]2 + , + ∞ [ , r > and u ∈ H s ( D ) . Then there exists T > such that for all v in the ball B ( u , r ) ⊂ H s ( D ) there exists a unique v ∈ C ([0 , T ] , H s ( D )) solving the Cauchy problem: ( ∂ t v + v∂ x v + H h D i α v = 0 v (0 , · ) = v ( · ) , (1.1) where H is the Hilbert transform defined by its symbol : H ( ξ ) = − i sgn( ξ ) . Moreover for all
R > , the flow map:B (0 , R ) → C ([0 , T ] , H s ( D )) v v is not uniformly continuous.Considering a weaker control norm we get, for all ǫ ′ > the flow map:B (0 , R ) → C ([0 , T ] , H s − α − + + ǫ ′ ( D )) v v is not C . We shall prove a stronger result (see Theorem 3.1) showing that for a dispersiveperturbation of order α <
2, the non-linear transport term dominates the flow’sevolution locally and this happens independently of the dimension. This limitedregularity of the flow implies that the Cauchy problem can not be solved by a Pi-card fixed point scheme and thus those equations are quasi-linear. • The results in [22] suggest that the result obtained here are sub-optimalbecause it suggests that the change to the semi-linear type equations happensfor α = 3, and the flow associated to the Benjamin-Ono equation on R failsto be uniformly continuous as shown in [16]. In [21] we show that the flowmap associated to the following equation ∂ t v + Re( v ) ∂ x v + i∂ x v = 0 , is Lipschitz from bounded sets of H s ( R ) to C ([0 , T ] , H s ( R )) under the extrahypothesis of L control on the data. Showing that the lack of regularityobtained in [22] for α ≥ L norms in Sobolev spaces on R . • This optimality is also confirmed by the results in [19] where L. Molinetproves that the flow map has Lipschitz regularity for the Benjamin-Onoequation on the torus in H s ( T ) for s ≥
0, which are the Sobolev spaces offunctions with zero mean value. • In our work [21] we generalize the result on the Benjamin-Ono equationand prove that the flow map associated to the Burgers equation with a nonlocal term of the form D α − ∂ x u , α ∈ ]1 , + ∞ [ is Lipschitz from bounded The use of the Hilbert transform insures that we always work with real valued functions whenthe initial data is real valued. ets of H s ( T ; R ) to C ([0 , T ] , H s − (2 − α ) + ( T ; R )) , s > . Thus prov-ing that the result obtained here is optimal for α ∈ ]1 , α ∈ [0 , + ∞ [ the flow map is not Lipschitz from bounded sets of H s ( T ; R ) to C ([0 , T ] , H s − (2 − α ) + ǫ ( T ; R )) , ǫ > R for α < H s ( T ; R ), s > which is due to thelack of dispersive estimates on T .It’s important to note that those results agree with Bourgain’s resultson the well posedness for the periodic Kdv equation in [10] and Molinet’sresults in in [19]. Indeed in [10] the contraction method is applied on initialdata in H s and then a gauge transform is used to deduce well posedness forgeneral data. It’s exactly this gauge transform that we use to prove the lackof regularity of the flow map when passing from H s to H s .Finally Theorem 3.1 contains applications to different classes of equations:-Firstly the Whitham equation on R : ( ∂ t u + u∂ x u − Lu x = 0 ,Lf ( x ) = R e ix · ξ p ( x, ξ ) ˆ f ( ξ ) dξ, is quasi-linear for p ∈ S α , α < p ) ∈ S (See (A.2) for the defini-tion of the symbol classes).-The second and main application is the water waves system with and withoutsurface tension. We follow here the presentation in [2] and [5].1.1. Assumptions on the domain.
We consider a domain with free boundary, ofthe form: n ( t, x, y ) ∈ [0 , T ] × R d × R : ( x, y ) ∈ Ω t o , where Ω t is the domain located between a free surfaceΣ t = n ( x, y ) ∈ R d × R : y = η ( t, x ) o and a given (general) bottom denoted by Γ = ∂ Ω t \ Σ t . More precisely we assumethat initially ( t = 0) we have the hypothesis H t given by: • The domain Ω t is the intersection of the half space, denoted by Ω ,t , locatedbelow the free surface Σ t ,Ω ,t = n ( x, y ) ∈ R d × R : y < η ( t, x ) o and an open set Ω ⊂ R d +1 such that Ω contains a fixed strip around Σ t ,which means that there exists h > n ( x, y ) ∈ R d × R : η ( t, x ) − h ≤ y ≤ η ( t, x ) o ⊂ Ω . We shall assume that the domain Ω (and hence the domain Ω t = Ω ,t ∩ Ω )is connected.1.2. The equations.
We consider an incompressible inviscid liquid, having unitdensity. The equations of motion are given by the Euler system on the velocity field v : ( ∂ t v + v · ∇ v + ∇ P = − ge y div v = 0 in Ω t , (1.2) here − ge y is the acceleration of gravity ( g >
0) and where the pressure term P can be recovered from the velocity by solving an elliptic equation. The problem isthen coupled with the boundary conditions: v · n = 0 on Γ ,∂ t η = p |∇ η | v · ν on Σ t ,P = − κH ( η ) on Σ t , (1.3)where n and ν are the exterior normals to the bottom Γ and the free surface Σ t , κ is the surface tension and H ( η ) is the mean curvature of the free surface: H ( η ) = div (cid:18) ∇ η p |∇ η | (cid:19) . We take κ = 1 for the case with surface tension and κ = 0 in the case of gravitywater waves (without surface tension). The first condition in (1.3) expresses in factthat the particles in contact with the rigid bottom remain in contact with it. As nohypothesis is made on the regularity of Γ , this condition is shown to make sense ina weak variational meaning due to the hypothesis H t , for more details on this werefer to Section 2 in [2] and Section 3 in [5].The fluid motion is supposed to be irrotational and Ω t is supposed to be simplyconnected thus the velocity v field derives from some potential φ i.e v = ∇ φ and: ( ∆ φ = 0 in Ω ,∂ n φ = 0 on Γ . The boundary condition on φ becomes: ∂ n φ = 0 on Γ ,∂ t η = ∂ y φ − ∇ η · ∇ φ on Σ t ,∂ t φ = − gη + κH ( η ) − |∇ x,y φ | on Σ t . (1.4)Following Zakharov [24] and Craig-Sulem [12] we reduce the analysis to a systemon the free surface Σ t . If ψ is defined by ψ ( t, x ) = φ ( t, x, η ( t, x )) , then φ is the unique variational solution of∆ φ = 0 in Ω t , φ | y = η = ψ, ∂ n φ = 0 on Γ . Define the Dirichlet-Neumann operator by( G ( η ) ψ )( t, x ) = p |∇ η | ∂ n φ | y = η = ( ∂ y φ )( t, x, η ( t, x )) − ∇ η ( t, x ) · ( ∇ φ )( t, x, η ( t, x )) . For the case with rough bottom we refer to [1], [2] and [5] for the well posedness ofthe variational problem and the Dirichlet-Neumann operator. Now ( η, ψ ) (see forexample [12]) solves: ∂ t η = G ( η ) ψ, (1.5) ∂ t ψ = − gη + κH ( η ) + 12 |∇ ψ | + 12 ∇ η · ∇ ψ + G ( η ) ψ |∇ η | . .3. Gravity water waves: Pressure and Taylor Coefficients.
Here we givea quick review of the ideas in [4]. Recall that by definition for gravity water waveswe work with κ = 0 and we define the Taylor coefficient a ( t, x ) = − ( ∂ y P )( t, x, η ( t, x )) . The stability of the waves is dictated by the Taylor sign condition, which is theassumption that there exists a positive constant c such that a ( t, x ) ≥ c > . (1.6)In [5] this condition is needed in the proof of the well posedness of the Cauchy prob-lem and it is shown to be locally propagated by the flow.Now we will show how to define P from the Zakharov formulation. Let R be thevariational solution of∆ R = 0 in Ω t , R | y = η = ηg + 12 |∇ x,y φ | | y = η . We define the pressure P in the domain Ω by P ( x, y ) = R ( x, y ) − gy − |∇ x,y φ ( x, y ) | . In [4] Alazard, Burq, and Zuily show that to a solution ( η, ψ ) ∈ C ([0 , T ] , H s + ( R d ) × H s + ( R d )) , s > d + of the Zakharov/Craig-Sulem system (1.5) corresponds a uniquesolution v of the Euler system.1.4. Quasi-linearity of the water Wave system.
In [2] and [5], Alazard, Burq,and Zuily perform a paralinearization and symmetrization of the the water wavessystem that take the form: ∂ t u + T V . ∇ u + iT γ u = f, where γ is of order in the case with surface tension and in the case without. Theterms V and γ verify the conditions required by Theorem 3.1 and thus the paralin-earization of the water-waves system are quasi-linear in the considered thresholds ofregularity. From this we will deduce the following two theorems.First in the case of water waves with surface tension, i.e κ = 1, where the well-posedness of the Cauchy problem is proved in [2] we complete it by the following. Theorem 1.2.
Fix the dimension d ≥ and consider two real numbers r > , s ∈ ]2 + d , + ∞ [ and ( η , ψ ) ∈ H s + ( R d ) × H s ( R d ) such that ∀ ( η ′ , ψ ′ ) ∈ B (( η , ψ ) , r ) ⊂ H s + ( R d ) × H s ( R d ) the assumption H t =0 is satisfied. Then there exists T > such that the Cauchyproblem (1.5) with initial data ( η ′ , ψ ′ ) ∈ B (( η , ψ ) , r ) has a unique solution ( η ′ , ψ ′ ) ∈ C ([0 , T ]; H s + ( R d ) × H s ( R d )) and such that the assumption H t is satisfied for t ∈ [0 , T ] .Moreover ∀ R > the flow map:B (0 , R ) → C ([0 , T ] , H s + ( R d ) × H s ( R d ))( η ′ , ψ ′ ) ( η ′ , ψ ′ ) is not uniformly continuous. e show that at least a loss of derivative is necessary to have Lipschitz controlover the flow map, i.e for all ǫ ′ > the flow mapB (0 , R ) → C ([0 , T ] , H s + ǫ ′ ( R d ) × H s − + ǫ ′ ( R d ))( η ′ , ψ ′ ) ( η ′ , ψ ′ ) is not C . Remark 1.1.
In our work [21] we improve the previous result by showing that forthe Gravity Capillary equation on T , the loss of derivative is sufficient to haveLipschitz control over the flow map, under an extra symmetry hypothesis on thedata. Now we turn to gravity water waves, i.e κ = 0 where the well posedness of theCauchy problem is proved in [5]. It is well known that the vertical and horizon-tal traces of the velocity on the free boundary play an important role in the wellposedness of the Cauchy problem and are given by: B = ( ∂ y φ ) | y = η = ∇ η · ∇ ψ + G ( η ) ψ |∇ η | , (1.7) V = ( ∇ x φ ) | y = η = ∇ ψ − B ∇ η. Theorem 1.3.
Fix the dimension d ≥ and consider two real numbers r > , s ∈ ]2 + d , + ∞ [ and ( η , ψ ) ∈ H s + ( R d ) × H s + ( R d ) and consider ( η ′ , ψ ′ ) ∈ B (( η , ψ ) , r ) ⊂ H s + ( R d ) × H s + ( R d ) such that we have:(1) V ′ ∈ H s ( R d ) , B ′ ∈ H s ( R d ) , (2) H t =0 is satisfied,(3) there exits a positive constant c such that, ∀ x ∈ R d , a ′ ( x ) ≥ c > .Then there exists T > such that the Cauchy problem (1.5) with initial data ( η ′ , ψ ′ ) has a unique solution ( η ′ , ψ ′ ) ∈ C ([0 , T ]; H s + ( R d ) × H s + ( R d )) such that for t ∈ [0 , T ] the assumption H t is satisfied, ∀ x ∈ R d , a ′ ( t, x ) ≥ c and ( V ′ , B ′ ) ∈ C ([0 , T ]; H s ( R d ) × H s ( R d )) . Moreover ∀ R > , the flow map:B (0 , R ) → C ([0 , T ] , H s + ( R d ) × H s + ( R d ))( η ′ , ψ ′ ) ( η ′ , ψ ′ ) is not uniformly continuous.Considering a weaker control norm we get: For all ǫ ′ > , the flow map:B (0 , R ) → C ([0 , T ] , H s − + ǫ ′ ( R d ) × H s − + ǫ ′ ( R d ))( η ′ , ψ ′ ) ( η ′ , ψ ′ ) is not C . Remark 1.2.
It is worth noticing that a previous result was obtained on the regu-larity of the flow map for the two dimensional gravity-capillary water waves (i.e withsurface tension) in [11] where they have proved that the flow is not C with respectto initial data ( η , ψ ) ∈ H s + ( R ) × H s ( R ) for s < .This result is in contrast with our result which holds for s > and this can indeedbe seen in the fact that in [11] the lack of regularity of the flow is shown to be Here we are slightly above the threshold of well-posedness of 1 + d proved in [5]. rimarily due to the influence of surface tension. Though in our work the lack ofregularity of the flow is shown to be due to the hydrodynamic term (the non-lineartransport term). Remark 1.3.
As the Cauchy problem for the water waves system on T d is solvedby the same particularization and symmetrization (see [6] ) and our technical gener-alization in Section 3.2 is proved on D d the previous results for the water waves on R d extend tautologically to T d . Strategy of the proof.
We explain the key ideas at the level of the equation(1.1), ∂ t v + v∂ x v + H h D i α v = 0 . The point of start is to adapt the classic proof of the quasi-linearity of the Burgersequation, presented to me in a personal note of C. Zuily [25], that we will recall here.1.5.1.
Quasi-linearity of the Burgers equation.
The result of quasi-linearity of theBurgers equation is that the flow map taken point-wise in time fails to be uniformlycontinuous. Such a result is obtained by constructing two families of solutions u and v from some initial data u and v depending on parameters λ and ǫ such thatlim λ → + ∞ ǫ → (cid:13)(cid:13) u − v (cid:13)(cid:13) H s = 0 and k ( u − v )( t, · ) k H s ≥ c > , with t > u ( t, · ) solution to the Burgers equation withinitial data u . Put χ ( t, x ) = x + tu ( x )the characteristic flow associated to the problem, which is a diffeomorphism in the x variable. Then, u ( t, · ) = u ◦ χ ( t, x ) − . The action of χ − on the graph of u is given by the following Figure 1 that alsoshows the shock formation phenomena.Then u and v are chosen as a high frequency compactly supported ansatz depend-ing on ( λ, ǫ ): u ( x ) = λ − s ω ( λx ) , v ( x ) = u ( x ) + ǫω ( x ) , with ω ∈ C ∞ , where ǫ represents a change in the initial speed of transport, and ( ǫ, λ ) verify: • ǫ → H s norm of the sequences of initialdata goes to 0. • λ → + ∞ is the usual ansatz parameter hypothesis. • ǫλ → + ∞ insuring that the change of transport speed is enough to havedisjoint supports at positive time.Now if we put χ and ˜ χ to be the characteristic flows associated to the solutions u and v then: ( u − v )( t, x ) = u ( χ ( t, x ) − ) − v ( ˜ χ ( t, x ) − )= u ( χ ( t, x ) − ) − u ( ˜ χ ( t, x ) − ) + O H s ( ǫ ) . Then using the compactly supported property of u and the change of speed weprove that u ( χ ( t, x ) − ) and u ( ˜ χ ( t, x ) − ) have disjoint supports which is illustratedby Figure 3. We then prove that (cid:13)(cid:13) u ( χ ( t, x ) − ) (cid:13)(cid:13) H s ≥ c > C ( H s ( D ) , C ([0 , T ] , H s − ǫ ( D ))), we get it from theestimate (cid:13)(cid:13) u ( χ ( t, x ) − ) (cid:13)(cid:13) H s − µ ≥ cλ − µ . x P in Q in P in Q in y = u y x P ( t ) Q ( t ) P ( t ) Q ( t ) y = u ( t ) y x P ( t ) Q ( t ) P ( t ) Q ( t ) y = u ( t )0 t t t (1) (2) T Figure 1.
The lines (1) and (2) are the characteristic curves from Q in and P in . T is the time of formation of the shock wave.y x v u ∼ λ − s ∼ ǫ Figure 2.
Graph of the ansatz.1.5.2.
Quasi-linearity of problem (2.1) . Now if we adapt the proof to our currentproblem (1.1) we get:( u − v )( t, x ) = f ( t, χ ( t, x ) − ) − g ( t, ˜ χ ( t, x ) − )= f ( t, χ ( t, x ) − ) − f ( t, ˜ χ ( t, x ) − ) + O H s (cid:0) ǫ + t ǫλ α (cid:1) , x v u ∼ λ − s ∼ ǫ t u ◦ χ u ◦ ˜ χ Figure 3.
Transport of the ansatz.where f and g are solutions to ∂ t f + ( H h D i α ) ∗ f = 0 (1.8) ∂ t g + ^ ( H h D i α ) ∗ g = 0 (1.9)and ( · ) ∗ and f ( · ) ∗ are the change of variables by the characteristic flows defined for asymbol a byOp( a ) ∗ ( u ◦ χ ) = (Op( a ) u ) ◦ χ i.e Op( a ) ∗ ( u ) = (Op( a )[ u ◦ χ − ]) ◦ χ, and analogously for f ( · ) ∗ , which we prove that they are well posed in Appendix B.The first immediate problem we face is the extra term t ǫλ α which diverges,to remedy this we give up control of the flow map punctually in time and use aconveniently chosen sequence of small time ( τ ) to control τ ǫλ α : τ → , but still insure λǫτ → + ∞ . The second, deeper problem we face is that we lose control over the support ofthe solution. Indeed (1.8) and (1.9) are obtained by pull-back of the linear equation ∂w + H h D i α w = 0 (1.10)which is a non local-dispersive equation that is expected to disperse the support ofthe solution and the L ∞ norm. This phenomena is thus expected to oppose thephenomena illustrated by the previous Figures (1) and (2) and indeed does so forthe KdV equation on R .To remedy this, the idea is not to use u and v as initial data but by profiting ofthe time reversibility of the equations use the backward in time solutions u and v defined by: ω solution of (1.10) ,ω ( τ, · ) = u ,ω (0 , · ) = u , ω ′ solution of (1.10) ,ω ′ ( τ, · ) = v ,ω ′ (0 , · ) = v . This idea fundamentally depends on the local reversibility in time of the linearised equationsand thus fails for the fractional Burgers equation. his gives us:( u − v )( τ, x ) = u ( χ (0 , τ, x )) − u ( ˜ χ (0 , τ, x )) + O H s (cid:0) ǫ + τ ǫλ α + τ λ α − (cid:1) . We then prove that this gives the desired result, in the threshold α ∈ [0 , (cid:13)(cid:13) u ( χ − ( t, x )) (cid:13)(cid:13) H s ≥ c > u and the change of speed we provethat u ( χ (0 , τ, x )) and u ( ˜ χ (0 , τ, x )) have disjoint supports.1.6. Acknowledgement.
I would like to express my sincere gratitude to my thesisadvisor Thomas Alazard. I would also like to thank Claude Zuily for his insightfulnote on the Burgers equation that helped me understand the problem.2.
Study of the model equation
In this section we give a full proof of Theorem 1.1.2.0.1.
Prerequisites on the Cauchy Problems.
For a real number α ∈ [0 , : ( ∂ t u + u∂ x u + H h D i α u = 0 u (0 , · ) = u ( · ) ∈ H s ( D ) , s > , (2.1)It is well known that the problem is well posed in Sobolev spaces, this can besummarized in the following Theorem: Theorem 2.1.
Consider two real numbers, s ∈ ] , + ∞ [ and r > . Fix u ∈ H s ( D ) .Then there exists T > , such that for all v ∈ B ( u , r ) ⊂ H s ( D ) , the problem (2.1) with initial data v has a unique solution v ∈ C ([0 , T ] , H s ( D )) , the map v v is continuous from B ( u , r ) to C ([0 , T ] , H s ( D )) and maps real functions into realfunctions. Moreover we have the estimates: ∀ ≤ µ ≤ s, k v ( t ) k H µ ( D ) ≤ C µ k v k H µ ( D ) . (2.2) Taking two different solutions u, v , such that u ∈ H s +1 ( D ) : ∀ ≤ µ ≤ s, k ( u − v )( t ) k H µ ( D ) ≤ k u − v k H µ ( D ) e C µ R t k u ( s ) k Hµ +1( D ) ds. (2.3) We will also need to remark that fixing the initial data at 0 is an arbitrary choice,that is all of the previous conclusions hold for the Cauchy problem defined for t ≤ T : ( ∂ t v + v∂ x v + H h D i α v = 0 v ( t , · ) = v ( · ) ∈ H s ( D ) , s > . (2.4) Remark 2.1.
Note that the previous Theorem holds for the Cauchy problem asso-ciated to the Burgers equation: ( ∂ t u + u∂ x u = 0 u (0 , · ) = u ( · ) ∈ H s ( D ) , s > , (2.5) Though we have some extra estimates in H¨older type spaces: ∀ ≤ k < s − , k u ( t ) k W k, ∞ ( D ) ≤ C k k u k W k, ∞ ( D ) , (2.6) Taking two different solution u, v , such that u ∈ H s +1 ( D ) : ∀ ≤ k < s − , k ( u − v )( t ) k W k, ∞ ( D ) ≤ k u − v k W k, ∞ ( D ) e C k R t k u ( s ) k Wk +1 , ∞ ( D ) ds . Recall that D = Op( | ξ | ). emark 2.2. The evolution PDE (2.1) , does not have a scaling because of theinhomogeneous term H h D i α . But for the purpose of our study of the local Cauchyproblem, the small frequency part does not play an important role. So in order to havea better idea on the main terms that locally drive the evolution we can heuristicallyreplace it with H | D | α in order to compute the scaling. By doing so we get that thechange of scale, u λ α − u ( λx ) gives the solution λ α − u ( λ α t, λx ) . Thus giving the scaling in Sobolev spaces: s c = 1+ − α , thus we prove quasi-linearityin the subcritical regime of the problem. Notation 2.1.
In order not to be confused with the pull-back symbol, henceforth theconjugate of a symbol a will be written as a ⊤ . As the linearized equation is a hyperbolic pseudo-differential equation we recallthe result on the Cauchy problem associated to this type of equations:
Theorem 2.2.
Consider ( a t ) t ∈ R a family of symbols in S β ( D d ) , β ∈ R , such that t a t is continuous and bounded from R to S β ( D d ) and such that Re( a t ) = a t + a ⊤ t is bounded in S ( D d ) , and take T > . Then for all s ∈ R , u ∈ H s ( D d ) and f ∈ C ([0 , T ]; H s ( D d )) the Cauchy problem: ( ∂ t u + Op( a ) u = f ∀ x ∈ D d , u (0 , x ) = u ( x ) (2.7) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimates: k u ( t ) k H s ( D d ) ≤ e Ct k u k H s ( D d ) + 2 Z t e C ( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s ( D d ) dt ′ , where C depends on a finite symbol semi-norm of Re( a t ) . We will also need to remarkthat fixing the initial data at 0 is an arbitrary choice. More precisely, ∀ ≤ t ≤ T and all data u ∈ H s ( D d ) the Cauchy problem: ( ∂ t u + Op( a ) u = f ∀ x ∈ D d , u ( t , x ) = u ( x ) (2.8) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimate: k u ( t ) k H s ( D d ) ≤ e C | t − t | k u k H s ( D d ) + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s ( D d ) dt ′ (cid:12)(cid:12)(cid:12)(cid:12) . Proof of Theorem 1.1.
To prove the theorem we will show that there existsa positive constant C and two sequences ( u λǫ,τ ) and ( v λǫ,τ ) solutions of 1.1 on [0 , t ∈ [0 , λ,ǫ,τ (cid:13)(cid:13)(cid:13) u λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D )) + (cid:13)(cid:13)(cid:13) v λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D )) ≤ C, ( u λǫ,τ ) and ( v λǫ,τ ) satisfy initiallylim λ → + ∞ ǫ,τ → (cid:13)(cid:13)(cid:13) u λǫ,τ (0 , · ) − v λǫ,τ (0 , · ) (cid:13)(cid:13)(cid:13) H s ( D ) = 0 , but, lim inf λ → + ∞ ǫ,τ → (cid:13)(cid:13)(cid:13) u λǫ,τ − v λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D )) ≥ c > . onsidering a weaker control norm we want to get, for all δ > λ → + ∞ ǫ,τ → (cid:13)(cid:13) u λǫ,τ − v λǫ,τ (cid:13)(cid:13) L ∞ ([0 , ,H s − α − δ ( D )) (cid:13)(cid:13) u λǫ,τ (0 , · ) − v λǫ,τ (0 , · ) (cid:13)(cid:13) H s ( D ) = + ∞ . Definition of the Ansatz. • For D = R , take ω ∈ C ∞ ( R ) , ω ( x ) = 1 if | x | ≤ , ω ( x ) = 0 if | x | ≥ • For D = T , we see functions on T = R / π Z as 2 π periodic function on R andwe take ω ∈ C ∞ ( T ) as the periodic extension of the function defined above.Let ( λ, ǫ ) be two positive real sequences such that λ → + ∞ , ǫ → , λǫ → + ∞ . (2.9)Put • for D = R , u ( x ) = λ − s ω ( λx ) , v ( x ) = u ( x ) + ǫω ( x ) , • for D = T , u and v as the periodic extensions of the functions definedabove.Take t > τ ) , <τ ≤ t and τ → l, l ′ be the solutions to the Cauchy problem on [0 , t ]: ( ∂ t l + H h D i α l = 0 ,l ( τ, · ) = u , ( ∂ t l ′ + H h D i α l ′ = 0 ,l ′ ( τ, · ) = v . Put u ( x ) = l (0 , x ) and define analogously v ( x ) = l ′ (0 , x ).Define u and v as the solution given by Theorem 2.1 with initial data u and v on the intervals [0 , T ] and [0 , T ′ ]. Taking 0 < δ < s − , u and v are uniformlybounded in H + δ ( D ) when λ → + ∞ and thus by Theorem 2.2, u and v are alsouniformly bounded in H + δ ( D ) and thus by the Sobolev injection Theorem they arebounded in ˙ W , ∞ ( D ). Thus we can take a uniform 0 < T on which all the solutionsare well defined and we take 0 < t ≤ T .2.1.2. Change of variables by transport.
Put ( ddt χ ( t, s, x ) = u ( t, χ ( t, s, x )) χ ( s, s, x ) = x , and define analogously ˜ χ from v .We recall that from the Cauchy-Lipschitz Theorem we have as u and v are H + ∞ ( D ) functions, then u , v are H + ∞ ( D ) and u and v are H + ∞ ( D ) with respectto the x variable thus χ, ˜ χ ∈ C ([0 , T ] , C ∞ ). And they are both diffeomorphismsin the x variable. Heuristically, if the existence time of the solution with initial data ω is [0 , T ] then the existencetime of the solution with initial data u is ∼ T λ s − which tends to infinity with λ , thus we are”dilating” the time scale of the problem with initial data ω and ”zooming” for short time and inthe ˙ H s ( D ) norm. In this part of the evolution, we prove that the Burgers transport term is moreimportant and gives this quasi-linear character to the PDE. y the estimate (2.2) u and v are uniformly bounded in ˙ W , ∞ ( D ) because theirSobolev norms are dominated by those of u and v thus by those of u and v byTheorem 2.2. By classic manipulations of ODEs we get the estimates: ( ∃ C > , ∀ t ′ , t ≤ t , ∀ x, C − ≤ | ∂ x χ ( t, t ′ , x ) | ≤ C ∀ ≤ k < ⌊ s − ⌋ , (cid:13)(cid:13) ∂ kx χ ( t, t ′ , x ) (cid:13)(cid:13) L ∞ ≤ C k u k W k, ∞ . (2.10)Analogous estimates hold for ˜ χ using v .The classic transport computation reads: ∂ t ( u ( t, χ ( t, , x ))) = ( ∂ t u )( t, χ ( t, , x )) + ∂ t ( χ ( t, , x ))( ∂ x u )( t, χ ( t, , x ))= − ( H h D i α u )( t, χ ( t, , x ))= − ( H h D i α ) ∗ ( u ( t, χ ( t, , x ))) ,u (0 , χ (0 , , x )) = u (0 , x ) = u ( x ) . where ( · ) ∗ is the change of variables by χ ( t, , x ) as presented in Theorem A.3.Thus if we put f the solution to the following Cauchy problem, which is well posedby Appendix B: ( ∂ t f + ( H h D i α ) ∗ f = 0 ∀ x ∈ D , f (0 , x ) = u ( x ) (2.11)we get: u ( t, χ ( t, , x )) = f ( t, x ) ⇔ u ( t, x ) = f ( t, χ (0 , t, x )) . (2.12)Analogously, if we put g the solution to the well posed Cauchy problem, ( ∂ t g + ^ ( H h D i α ) ∗ g = 0 ∀ x ∈ D , g (0 , x ) = v ( x ) (2.13)where f ( · ) ∗ is the change of variables by ˜ χ ( t, , x ), we get v ( t, x ) = g ( t, ˜ χ (0 , t, x )) ⇔ v ( t, ˜ χ ( t, , x )) = g ( t, x ) . (2.14)Returning to the ODEs defining χ and ˜ χ , for a generic initial time 0 ≤ t ′ ≤ t weget: ( χ ( t, t ′ , x ) = x + R tt ′ f ( s, x ) ds, ˜ χ ( t, t ′ , x ) = x + R tt ′ g ( s, x ) ds, (2.15) Proposition 2.1.
There exists
C > independent of ( τ, ǫ, λ ) such that: ∀ h ∈ H s ( D ) , ∀ ( t, t ′ ) ≤ t ,C − k h k H s ≤ (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) (cid:13)(cid:13) H s ≤ C k h k H s ,C − k h k H s ≤ (cid:13)(cid:13) h ◦ ˜ χ ( t, t ′ , x ) (cid:13)(cid:13) H s ≤ C k h k H s . Proof.
We will start by proving the upper bound for the estimate on the compositionwith χ . As u is bounded in ( τ, ǫ, λ ) on C ([0 , T ] , H s ( D )) then there exists a uniquesolution H ∈ C ([0 , T ] , H s ( D )) to ( ∂ t H + u∂ x H = 0 ,H ( t, x ) = h ( x ) , and H is bounded in ( τ, ǫ, λ ) on C ([0 , T ] , H s ( D )). The desired bound come from thefact that we have the explicit formula for H : H ( t ′ , x ) = h ◦ χ ( t, t ′ , x ) . ow to get the lower bound it suffices to write by the upper bound computations: k h k H s = (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) ◦ χ ( t ′ , t, x ) (cid:13)(cid:13) H s ≤ C (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) (cid:13)(cid:13) H s . We get analogously the estimates on the composition with ˜ χ . (cid:3) Key Lemma and proof of the Theorem.
Lemma 2.1.
Take ǫ ′ > sufficiently small, as ≤ α < we can find a sequence ( τ, ǫ, λ ) such that: τ → ,ǫ → ,λ → + ∞ , τ λ ( α − + → ,ǫ − λ − α − + + ǫ ′ → + ∞ ,λǫτ → + ∞ ,λ α ǫτ → . (2.16) Then there exists c > such that:(1) For ν ≥ and ∀ ( τ, ǫ, λ ) , (cid:13)(cid:13) u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x ) (cid:13)(cid:13) H s − ν > cλ − ν . (2) For ν ≥ u ( τ, x ) − v ( τ, x ) = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x )+ O H s − ν (cid:0) ǫ + ( τ λ ( α − + + τ λ α − ( s − ν ) ) λ − ν + τ ǫλ α − ν (cid:1) . (2.17)We will now show that this Lemma implies the Theorem 1.1. We have by com-bining the estimates (1) and (2) for ν = s : ∀ ( τ, ǫ, λ ) , k u ( τ, x ) − v ( τ, x ) k H s > c > τ,ǫ,λ k u ( τ, x ) − v ( τ, x ) k H s > c > . Also by Theorem 2.2: ∃ C > , (cid:13)(cid:13) u ( x ) − v ( x ) (cid:13)(cid:13) H s ≤ Cǫ, thus (cid:13)(cid:13) u ( x ) − v ( x ) (cid:13)(cid:13) H s → , which gives the non uniform continuity in the desired norms.Now for the control in a weaker norm we write: k u ( τ, x ) − v ( τ, x ) k H s − α − ǫ ′ k u ( x ) − v ( x ) k H s ≥ cǫ − λ − α − + + ǫ ′ → + ∞ , which gives the desired result.2.1.4. Proof of point 1 of Lemma 2.1.
We first prove that there exists c > (cid:13)(cid:13) u ◦ χ (0 , τ, x ) (cid:13)(cid:13) H s − ν > cλ − ν , indeed by Proposition 2.1 and change of variable: (cid:13)(cid:13) u ◦ χ (0 , τ, x ) (cid:13)(cid:13) H s − ν ≥ C − (cid:13)(cid:13) u (cid:13)(cid:13) H s − ν ≥ C − λ − ν k ω k H s − ν . (2.18)Now we will show that u ◦ χ (0 , τ, x ) and u ◦ ˜ χ (0 , τ, x ) have disjoint supports whichwill suffice to conclude given (2.18). Put y = χ (0 , τ, x ), thus x = χ ( τ, , y ). On thesupport of u ◦ χ (0 , τ, x ) we have: • If D = R , λ | y | ≤ • If D = T , ∀ k ∈ N , πk − ≤ λ | y | ≤ πk + 1.We compute by the Taylor formula, since x = ˜ χ ( τ, , y ):˜ χ (0 , τ, x ) = ˜ χ (0 , τ, ˜ χ ( τ, , y )) (2.19)+ ( χ ( τ, , y ) − ˜ χ ( τ, , y )) Z ∂ y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr = y + ( χ ( τ, , y ) − ˜ χ ( τ, , y )) Z ∂ y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr. irst by (2.15), ∂ y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y ))= 1 + Z τ ∂ y [ g ( t, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y ))] dt. Thus by estimates of Theorem B.1, taking 0 < δ < s −
32 6 : ∂ y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) = 1 + O L ∞ (cid:0) τ [1 + (cid:13)(cid:13) v (cid:13)(cid:13) H
32 + δ + (cid:13)(cid:13) u (cid:13)(cid:13) H
32 + δ ] (cid:1) = 1 + O L ∞ ( τ ) , Which gives Z ∂ y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr = 1 + O L ∞ (cid:0) τ (cid:1) . (2.20)Now we estimate χ ( τ, , y ) − ˜ χ ( τ, , y ), by (2.15) : χ ( τ, , y ) − ˜ χ ( τ, , y ) = Z τ f ( t, y ) − g ( t, y ) dt. (2.21)Taking 0 < δ < s − , by estimates of Theorem B.1: f ( t, y ) = f (0 , y ) + Z t ∂ t f ( r, y ) dr = u ( y ) + tO L ∞ ( (cid:13)(cid:13) u (cid:13)(cid:13) H
12 + α + δ )= u ( y ) + O L ∞ (cid:0) tλ − (cid:1) . Analogously we get: g ( t, y ) = v ( y ) + O L ∞ (cid:0) tǫ (cid:1) . Consider µ the solution to the Cauchy problem: ( ∂ t µ + H h D i α µ = 0 ∀ y ∈ D , µ ( τ, y ) = ω ( y ) . (2.22)By definition: u ( y ) − v ( y ) = − ǫµ (0 , y ) = − ǫω ( y ) + ǫ Z τ ∂ t µ ( t, y ) dt = − ǫω ( y ) + O L ∞ (cid:0) ǫτ (cid:1) . Thus, χ ( τ, , y ) − ˜ χ ( τ, , y ) = − ǫτ ω ( y ) + O L ∞ (cid:0) τ ǫ (cid:1) , and finally we get in (2.19),˜ χ ( τ, , x ) − y = − ǫτ ω ( y ) + O L ∞ (cid:0) τ ǫ (cid:1) . We get for x ∈ supp u ◦ χ (0 , τ, · ): • For D = R : λ | ˜ χ (0 , τ, x ) | ≥ τ ǫλ − o L ∞ (cid:0) τ ǫλ (cid:1) ≥ , by hypothesis τ ǫλ → + ∞ , which gives the desired result. • For D = T given an adequate choice of τ, ǫ and λ :2 nπ + 1 ≤ λ | ˜ χ (0 , τ, x ) | ≤ n + 1) π − , Which again gives the desired result. Recall the notation O k k in A.1. .1.5. Proof of point 2 of Lemma 2.1.
We start by writing: u ( t, x ) − v ( t, x ) = f ( t, χ (0 , t, x )) − g ( t, ˜ χ (0 , t, x ))= f ( t, χ (0 , t, x )) − f ( t, ˜ χ (0 , t, x )) | {z } (1) +( f − g )( t, ˜ χ (0 , t, x )) . Term (1) resembles the main term in the usual transport estimates we used in point1 of the Lemma but with a main difference of f being some dispersed data andnot compactly supported. The main trick here was to construct from u , v thedefocused data in the past u , v and use this as the initial data for f and g . u ( τ, x ) − v ( τ, x ) = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x )+ ( f − u )( τ, χ (0 , τ, x )) − ( f − u )( τ, ˜ χ (0 , t, x )) + ( f − g )( τ, ˜ χ (0 , τ, x )) . The idea is then to see that by definition of l : l ( τ, x ) = u ( x ) and we get: u ( τ, x ) − v ( τ, x ) = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x )+ ( f − l )( τ, χ (0 , τ, x )) − ( f − l )( τ, ˜ χ (0 , t, x )) | {z } (1) + ( f − g )( τ, ˜ χ (0 , τ, x )) | {z } . We start by estimating (1), by Proposition 2.1: k ( f − l )( τ, χ (0 , τ, x )) k H s ≤ C k ( f − l )( τ, · ) k H s . Now f − l solve: ( ∂ t ( f − l ) + H h D i α ( f − l ) = ( H h D i α − H h D i α ∗ ) f ∀ x ∈ D , ( f − l )(0 , x ) = 0 . (2.23)Thus we have the estimates: k f − l ( τ, · ) k H ν ≤ C (cid:13)(cid:13) ( h D i α −h D i α ∗ ) f (cid:13)(cid:13) L ([0 ,τ ] ,H ν ) ≤ Cτ (cid:13)(cid:13) ( h D i α −h D i α ∗ ) f (cid:13)(cid:13) L ∞ ([0 ,τ ] ,H ν ) By Theorem A.3 and the Kato-Ponce commutator estimates A.4, k f − l ( τ, · ) k H ν ≤ Cτ k ( Id − Dχ (0 , t, χ ( t, , x ))) k L ∞ k f k L ∞ ([0 ,τ ] ,H ν + α ) + Cτ k Id − Dχ (0 , t, χ ( t, , x )) k L ∞ ([0 ,τ ] ,W ν, ∞ ) k f k L ∞ ([0 ,τ ] ,H α ) + Cτ k Id − Dχ (0 , t, χ ( t, , x )) k L ∞ ([0 ,τ ] ,W , ∞ ) k f k L ∞ ([0 ,τ ] ,H ν + α − ) , Using Theorem B.3 and taking 0 < δ < s − α − : k f − l ( τ, · ) k H ν ≤ C ( τ λ ( α − + + τ λ α − ν ) λ ν − s . (2.24)Thus we get k ( f − l )( τ, χ (0 , τ, x )) k H ν ≤ C ( τ λ ( α − + + τ λ α − ν ) λ ν − s . Analogously we get k ( f − l )( τ, ˜ χ (0 , τ, x )) k H ν ≤ C ( τ λ ( α − + + τ λ α − ν ) λ ν − s , which gives k (1) k H ν ≤ C ( τ λ ( α − + + τ λ α − ν ) λ ν − s . (2.25)Now we estimate (2) in the same manner, by Proposition 2.1: k ( f − g )( τ, ˜ χ (0 , τ, x )) k H ν ≤ k ( f − g )( τ, · ) k H ν Like the ones used in proving the quasi-linearity of the Burgers equation. − g solve: ( ∂ t ( f − g ) + H h D i α ∗ ( f − g ) + ( H h D i α ∗ − ^ H h D i α ∗ ) g = 0 ∀ x ∈ D , ( f − g )(0 , x ) = ( u − v )( x ) . (2.26)By Theorem A.3 and the Kato-Ponce commutator estimates A.4, k f − g ( τ, · ) k H ν ≤ C k u − v k H ν + Cτ k ( D ˜ χ (0 , t, ˜ χ ( t, , x )) − Dχ (0 , t, χ ( t, , x ))) k L ∞ k g k L ∞ ([0 ,τ ] ,H ν + α ) + Cτ k D ˜ χ (0 , t, ˜ χ ( t, , x )) − Dχ (0 , t, χ ( t, , x )) k L ∞ ([0 ,τ ] ,W ν, ∞ ) k g k L ∞ ([0 ,τ ] ,H α ) + Cτ k D ˜ χ (0 , t, ˜ χ ( t, , x )) − Dχ (0 , t, χ ( t, , x )) k L ∞ ([0 ,τ ] ,W , ∞ ) k g k L ∞ ([0 ,τ ] ,H ν + α − ) . Using Theorem B.3 and taking 0 < δ < s − α − : k f − g ( τ, · ) k H ν ≤ C ( ǫ + τ ǫλ α λ ν − s + ǫτ + τ λ ( α − + λ ν − s ) , which gives k (2) k H ν ≤ C ( ǫ + τ ǫλ α λ ν − s + ǫτ + τ λ ( α − + λ ν − s ) , (2.27)finishing the proof of Lemma 2.1 and Theorem 1.1.3. A technical generalization
The techniques used in the previous proof will be generalized but with some carein the estimates due to the non linearity we add to the dispersive term. This extra”complication” is crucial for our application to the Water Waves system.
Theorem 3.1.
Consider three real numbers α ∈ [0 , , s ∈ ]2 + d , + ∞ [ and T > .Consider a elliptic skew symmetric C symbol a : [0 , T ] × H s ( D d ) → Γ α ( D d ) i.e such that Re( a t ) = a t + a ⊤ t is bounded in Γ ( D d ) , ∃ C > , ∀ ( t, x ) ∈ [0 , T ] × D d , ∀ ξ, | ξ | ≥ , | a ( t, xξ ) | ≥ C | ξ | α . Consider a C function V ( t, x, u ) : [0 , T ] × D d × H s ( D d ) → H s ( D d ; R d ) and afunction F ∈ L ∞ (cid:0) [0 , T ] , W , ∞ (cid:0) H s ( D d ) , H s ( D d ) (cid:1)(cid:1) .Suppose that the following hypothesis H1 is verified, there exists ω ∈ C ∞ c ( D d ) supported in B (0 , such that ∀ ( t, x ) ∈ [0 , T ] × supp ω, C − x t ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t D u V ( s, x, ω ( x )] ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C x t. ( H1 ) for a constant C x > when ω ( x ) = 0 .Fix u ∈ H s ( D d ) and take r > , then there exists T > such that for all v inthe ball B ( u , r ) ⊂ H s ( D d ) the Cauchy problem: ( ∂ t v + T V ( t,x,v ) · ∇ v + T a ( t,v ) v = F ( t, v ) v (0 , · ) = v ( · ) , (3.1) has a unique solution v ∈ C ([0 , T ] , H s ( D d )) . Moreover ∀ R > , the flow map:B (0 , R ) → C ([0 , T ] , H s ( D d )) v v is not uniformly continuous. Recall the notation a ⊤ for the adjoin of an operator a . onsidering a weaker control norm we get, for all ǫ > the flow map:B (0 , R ) → C ([0 , T ] , H s − α − + + ǫ ( D d )) v v is not C . In the proof of quasi-linearity of the water waves systems we will need the followingslight generalization to systems given by the following corollary.
Corollary 3.1.
Consider a positive integer n ≥ and three real numbers α ∈ [0 , , s ∈ ]2 + d , + ∞ [ and T > .Consider a C elliptic symbol a : [0 , T ] × H s ( D d ; R n ) → Γ α ( D d ; M n ( R )) skewsymmetric.Consider a C function V ( t, x, u ) : [0 , T ] × D d × H s ( D d ; C n ) → H s ( D d ; R n ) and afunction F ∈ L ∞ (cid:0) [0 , T ] , W , ∞ (cid:0) H s ( D d ; R n ) , H s ( D d ; R n ) (cid:1)(cid:1) .Suppose that the following hypothesis H1 is verified, there exists ω ∈ C ∞ c ( D d ; R n ) supported in B (0 , such that ∀ ( t, x ) ∈ [0 , T ] × supp ω, C − x t ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t D u V ( s, x, ω ( x )] ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C x t, ( H1 ) for a constant C x > when ω ( x ) = 0 .Fix u ∈ H s ( D d ; R n ) and take r > , then there exists T ′ > such that for all v in the ball B ( u , r ) ⊂ H s ( D d ; R n ) the Cauchy problem: ( ∀ i ∈ [1 , .., n ] , ∂ t v i + V ( t, x, v ) · ∇ v i + ( T a ( t,v ) v ) i = F i ( t, v ) ,v (0 , · ) = v ( · ) , has a unique solution v ∈ C ([0 , T ′ ] , H s ( D d ; R n )) . Moreover ∀ R > , the flow map:B (0 , R ) → C ([0 , T ′ ] , H s ( D d ; R n )) v v is not uniformly continuous.Considering a weaker control norm we get, for all ǫ ′ > the flow map:B (0 , R ) → C ([0 , T ′ ] , H s − α − + + ǫ ′ ( D d ; R n )) v v is not C . Prerequisites on the Cauchy problem.
We consider the Cauchy problemassociated to Theorem 3.1: ( ∂ t u + T V ( t,x,u ) · ∇ u + T a u = F ( t, u ) u (0 , · ) = u ( · ) ∈ H s ( D d ) , s > d , (3.2) Theorem 3.2.
Consider ≤ α < , T > , an elliptic C symbol a : [0 , T ] × H s ( D d ) → Γ α ( D d ) skew symmetrici.e such that Re( a t ) = a t + a ⊤ t is bounded in Γ ( D d ) . Consider a C function V ( t, x, u ) : [0 , T ] × D d × H s ( D d ) → H s ( D d ; R d ) and afunction F ∈ L ∞ (cid:0) [0 , T ] , W , ∞ (cid:0) H s ( D d ) , H s ( D d ) (cid:1)(cid:1) .Consider s > d , r > and u ∈ H s ( D d ) such that : ∀ v ∈ B ( u , r ) , T k∇ x v k L ∞ < . hen the problem (3.2) with initial data v has a unique solution v ∈ C ([0 , T ] , H s ( D d )) and the map v v is continuous from B ( u , r ) to C ([0 , T ] , H s ( D d )) . Moreover wehave the estimates: ∀ ≤ µ ≤ s, k v ( t ) k H µ ( D d ) ≤ C µ k v k H µ ( D d ) . (3.3) Taking Two different solution v, v ′ , such that v ∈ H s + β ( D d ) : ∀ ≤ µ ≤ s, (cid:13)(cid:13) ( v − v ′ )( t ) (cid:13)(cid:13) H µ ( D d ) ≤ (cid:13)(cid:13) v − v ′ (cid:13)(cid:13) H µ ( D d ) e C µ R t k v ( s ) k Hµ + α ( D d ) ds. (3.4)We will also work with hyperbolic paradifferential equations and we summarizethe properties needed in the following Theorem: Theorem 3.3.
Consider ( a t ) t ∈ R a family of symbols in Γ β ( D d ) with β ∈ R , such that t a t is continuous and bounded from R to Γ β ( D d ) and such that Re( a t ) = a t + a ⊤ t is bounded in Γ ( D d ) , and take T > . Then for all initial data u ∈ H s ( D d ) , and f ∈ C ([0 , T ]; H s ( D d )) the Cauchy problem: ( ∂ t u + T a u = f ∀ x ∈ D d , u (0 , x ) = u ( x ) (3.5) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimates: k u ( t ) k H s ( D d ) ≤ e Ct k u k H s ( D d ) + 2 Z t e C ( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s ( D d ) dt ′ , where C depends on a finite symbol semi-norm M (Re( a t )) . Remark 3.1.
We will also need to remark that fixing the initial data at 0 is anarbitrary choice. More precisely, ∀ ≤ t ≤ T and all data u ∈ H s ( D d ) the Cauchyproblem: ( ∂ t u + T a u = f ∀ x ∈ D d , u ( t , x ) = u ( x ) (3.6) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimate: k u ( t ) k H s ( D d ) ≤ e C | t − t | k u k H s ( D d ) + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s ( D d ) dt ′ (cid:12)(cid:12)(cid:12)(cid:12) . Proof of Theorem 3.1.
As for Theorem 1.1, for the proof we will show thatthere exists a positive constant C and two sequences ( u λǫ,τ ) and ( v λǫ,τ ) solutions of(3.1) on [0 ,
1] such that for every t ∈ [0 , λ,ǫ,τ (cid:13)(cid:13)(cid:13) u λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D d )) + (cid:13)(cid:13)(cid:13) v λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D d )) ≤ C, ( u λǫ,τ ) and ( v λǫ,τ ) satisfy initiallylim λ → + ∞ ǫ,τ → (cid:13)(cid:13)(cid:13) u λǫ,τ (0 , · ) − v λǫ,τ (0 , · ) (cid:13)(cid:13)(cid:13) H s ( D d ) = 0 , but, lim inf λ → + ∞ ǫ,τ → (cid:13)(cid:13)(cid:13) u λǫ,τ − v λǫ,τ (cid:13)(cid:13)(cid:13) L ∞ ([0 , ,H s ( D d )) ≥ c > . Considering a weaker control norm we want to get, for all ǫ ′ > λ → + ∞ ǫ,τ → (cid:13)(cid:13) u λǫ,τ − v λǫ,τ (cid:13)(cid:13) L ∞ ([0 , ,H s − α − ǫ ′ ( D d )) (cid:13)(cid:13) u λǫ,τ (0 , · ) − v λǫ,τ (0 , · ) (cid:13)(cid:13) H s ( D d ) = + ∞ . .2.1. Definition of the Ansatz.
Let ( λ, ǫ ) be two positive real sequences such that λ → + ∞ , ǫ → , λǫ → + ∞ , (3.7)and put • on R d , u ( x ) = λ d − s ω ( λx ) , v ( x ) = u ( x ) + ǫω ( x ) , • on T d , u and v as the periodic extensions of the functions defined above.Take t > τ ) , < τ ≤ t .Now let l be the solutions to the Cauchy problem on [0 , t ]: ( ∂ t l + T a ( t,l ) l = F ( t, l ) ∀ x ∈ D d , l ( τ, x ) = u ( x ) . (3.8)Put u ( x ) = l (0 , x ) and define l ′ to be the solutions to the Cauchy problem on [0 , t ]: ( ∂ t l ′ + T a ( t,l ) l ′ = F ( t, l ′ ) ∀ x ∈ D d , l ( τ, x ) = v ( x ) . (3.9)and put v = l (0 , x ). Remark 3.2.
It’s important to notice that we use the same term T a ( t,l ) in (3.8) and (3.9) and thus ( l, l ′ ) have Lipschitz dependence on the data ( u , v ) . Define u and v as the solution given by Theorem 3.2 with initial data u and v onthe intervals [0 , T ], [0 , T ′ ]. Taking 0 < δ < s − − d , u and v are uniformly boundedin H d + δ ( D d ) and thus by Theorem 3.3, u and v are also uniformly boundedin H d + δ ( D d ) and thus by the Sobolev injection Theorems they are bounded in˙ W , ∞ ( D d ). Thus we can take a uniform 0 < T on which all the solutions are welldefined and we take 0 < t ≤ T .3.2.2. Change of variables by transport.
Put ( ddt χ ( t, s, x ) = V (cid:0) t, χ ( t, s, x ) , u ( t, χ ( t, s, x )) (cid:1) ,χ ( s, s, x ) = x, and define analogously ˜ χ from v . We recall that from the Cauchy-Lipschitz Theo-rem as u and v are H + ∞ ( D d ) functions, then u , v are H + ∞ and u and v are H + ∞ ( D d ) with respect to the x variable thus χ, ˜ χ ∈ C ([0 , T ] , C ∞ ( D d )). And theyare both diffeomorphisms in the x variable.By the estimate (2.2) u and v are uniformly bounded in ˙ W , ∞ ( D d ) say by M > u and v thus by those of u and v by Theorem 3.3. By classic manipulations of ODEs we get the estimates: ( ∃ C > , ∀ ( t ′ , t ) ∈ [0 , t ] , ∀ x ∈ D d , C − ≤ | Dχ ( t, t ′ , x ) | ≤ C ∀ ≤ k < ⌊ s − d ⌋ , (cid:13)(cid:13) D k χ ( t, t ′ , x ) (cid:13)(cid:13) L ∞ ≤ C k u k W k, ∞ (3.10)Analogous estimates hold for ˜ χ using v .Now we compute the analogue of the classic transport computation but with theparacomposition operator which reads: ∂ t ( χ ( t, , x ) ∗ u ( t, x )) = χ ( t, , x ) ∗ ∂ t u + T ∂ t χ ( t, ,x ) · χ ( t, , x ) ∗ ∇ u ( t, x ) + R ( t, u )= − χ ( t, , x ) ∗ ( T a ( t,u ) u )( t, x ) + χ ( t, , x ) ∗ F ( t, u ) + R ( t, u )= − T a ( t,u ) ∗ χ ( t, , x ) ∗ u ( t, x ) + χ ( t, , x ) ∗ F ( t, u ) + R ( t, u ) + R ′ ( t, u ) ,χ (0 , , x ) ∗ u (0 , x ) = u (0 , x ) = u ( x ) . here ( · ) ∗ is the change of variables by χ ( t, , x ) as presented in Theorem A.8. Wecan assemble the terms R, R ′ and F in a new term F ′ verifying the same hypothesisas F, thus without loss of generality henceforth we will keep the generic notation Ffor all the terms verifying the same hypothesis.Thus if we put f the solution to the Cauchy problem, which is well posed by Ap-pendix B: ( ∂ t f + T a ( t,u ) ∗ f = χ ( t, , x ) ∗ F ( t, u ) ∀ x ∈ D d , f (0 , x ) = u ( x ) (3.11)we get: χ ( t, , x ) ∗ u ( t, x ) = f ( t, x ) . (3.12)Analogously, if we put g the solution to the well posed Cauchy problem, ( ∂ t g + T ^ a ( t,v ) ∗ g = ˜ χ ( t, , x ) ∗ F ( t, v ) ∀ x ∈ D d , g (0 , x ) = v ( x ) (3.13)where f ( · ) ∗ is the change of variables by ˜ χ ( t, , x ), we get˜ χ ( t, , x ) ∗ v ( t, x ) = g ( t, x ) . (3.14)Returning to the ODEs defining χ and ˜ χ we get: ( χ ( t, t ′ , x ) = x + R tt ′ V ( s, χ ( s, t ′ , x ) , f ( s, x ))) ds, ˜ χ ( t, t ′ , x ) = x + R tt ′ V ( s, ˜ χ ( s, t ′ , x ) , g ( s, x )) ds. (3.15) Proposition 3.1.
There exists
C > independent of ( τ, ǫ, λ ) such that: ∀ h ∈ H s ( D d ) , ∀ ( t, t ′ ) ≤ t ,C − k h k H s ≤ (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) (cid:13)(cid:13) H s ≤ C k h k H s ,C − k h k H s ≤ (cid:13)(cid:13) h ◦ ˜ χ ( t, t ′ , x ) (cid:13)(cid:13) H s ≤ C k h k H s . Proof.
We will start by proving the upper bound for the estimate on the compositionwith χ . As u is bounded in ( τ, ǫ, λ ) on C ([0 , T ] , H s ( D d )) then there exists a uniquesolution H ∈ C ([0 , T ] , H s ( D )) to ( ∂ s H ( s, x ) + V ( s, x, u ) · ∇ H ( s, x ) = 0 H ( t, x ) = h ( x )and H is bounded in ( τ, ǫ, λ ) on C ([0 , T ] , H s ( D d )). The desired bounds come fromthe fact that we have the explicit formula for H : H ( t ′ , x ) = h ◦ χ ( t, t ′ , x ) . Now to get the lower bound it suffices to write by the upper bound computations: k h k H s = (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) ◦ χ ( t ′ , t, x ) (cid:13)(cid:13) H s ≤ C (cid:13)(cid:13) h ◦ χ ( t, t ′ , x ) (cid:13)(cid:13) H s . We get analogously the estimates on the composition with ˜ χ . (cid:3) .2.3. Key Lemma and proof of the Theorem.
Lemma 3.1. As ≤ α < we can find a sequence ( τ, ǫ, λ ) such that for all ǫ ′ > sufficiently small: τ → ,τ λ α − → ,ǫ = o ( τ λ α − ) , τ ǫλ α → ǫ − λ − α − + − ǫ ′ → + ∞ ,τ λǫ → + ∞ . (3.16) Then there exists c > such that:(1) ∀ ( τ, ǫ, λ, ν ) , (cid:13)(cid:13) u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x ) (cid:13)(cid:13) H s − ν > cλ − ν . (2) For δ such that < δ < s − − d : u ( τ, x ) − v ( τ, x ) = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x ) + O H s − ν (cid:0) ǫ + τ λ α − λ − ν + τ ǫλ α λ − ν (cid:1) . We will now show that this Lemma implies the Theorem. We have by combiningthe estimates (1) and (2) for ν = s : ∀ ( τ, ǫ, λ ) , k u ( τ, x ) − v ( τ, x ) k H s > c > τ,ǫ,λ k u ( τ, x ) − v ( τ, x ) k H s > c > ∃ C > , (cid:13)(cid:13) u ( x ) − v ( x ) (cid:13)(cid:13) H s ≤ Cǫ thus (cid:13)(cid:13) u ( x ) − v ( x ) (cid:13)(cid:13) H s → , which gives the non uniform continuity in the desired norms. Now for the controlin a weaker norm we write: k u ( τ, x ) − v ( τ, x ) k H s − α − − ǫ ′ k u ( x ) − v ( x ) k H s ≥ cǫ − λ − α − + − ǫ ′ → + ∞ , which gives the desired result.3.2.4. Proof of point 1 of Lemma 3.1.
We first prove that ∃ c > (cid:13)(cid:13) u ◦ χ (0 , τ, x ) (cid:13)(cid:13) H s > cλ − ν , indeed by Proposition 2.1 and change of variable: (cid:13)(cid:13) u ◦ χ (0 , τ, x ) (cid:13)(cid:13) H s − ν ≥ C − (cid:13)(cid:13) u (cid:13)(cid:13) H s − ν ≥ C − λ − ν k ω k H s − ν . (3.17)Now we will show that u ◦ χ (0 , τ, x ) and u ◦ ˜ χ (0 , τ, x ) have disjoint supports whichwill suffice to conclude given (3.17). Put y = χ (0 , τ, x ), thus x = χ ( τ, , y ). On thesupport of u ◦ χ (0 , τ, x ) we have: • If D d = R d , λ | y | ≤ • If D d = T d , ∀ k ∈ N , πk − ≤ λ | y | ≤ πk + 1.We then compute by the Taylor formula:˜ χ (0 , τ, x ) = ˜ χ (0 , τ, ˜ χ ( τ, , y ))+ Z D y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr [ χ ( τ, , y ) − ˜ χ ( τ, , y )](3.18)= y + Z D y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr [ χ ( τ, , y ) − ˜ χ ( τ, , y )] . First, D y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y ))= Id + Z τ D y [ V ( t, ˜ χ (0 , τ, rχ ( τ, , y )+(1 − r ) ˜ χ ( τ, , y )) , g ( t, rχ ( τ, , y )+(1 − r ) ˜ χ ( τ, , y )))] dt. hus by estimates of Theorem B.1 taking 0 < δ < s − d − D y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) = Id + O L ∞ ( τ ( (cid:13)(cid:13) v (cid:13)(cid:13) H d δ + (cid:13)(cid:13) u (cid:13)(cid:13) H d δ ))= Id + O L ∞ ( τ ) , Which gives Z D y ˜ χ (0 , τ, rχ ( τ, , y ) + (1 − r ) ˜ χ ( τ, , y )) dr = Id + O L ∞ ( τ ) . (3.19)Now we estimate χ ( τ, , y ) − ˜ χ ( τ, , y ), by (3.15) : χ ( τ, , y ) − ˜ χ ( τ, , y ) = Z τ V ( t, χ ( t, , y ) , f ( t, y )) − V ( t, ˜ χ ( t, , y ) , g ( t, y )) dt (3.20)= Z τ Z D u V ( t, χ ( t, , y ) , rf ( t, y ) + (1 − r ) g ( t, y ))[ f ( t, y ) − g ( t, y )] dtdr + Z τ Z D x V ( t, rχ ( t, , y ) + (1 − r ) ˜ χ ( t, , y ) , g ( t, y ))[ χ ( t, , y ) − ˜ χ ( t, , y )] dtdr. Taking 0 < δ < s − α − d ,by estimates of Theorem B.1: f ( t, y ) = f (0 , y ) + Z t ∂ t f ( r, y ) dr = u ( y ) + O L ∞ ( t ( (cid:13)(cid:13) u (cid:13)(cid:13) H d α + δ )) = u ( y ) + O L ∞ ( tλ − ) . Analogously we get: g ( t, y ) = v ( y ) + O L ∞ ( tǫ ) . Now ( u − v )( y ) = ( l − l ′ )(0 , y ) is the evaluation of the solution of the followingCauchy problem at t = 0: ( ∂ t ( l − l ′ ) + T a ( t,l ) ( l − l ′ ) = F ( t, l ) − F ( t, l ′ ) ∀ y ∈ D d , ( l − l ′ )( τ, y ) = − ǫω ( y ) . (3.21)thus by estimates of Theorem B.1: u ( y ) − v ( y ) = ( l − l ′ )(0 , y ) = − ǫω ( y ) + Z τ ∂ t ( l − l ′ )( t, y ) dt = − ǫω ( y ) + O L ∞ ( τ ( (cid:13)(cid:13) v (cid:13)(cid:13) H α + d δ + (cid:13)(cid:13) u (cid:13)(cid:13) H α + d δ ))= − ǫω ( y ) + O L ∞ ( τ ǫ ) . Thus, χ ( τ, , y ) − ˜ χ ( τ, , y )= − ǫ [ Z τ Z D u V ( t, χ ( t, , y ) , rf ( t, y ) + (1 − r ) g ( t, y )) dtdr ][ ω ( y )] + O L ∞ ( τ ǫ )+ Z τ Z D x V ( t, rχ ( t, , y ) + (1 − r ) ˜ χ ( t, , y ) , g ( t, y )) dr [ χ ( t, , y ) − ˜ χ ( t, , y ) | {z } ∗ ] dt. Iterating the computation in (*): χ ( τ, , y ) − ˜ χ ( τ, , y )= − ǫ [ Z τ Z D u V ( t, χ ( t, , y ) , rf ( t, y ) + (1 − r ) g ( t, y )) dtdr ][ ω ( y )] + O L ∞ ( τ ǫ )= − ǫ [ Z τ D u V ( t, y, dt ][ ω ( y )] + O L ∞ ( τ ǫ + ǫ τ ) . nd finally we get in (3.18),˜ χ ( τ, , x ) − y = − ǫ [ Z τ D u V ( t, y, dt ][ ω ( y )] + O L ∞ ( τ ǫ + ǫ τ ) . We get for x ∈ supp u ◦ χ (0 , τ, · ): • For D d = R d : λ | ˜ χ (0 , τ, x ) |≥ ǫλ (cid:12)(cid:12)(cid:12)(cid:12)Z τ Z D u V ( t, y, drdt [ ω ( y )] (cid:12)(cid:12)(cid:12)(cid:12) − o L ∞ ( τ λǫ ) ≥ , which gives the desired result. • For D d = T d given an adequate choice of τ, ǫ and λ we get:2 nπ + 1 ≤ λ | ˜ χ (0 , τ, x ) | ≤ n + 1) π − , Which again gives the desired result.3.2.5.
Proof of point 2 of Lemma 3.1.
We start by writing: u ( t, x ) − v ( t, x ) = χ (0 , t, x ) ∗ f ( t, x ) − ˜ χ (0 , t, x ) ∗ g ( t, x ) + R ( f ) − R ( g )where R is a regularizing operator of order 2, u ( t, x ) − v ( t, x ) = χ (0 , t, x ) ∗ f ( t, x ) − ˜ χ (0 , t, x ) ∗ f ( t, x ) | {z } (1) + ˜ χ (0 , t, x ) ∗ ( f − g )( t, x ) + R ( f ) − R ( g ) . Term (1) resembles the main term in the usual transport estimates we used in point1 of the Lemma but with a main difference is f being some dispersed data andnot compactly supported and the use of the paracomposition operator. Again, themain trick here was to construct from u , v the defocused data in the past u , v and use this as the initial data for f and g . u ( τ, x ) − v ( τ, x ) = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x )+ T ( u ) ′ ◦ χ (0 ,τ,x ) χ (0 , τ, x ) − T ( u ) ′ ◦ ˜ χ (0 ,τ,x ) ˜ χ (0 , τ, x )+ χ (0 , τ, x ) ∗ ( f − u )( τ, x ) − ˜ χ (0 , t, x ) ∗ ( f − u )( τ, x )+ ˜ χ (0 , τ, x ) ∗ ( f − g )( τ, x ) + R ( f ) − R ( g ) . = u ◦ χ (0 , τ, x ) − u ◦ ˜ χ (0 , τ, x )+ χ (0 , τ, x ) ∗ ( f − l )( τ, x ) − ˜ χ (0 , t, x ) ∗ ( f − l )( τ, x ) | {z } (1) + ˜ χ (0 , τ, x ) ∗ ( f − g )( τ, x ) | {z } (2) + R ( f ) − R ( g )+ T ( u ) ′ ◦ χ (0 ,τ,x ) χ (0 , τ, x ) − T ( u ) ′ ◦ ˜ χ (0 ,τ,x ) ˜ χ (0 , τ, x ) | {z } (3) . Where l is defined by (3.8) and R was modified to contain other regularizing opera-tors of order 2 that appear by symbolic calculus rules. The easiest part to estimateis the remainder one because of the gain of derivatives and Theorem (3.4): k R ( f ) − R ( g ) k H s ≤ Cǫ. Like the ones used in proving the quasi-linearity of the Burgers equation. e turn to estimating (1), by Theorem A.8: k χ (0 , τ, x ) ∗ ( f − l )( τ, x ) k H s ≤ C k ( f − l )( τ, · ) k H s . Now f − l solve: ( ∂ t ( f − l ) + T a ( t,l ) ( f − l ) = ( T a ( t,l ) − T a ( t,u ) ∗ ) f − F ( t, l ) + χ ( t, , x ) ∗ F ( t, u ) F ( t, f ) ∀ x ∈ D d , ( f − l )(0 , x ) = 0 . (3.22)Writing χ ( t, , x ) ∗ F ( t, u ) − F ( t, l ) = G ( f − l )where G is a continuous linear operator on H s we get the estimates by TheoremB.1: k f − l ( τ, · ) k H ν ≤ C [ (cid:13)(cid:13) ( T a ( τ,l ) − T a ( t,l ) ∗ ) f (cid:13)(cid:13) L ([0 ,τ ] ,H ν ) + (cid:13)(cid:13) ( T a ( τ,l ) ∗ − T a ( τ,u ) ∗ ) f (cid:13)(cid:13) L ([0 ,τ ] ,H ν ) ] ≤ C [ τ (cid:13)(cid:13) Id − Dχ − (cid:13)(cid:13) L ∞ k f k H ν + α + τ k u − l k L ∞ k f k H ν + α ] . As s > d k f − l ( τ, · ) k H ν ≤ C [ τ λ α − + τ λ α − ] λ ν − s , which gives k χ (0 , τ, x ) ∗ ( f − l )( τ, x ) k H ν ≤ Cτ λ α − λ ν − s , and k ˜ χ (0 , τ, x ) ∗ ( f − l )( τ, x ) k H ν ≤ Cτ λ α − λ ν − s . Thus we finally get k (1) k H ν ≤ Cτ λ α − λ ν − s . (3.23)Now we estimate (2) and (3) in the same manner, by Theorem A.8: k ˜ χ (0 , τ, x ) ∗ ( f − g )( τ, x ) k H ν ≤ C k ( f − g )( τ, · ) k H ν , And as s > d and by (3.15): (cid:13)(cid:13) T ( u ) ′ ◦ χ (0 ,τ,x ) χ (0 , τ, x ) − T ( u ) ′ ◦ ˜ χ (0 ,τ,x ) ˜ χ (0 , τ, x ) (cid:13)(cid:13) ≤ C k ( f − g )( τ, · ) k H ν . Now f − g solve: ∂ t ( f − g ) + T a ( t,u ) ∗ ( f − g ) − ( T a ( t,v ) ∗ − T ^ a ( t,v ) ∗ ) g − χ ( t, , x ) ∗ F ( t, u ) + ˜ χ ( t, , x ) ∗ F ( t, v ) = ( T a ( t,u ) ∗ − T a ( t,v ) ∗ ) g ∀ x ∈ D d , ( f − g )(0 , x ) = ( u − v )( x ) . (3.24)Here will need to be more careful as the nonlinearity in the dispersive term can bemore ”harmful” than the transport term when α ≥
1, which was not there in thetreatment of the model problem. More precisely we write: χ ( t, , x ) ∗ F ( t, u ) − F ( t, l ) = G ( f − g ) , where G is a continuous linear operator on H s and we get the estimates by applyingTheorem 3.4 with λ = ∂ u a. Thus we get k f − g ( τ, · ) k H ν ≤ C [ (cid:13)(cid:13)(cid:13) ( T a ( t,v ) ∗ − T ^ a ( t,v ) ∗ ) g (cid:13)(cid:13)(cid:13) L ([0 ,τ ] ,H ν ) + (cid:13)(cid:13) u − v (cid:13)(cid:13) H ν ] ≤ C [ τ (cid:13)(cid:13) Dχ − − D ˜ χ − (cid:13)(cid:13) L ∞ k g k H ν + ǫ ] ≤ C [ τ ǫλ α λ ν − s + ǫ ]which gives k (2) k H ν ≤ C ( τ ǫλ α λ ν − s + ǫ ) , (3.25) nd k (3) k H ν ≤ C ( τ ǫλ α λ ν − s + ǫ ) , (3.26)finishing the proof of Lemma 3.1 and Theorem 3.1.3.2.6. Estimate on the differential of the transported flow map.
Here we will givethe crucial estimate needed in section 3.2.5.
Theorem 3.4.
Consider two real numbers α ∈ [0 , , s ∈ ]2 + d , + ∞ [ .Consider two elliptic skew symmetric C symbols γ : [0 , T ] × H s ( D d ) → Γ α ( D d ) and ˜ γ : [0 , T ] × H s ( D d ) → Γ α ( D d ) .Consider a function F ∈ L ∞ ([0 , T ] , W , ∞ ( H s ( D d ) , H s ( D d ))) and a symbol λ ∈ Γ α ( D d ) .Fix g ∈ H s ( D d ) and take T > the existence time associated to the well-posedCauchy problem by Theorem 3.2: ( ∂ t g + T γ g = F ( t, g ) g (0 , · ) = g ( · ) . (3.27) Then for all h ∈ H s ( D d ) and f ∈ C ([0 , T ]; H s ( D d )) the problem ( ∂ t h + T ˜ γ h = T h T λ g + fh (0 , · ) = h ( · ) (3.28) has a unique solution h ∈ C ([0 , T ] , H s ( D d )) .Moreover we have the estimates: ∀ t ∈ [0 , T ] , k h ( t ) k H s ( D d ) ≤ C ( M ( Re (˜ γ )) , k g k H s ( D d ) ) k h k H s ( D d ) + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( k g k Hs ( D d ) ,M ( Re (˜ γ )))( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ (cid:12)(cid:12)(cid:12)(cid:12) . (3.29) Proof.
We start by noticing that the well posedness of the Cauchy problem is alreadyalready known for g ∈ H s + α ( D d ) and we have the immediate estimate ∀ t ∈ [0 , T ] , k h ( t ) k H s ( D d ) ≤ C ( t k g k H s + α ( D d ) ) k h k H s ( D d ) . (3.30)The goal of this proof is thus to significantly improve this estimate on g , we startby working with s ≥ α + d to justify the computations and with the improvedestimate the usual bootstrap argument will give the result for s > d .We start by getting an equation on T h T λ g where we ”morally” use the ellipticityto write T γ − ∂ t as an operator of order 0 at the price of an acceptable remainder: ∂ t g + T γ g = F ( t, g ) T h T λ T γ − ∂ t g + T h T λ g = T λ T γ − F ( t, g )as s > d , T h ∂ t (cid:0) T λ T γ − g (cid:1) + T h T λ g = T h T λ T γ − F ( t, g )where F was modified on each line to include terms which verify the same hypothesis.The ”gain” is that T λ T γ − is of order 0 and that the ”cost” of α derivative in thespatial variable is put on ∂ t .Getting back to h we have: ∂ t h + T ˜ γ h = − T h ∂ t (cid:0) T λ T γ − g (cid:1) + T h T λ T γ − F ( t, g ) + f. (3.31)The key idea is that now we now how to solve ∂ t h + h∂ t (cid:0) T λ T γ − g (cid:1) = 0 explicitlyso we make the change of unknowns: u = e T λ T γ − g h. (3.32) s s > d by the Sobolev embedding we have g ∈ W , ∞ and it’s clear that H s estimates on h are equivalent to ones on u i.e for all t ∈ [0 , T ]: C − ( k g k H s ) k u k H s ≤ k h k H s ≤ C ( k g k H s ) k u k H s . (3.33)Now we compute the P DE verified by u : ∂ t u + e T λ T γ − g T ˜ γ e − T λ T γ − g u = T ∂ t g u + e T λ T γ − g T h T λ T γ − F ( t, g ) (3.34)+ e T λ T γ − g f + e T λ T γ − g R where R verifies by A.6: k R k H s ≤ C k g k H s k h k H s . (3.35)Now we reduce the H s estimates on u to L , first by noticing that T e T λ T γ − g T ˜ γ T e − T λ T γ − g is an elliptic paradifferential operator of a symbol we willcall β . Then by making the change of variables φ = T | β | sα u . By the ellipticity of β and the immediate L estimate on u , an H s estimate on u is equivalent to an L estimate on φ i.e C − ( k g k H s ) k u k H s ≤ k φ k L ≤ C ( k g k H s ) k u k H s . (3.36)We then commute T | β | sα with equation (3.34) and get by assembling the differentterms verifying the same estimates as R in one term: ∂ t φ + e T λ T γ − g T ˜ γ e − T λ T γ − g φ = T | β | sα e T λ T γ − g f + T | β | sα R. (3.37)Now to finally get the L estimate on φ we need to do the classic energy esti-mate but with the adequate choice of basis in which e T λ T γ − g T ˜ γ e − T λ T γ − g is skewsymmetric. For this the energy estimate is made on ∂ t k φ k L (cid:0) e − TλTγ − g ) dx (cid:1) , and the residual term k φ k L (cid:0) ∂ t e − TλTγ − g ) dx (cid:1) is controlled as s > d . Combining this estimates and the Gronwall lemma weget k φ k L ≤ C ( M ( Re (˜ γ )) , k g k H s ) k φ k L + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( k g k Hs ( D d ) ,M ( Re (˜ γ )))( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ (cid:12)(cid:12)(cid:12)(cid:12) , which concludes the proof by (3.33) and (3.36). (cid:3) Quasi-linearity of the Water-Waves system with surface tension
In this section we always have κ = 1.4.1. Prerequisites from the Cauchy problem.
We start by recalling the aprioriestimates given by Proposition 5 . Proposition 4.1. (From [2] ) Let d ≥ be the dimension and consider a real number s > d . Then there exists a non decreasing function C such that, for all T ∈ ]0 , and all solution ( η, ψ ) of (1.5) such that ( η, ψ ) ∈ C ([0 , T ]; H s + ( R d ) × H s ( R d )) and H t is verified for t ∈ [0 , T ] ,we have k ( η, ψ ) k L ∞ (0 ,T ; H s + 12 × H s ) ≤ C (( η , ψ ) H s + 12 × H s ) + T C ( k ( η, ψ ) k L ∞ (0 ,T ; H s + 12 × H s ) ) . he proof will rely on the para-linearised and symmetrized version of (1.5) givenby Proposition 4 . . m ⊂ Γ m given by: Definition 4.1. (From [2] ) Given m ∈ R , Σ m denotes the class of symbols a of theform a = a ( m ) + a ( m − with a ( m ) = F ( ∇ η ( t, x ) , ξ ) a ( m − = X | k | =2 G α ( ∇ η ( t, x ) , ξ ) ∂ kx η ( t, x ) , such that(1) T a maps real valued functions to real-valued functions;(2) F is of class C ∞ real valued function of ( ζ, ξ ) ∈ R d × ( R d \ , homogeneous oforder m in ξ ; and such that there exists a continuous function K = K ( ζ ) > such that F ( ζ, ξ ) ≥ K ( ζ ) | ξ | m , for all ( ζ, ξ ) ∈ R d × ( R d \ ;(3) G α is a C ∞ complex valued function of ( ζ, ξ ) ∈ R d × ( R d \ , homogeneousof order m − in ξ . Σ m enjoys all the usual symbolic calculus properties modulo acceptable remindersthat we define by the following: Definition-Notation 4.1. (From [2] ) Let m ∈ R and consider two families ofoperators of order m, { A ( t ) : t ∈ [0 , T ] } , { B ( t ) : t ∈ [0 , T ] } . We shall say that A ∼ B if A − B is of order m − and satisfies the followingestimate: for all µ ∈ R , there exists a continuous function C such that for all t ∈ [0 , T ] , k A ( t ) − B ( t ) k H µ → H µ − m + 32 ≤ C ( k η ( t ) k H s + 12 ) . In the next Proposition we recall the different symbols that appear in the para-linearisation and symmetrisation of the equations.
Proposition 4.2. (From [2] )We work under the hypothesis of Proposition 4.1. Put λ = λ (1) + λ (0) , l = l (2) + l (1) with, λ (1) = q (1 + |∇ η | ) | ξ | − ( ∇ η · ξ ) ,λ (0) = |∇ η | λ (1) (cid:26) div (cid:18) α (1) ∇ η (cid:19) + i∂ ξ λ (1) · ∇ α (1) (cid:27) ,α (1) = √ |∇ η | (cid:18) λ (1) + i ∇ η · ξ (cid:19) . (4.1) l (2) = (1 + |∇ η | ) − (cid:18) | ξ | − ( ∇ η · ξ ) |∇ η | (cid:19) ,l (1) = − i ( ∂ x · ∂ ξ ) l (2) . (4.2) ow let q ∈ Σ , p ∈ Σ , γ ∈ Σ be defined by q = (1 + |∇ η | ) − ,p = (1 + |∇ η | ) − p λ (1) + p ( − ) ,γ = p l (2) λ (1) + s l (2) λ (1) Reλ (0) − i ∂ ξ · ∂ x ) p l (2) λ (1) ,p ( − ) = 1 γ ( ) n ql (1) − γ ( ) p ( ) + i∂ ξ γ ( ) · ∂ x p ( ) o . Then T q T λ ∼ T γ T q , T q T l ∼ T γ T p , T γ ∼ ( T γ ) ⊤ . Now we can write the para-linearization and symmetrization of the equations(1.5) after a change of variable:
Corollary 4.1. (From [2] )Under the hypothesis of Proposition 4.1, introduce theunknowns U = ψ − T B η , Φ = T p η and Φ = T q U, where we recall, ( B = ( ∂ y φ ) | y = η = ∇ η ·∇ ψ + G ( η ) ψ |∇ η | ,V = ( ∇ x φ ) | y = η = ∇ ψ − B ∇ η. Then Φ , Φ ∈ C ([0 , T ]; H s ( R d )) and ( ∂ t Φ + T V · ∇ Φ − T γ Φ = f ,∂ t Φ + T V · ∇ Φ + T γ Φ = f , (4.3) with f , f ∈ L ∞ (0 , T ; H s ( R d )) , and f , f have C dependence on ( U, θ ) verifying: k ( f , f ) k L ∞ (0 ,T ; H s ( R d )) ≤ C ( k ( η, ψ ) k L ∞ (0 ,T ; H s + 12 × H s ( R d )) ) . Proof of Theorem 1.2.
Corollary 4.1 shows that the para-linearization andsymmetrization of the equations (1.5) are of the form of the equations treated inTheorem 3.1. The goal of the proof is thus to mainly show that the previous changeof unknowns preserves the quasi-linear structure of the equations. This we will beproved but with a slightly different change of unknowns that will satisfy the sametype of equations.4.2.1.
Reducing the problem around 0.
Fix
T > r > r small. Henceforth we will beworking on B(0 , r ) ⊂ C ([0 , T ]; H s + ( R d ) × H s ( R d )) and without loss of generalitywe suppose that H t is always verified on [0 , T ] on that set.4.2.2. New change of unknowns.
Lemma 4.1.
Under the hypothesis of Proposition 4.1, fix ǫ > and introduce theunknowns U = ψ − T B η, ˜Φ = [ T p + ǫ ( I − T )] η and ˜Φ = [ T q + ǫ ( I − T )] U. Then ˜Φ , ˜Φ ∈ C ([0 , T ]; H s ( R d )) and ( ∂ t ˜Φ + T V · ∇ ˜Φ − T γ ˜Φ = ˜ f ,∂ t ˜Φ + T V · ∇ ˜Φ + T γ ˜Φ = ˜ f , (4.4) U is commonly called the ”good” unknown of Alinhac. ith ˜ f , ˜ f ∈ L ∞ (0 , T ; H s ( R d )) , and ˜ f , ˜ f have C dependence on ( U, θ ) verifying: (cid:13)(cid:13)(cid:13) ( ˜ f , ˜ f ) (cid:13)(cid:13)(cid:13) L ∞ (0 ,T ; H s ( R d )) ≤ C ( k ( η, ψ ) k L ∞ (0 ,T ; H s + 12 × H s ( R d )) ) . Proof.
The Lemma simply follows from the fact that I − T is a regularizing operator. (cid:3) The new change of unknowns locally preserves the structure of the equations:
To apply Theorem 3.1 we simply note that DV (0 , h, k ) = ∇ h . Thus proof ofTheorem 1.2 in the threshold s > d will then follow from Theorem 3.1 combinedwith Lemma 4.1 and the following Lemma. Lemma 4.2.
Let d ≥ and s > d . There exists r, ǫ > such that: ˜Φ : B (0 , r ) → C ([0 , T ]; H s ( R d ))( η, ψ ) ( ˜Φ , ˜Φ ) is a C ∞ diffeomorphism upon it’s image and ˜Φ(0) =0.Proof. ˜Φ( η, ψ ) = (cid:18) T p + ǫ ( I − T ) 00 T q + ǫ ( I − T ) (cid:19)| {z } (1) (cid:18) I − T B I (cid:19)| {z } (2) (cid:18) ηψ (cid:19) (2) being clearly a diffeomorphism we will concentrate on (1).First we see that for r small enough T q + ǫ ( I − T ) is a perturbation of the T + ǫ ( I − T ),indeed by symbolic calculus rules: k T q + ǫ ( I − T ) − T − ǫ ( I − T ) k L ( H s ) = k T q − T k L ( H s ) ≤ M ( q − ≤ C ( k η k W , ∞ ) k η k W , ∞ ≤ C ( k η k H s ) k η k H s which gives the desired result.Now we turn to T p + ǫ ( I − T I ). First notice that for ǫ > T | ξ | + ǫ ( I − T I ) : C ([0 , T ]; H s + ( R d )) → C ([0 , T ]; H s ( R d ))is a C ∞ diffeomorphism. And now we see that T p + ǫ ( I − T I ) is a perturbation of T | ξ | + ǫ ( I − T I ) indeed by symbolic calculus rules: (cid:13)(cid:13)(cid:13) T p − T | ξ | (cid:13)(cid:13)(cid:13) L ( H s + 12 ,H s ) ≤ C ( k η k W , ∞ ) k η k W , ∞ ≤ C ( k η k H s ) k η k H s (cid:3) Now to conclude the proof of Theorem 1.2, we see that by Lemma 4.2, the equa-tions (4.4) verify the hypothesis of Corollary 3.1 in the threshold s > d thus wehave two sequences ( ∃ ( ˜Φ , ˜Φ ) ∈ C ([0 , T ]; H s ( R d )) solution of (4.4) , ∃ ( ˜Φ , ˜Φ ) ∈ C ([0 , T ]; H s ( R d )) solution of (4.4) , such that ∃ c > (cid:13)(cid:13)(cid:13) ( ˜Φ , ˜Φ )(0 , · ) − ( ˜Φ , ˜Φ )(0 , · ) (cid:13)(cid:13)(cid:13) H s → , (cid:13)(cid:13)(cid:13) ( ˜Φ , ˜Φ ) − ( ˜Φ , ˜Φ ) (cid:13)(cid:13)(cid:13) L ∞ ([0 ,T ] ,H s ) > c. ow putting ( η , ψ ) = ˜Φ − ( ˜Φ , ˜Φ ) and ( η , ψ ) = ˜Φ − ( ˜Φ , ˜Φ ) we get fromLemmas 4.1 and 4.2: ( ( η , ψ ) ∈ C ([0 , T ]; H s + ( R d ) × H s ( R d )) is a solution of (1.5) , ( η , ψ ) ∈ C ([0 , T ]; H s + ( R d ) × H s ( R d )) is a solution of (1.5) , such that (cid:13)(cid:13) ( η , ψ )(0 , · ) − ( η , ψ )(0 , · ) (cid:13)(cid:13) H s + 12 × H s → , (cid:13)(cid:13) ( η , ψ ) − ( η , ψ ) (cid:13)(cid:13) L ∞ ([0 ,T ] ,H s + 12 × H s ) > c. thus giving us the desired result. As the change of unknowns is a diffeomorphism(thus is Lipschitz) we get analogously the result on the control in weaker norms.5. Quasi-Linearity of the Gravity Water Waves
In this section we always have κ = 0. The proof will follow as in the previoussection but with some extra care, taking into account the lower regularity framework.5.1. Prerequisites from the Cauchy problem.
We start by recalling the aprioriestimates given by Proposition 4 . Proposition 5.1. (From [5] ) Let d ≥ be the dimension and consider a real number s > d . Then there exists a non decreasing function C such that, for all T ∈ ]0 , and all solution ( η, ψ ) of (1.5) such that: ( η, ψ ) ∈ C ([0 , T ]; H s + ( R d ) × H s + ( R d )) ,H t is verified for t ∈ [0 , T ] , ∃ c > , ∀ t ∈ [0 , T ] , a ( t, x ) ≥ c , we have k ( η, ψ, V, B ) k L ∞ ([0 ,T ]; H s + 12 × H s + 12 × H s × H s ) ≤ C ( k ( η , ψ , V , B ) k H s + 12 × H s + 12 × H s × H s )+ T C ( k ( η, ψ, V, B ) k L ∞ (0 ,T ; H s + 12 × H s + 12 × H s × H s ) ) . The proof will rely on the para-linearised and symmetrized version of (1.5) givenby Proposition 4 . .
10 of [5]. Given the low regularity threshold, η and thusΩ t are in W , ∞ ( R d ) for the gravity water waves by contrast to W , ∞ ( R d ) framework for the case with surface tension, the para-linearisation of (1.5) is done with thevariables V and B. This will only add a technical level to our proof of quasi-linearity. Proposition 5.2. (From [5] )Under the hypothesis of Proposition (5.1) , supposemoreover that k ( V , B ) k H s × H s < + ∞ thus by Proposition (5.1) this regularity ispropagated on [0 , T ] . Now introduce the unknowns ( ζ = ∇ η,U = V + T ζ B, where, ( B = ( ∂ y φ ) | y = η = ∇ η ·∇ ψ + G ( η ) ψ |∇ η | ,V = ( ∇ x φ ) | y = η = ∇ ψ − B ∇ η. Now define the symbols: λ = q (1 + |∇ η | ) | ξ | − ( ∇ η · ξ ) ,γ = √ aλ,q = p aλ . Recall B and V are defined by (1.7). et θ = T q ζ. Then θ, U ∈ C ([0 , T ]; H s ( R d )) and ( ∂ t U + T V · ∇ U + T γ θ = f ,∂ t θ + T V · ∇ θ − T γ U = f , (5.1) with f , f ∈ L ∞ (0 , T ; H s ( R d )) , and f , f have C dependence on ( U, θ ) verifying: k ( f , f ) k L ∞ (0 ,T ; H s ) ≤ C ( k ( η, ψ, V, B ) k L ∞ (0 ,T ; H s + 12 × H s + 12 × H s × H s ) )5.2. Proof of Theorem 1.3.
As in the proof of Theorem 1.2, Proposition (5.2)shows that the para-linearisation and symmetrisation of the Equations (1.5) are ofthe form of the equations treated in Theorem 3.1. Thus again, the goal of the proofis thus to mainly show that the previous change of unknowns preserves the quasi-linear structure of the equations. This we will be proved but with a slightly differentchange of unknowns that will satisfy the same type of equations but where we takeinto account the low frequencies. For concision we will omit the ( R d ) when writingthe functional spaces.5.2.1. Reducing the problem around 0.
Fix
T > r > r small.Put I s,T = n ( η, ψ ) ∈ C ([0 , T ]; H s + × H s + ) , ( V, B ) ∈ C ([0 , T ]; H s × H s ) , ∃ c > , a ≥ c o ,I s, = n ( η , ψ ) ∈ H s + × H s + , ( V , B ) ∈ H s × H s , ∃ c > , a ≥ c o , henceforth we will be working on B(0 , r ) ⊂ I s,T and without loss of generality wesuppose that H t is always verified on [0 , T ], on that set.5.2.2. New change of unknowns.
Lemma 5.1.
Consider ǫ > and ω ∈ C ∞ ( R d ) such that ω = 1 on B (0 , and ω = 0 on R d \ B (0 , . Under the hypothesis of Proposition (5.2) , introduce the unknowns ˜ ζ = (1 − ω ( D )) ∇ η, ˜ U = (1 − ω ( D ))( V + T ζ B ) ,aux = ω ( D ) ψ,aux = ω ( D ) η, where, ( B = ( ∂ y φ ) | y = η = ∇ η ·∇ ψ + G ( η ) ψ |∇ η | ,V = ( ∇ x φ ) | y = η = ∇ ψ − B ∇ η. and set ˜ θ = T q ˜ ζ + ǫ ( I − T ) , where q is defined in Proposition (5.2) .Then ˜ θ, U, aux , aux ∈ C ([0 , T ]; H s ) and ( ∂ t ˜ U + T V · ∇ ˜ U + T γ ˜ θ = f ′ ,∂ t ˜ θ + T V · ∇ ˜ θ − T γ ˜ U = f ′ , (5.2) with f ′ , f ′ ∈ L ∞ (0 , T ; H s ) , and f ′ , f ′ have C dependence on ( U, θ ) verifying: (cid:13)(cid:13) ( f ′ , f ′ ) (cid:13)(cid:13) L ∞ (0 ,T ; H s ) ≤ C ( k ( η, ψ, V, B ) k L ∞ (0 ,T ; H s + 12 × H s + 12 × H s × H s ) ) Proof.
Again the lemma simply follows from the fact that I − T and ω ( D ) areregularizing operators. (cid:3) .2.3. Decomposing the change of variable:
SetΦ : I s,T × H s → C ([0 , T ]; H s ) Φ : I s, × H s → H s ( η, ψ ) ( ˜ U , ˜ θ, aux , aux ) ( η, ψ ) ( ˜ U , ˜ θ, aux , aux )The goal is to prove that Φ is locally invertible and then the proof will follow fromTheorem 3.1.We write Φ = Φ ◦ Φ withΦ : I s,T → C ([0 , T ]; H s × H s − × H s × H s )( η, ψ ) ( ˜ U , ˜ ζ, aux , aux )and, Φ : C ([0 , T ]; H s × H s − × H s × H s ) → C ([0 , T ]; H s )( ˜ U , ˜ ζ, aux , aux ) ( ˜ U , ˜ θ, aux , aux )We define Φ and Φ analogously when Φ is defined on I s, . Lemma 5.2.
There exists r, r , ǫ > such that: Φ : B (0 , r ) ∩ I s,T → C ([0 , T ]; H s × H s − × H s × H s ) is a C ∞ diffeomorphism upon it’s image. Φ : B (0 , r ) ∩ C ([0 , T ]; H s × H s − × H s × H s ) → C ([0 , T ]; H s ) is a C ∞ diffeomorphism upon it’s image.Analogous result hold when Φ is defined on I s, . The proof of Theorem 1.3 follows as in the previous section from Corollary 3.1and the previous Lemma combined with the fact that Φ (0) = 0 thus we haveB(0 , r ) ∩ C ([0 , T ]; H s × H s − × H s × H s ) ⊂ Φ (cid:18) B(0 , r ) ∩ C ([0 , T ]; H s + × H s + ) (cid:19) . Also Φ (0) = 0 thus there exists r :B(0 , r ) ∩ C ([0 , T ]; H s ) ⊂ Φ (cid:18) B(0 , r ) ∩ C ([0 , T ]; H s × H s − × H s × H s (cid:19) . We now turn to the proof of the lemma.
Proof.
As all of the estimates used are punctual in time thus the proof is the samefor I s,T and I s, and we only write the one for I s,T . We start by Φ , first the part η (˜ ζ, aux ) is invertible with inverse F [Φ − (˜ ζ, aux )]( ξ ) = 1 d X j (1 − ω ( ξ )) F [ ∂ i ˜ ζ ]( ξ ) iξ j + ω ( ξ ) F [ aux ] . By the same argument ψ ((1 − ω ( D )) ∇ ψ, ω ( D ) ψ ) is invertible and we see that( ˜ U , aux ) is a perturbation of that map indeed: (cid:13)(cid:13)(cid:13) ((1 − ω ( D )) ∇ ψ, ω ( D ) ψ ) − ( ˜ U , aux ) (cid:13)(cid:13)(cid:13) L ( H s + 12 ,H s ) ≤ C ( k B k W , ∞ ) k η k H s + 12 ≤ C ( k ( η, ψ ) k H s + 12 ) k η k H s + 12 thus for r small enough we get the desired result.Now we turn to Φ . This operator is the identity on ˜ U , aux , aux thus we onlyhave to work on ˜ θ . Put a as the Taylor coefficient associated to the solution of theproblem (0,0). Now notice that for ǫ > T √ a | ξ | − + ǫ ( I − T I ) : C ([0 , T ]; H s − ) → C ([0 , T ]; H s ) s a C ∞ diffeomorphism. And now we see that T q + ǫ ( I − T ) is a perturbation of T √ a | ξ | − + ǫ ( I − T ) indeed by symbolic calculus rules: (cid:13)(cid:13)(cid:13)(cid:13) T q − T √ a | ξ | − (cid:13)(cid:13)(cid:13)(cid:13) L ( H s − ,H s ) ≤ C ( k η k H s ) k η k H s , which gives the result by taking r small. (cid:3) Appendix A. Pseudodifferential and Paradifferential operators
In this paragraph we review classic notations and results about pseudodifferentialand paradifferential calculus that we need in this paper. We follow the presentationsin [14] , [23], and [17] which give an accessible presentation.A.1.
Notations and functional analysis.
We present the definitions of the func-tional spaces that will be used.We will use the usual definitions and standard notations for the regular functions C k , C k for those with compact support, the distribution space D ′ , E ′ for those withcompact support, D ′ k , E ′ k for distributions of order k, Lebesgue spaces ( L p ), Sobolevspaces ( H s , W p,q ) and the Schwartz class S and it’s dual S ′ . All of those spacesare equipped with their standard topologies.We also recall the Landau notation the expression O k k ( X ) is used to denote anyquantity bounded in k k by CX , thus Y = O k k ( X ) is equivalent to k Y k ≤ CX .In the following presentation we will use D to denote generically T or R and ˆ D todenote their Pontryagin duals that is Z in the case of T and R in the case of R . Definition A.1 (Littlewood-Paley decomposition) . Pick P ∈ C ∞ ( R d ) so that P ( ξ ) = 1 for | ξ | < and 0 for | ξ | > . We define a dyadic decomposition ofunity by: for k ≥ , P ≤ k ( ξ ) = Φ (2 − k ξ ) , P k ( ξ ) = P ≤ k ( ξ ) − P ≤ k − ( ξ ) . Thus, P ≤ k ( ξ ) = X ≤ j ≤ k P j ( ξ ) and ∞ X j =0 P j ( ξ ) . Introduce the operator acting on S ′ ( R d ) : P ≤ k u = F − ( P ≤ k ( ξ ) u ) and u k = F − ( P k ( ξ ) u ) . Thus, u = X k u k . Finally put { k ≥ , C k = supp P k } the set of rings associated to this decomposition. Remark A.1.
An interesting property of the Littlewood-Paley decomposition is thateven if the decomposed function is merely a distribution the terms of the decomposi-tion are regular, indeed they all have compact spectrum and thus are entire functions.On classical functions spaces this regularization effect can be ”measured” by the fol-lowing inequalities due to Bernstein.
Proposition A.1 (Bernstein’s inequalities) . Suppose that a ∈ L p ( R d ) has its spec-trum contained in the ball {| ξ | ≤ λ } . Then a ∈ C ∞ and for all α ∈ N d and q ≥ p ,there is C α,p,q (independent if λ ) such that k ∂ αx a k L q ≤ C α,p,q λ | α | + dp − dq k a k L p . In particular, k ∂ αx a k L q ≤ C α λ | α | k a k L p , and for p = 2 , p = ∞ a k L ∞ ≤ Cλ d k a k L . Proposition A.2.
For all µ > , there is a constant C such that for all λ > and for all α ∈ W µ, ∞ with spectrum contained in {| ξ | ≥ λ } . one has the followingestimate: k a k L ∞ ≤ Cλ − µ k a k W µ, ∞ . Definition-Proposition A.1 (Sobolev spaces on R d ) . It is also a classical resultthat for s ∈ R : H s ( R d ) = ( u ∈ S ′ ( R d ) , | u | s = (cid:18) X q qs k u q k L (cid:19) < ∞ ) with the right hand side equipped with its canonical topology giving it a Hilbert spacestructure and | | s is equivalent to the usual norm on k k H s . Proposition A.3.
Let B be a ball with center 0. There exists a constant C suchthat for all s > and for all ( u q ) ∈ N ∈ S ′ ( R d ) verifying: ∀ q, supp ˆ u q ⊂ q B and (2 qs k u q k L ) q ∈ N is in L ( N ) then , u = X q u q ∈ H s ( R d ) and | u | s ≤ C − − s (cid:18) X q qs k u q k L (cid:19) . Remark A.2.
The previous definition and properties of the Littlewood-Paley de-composition and Sobolev spaces carries out naturally to T d . Here we rec all the usual Kato-Ponce [15] commutator estimates:
Proposition A.4.
Consider s > and f, g ∈ H s then k [ h D i s , f ] g k L ≤ C ( k f k W , ∞ k g k H s − + k f k H s k g k L ∞ ) . A.2.
Pseudodifferential operators.
We introduce here the basic definitions andsymbolic calculus results. We first introduce the classes of regular symbols.
Definition A.2.
Given m ∈ R , ≤ ρ ≤ and ≤ σ ≤ we denote the symbol class S mρ,σ ( D d × ˆ D d ) the set of all a ∈ C ∞ ( D d × ˆ D d ) such that for all multi-orders α, β wehave the estimate: (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12)(cid:12) ≤ C α,β (1 + | ξ | ) m − ρβ + σα .S mρ,σ ( D d × ˆ D d ) is a Fr´echet space with the topology defined by the family of semi-norms: M mβ,α ( a ) = sup D d × ˆ D d (cid:12)(cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ )(1 + | ξ | ) ρβ − m − σα (cid:12)(cid:12)(cid:12) . Set S m ( D d × ˆ D d ) = S m , ( D d × ˆ D d ) ,S −∞ ( D d × ˆ D d ) = \ m ∈ R S m ( D d × ˆ D d ) and S + ∞ ( D d × ˆ D d ) = [ m ∈ R S m ( D d × ˆ D d ) equipped with their canonically induced topology. For u ∈ S ( D d ) we haveOp( a ) u ( x ) = (2 π ) − d Z ˆ D d e ix.ξ a ( x, ξ )ˆ u ( ξ ) dξ = (2 π ) − d Z ˆ D d e ix.ξ a ( x, ξ ) Z D d e − iy.ξ u ( y ) dydξ = Z D d (cid:18) (2 π ) − n Z ˆ D d e i ( x − y ) .ξ a ( x, ξ ) dξ (cid:19) u ( y ) dy hus giving us the following Proposition. Proposition A.5.
For a ∈ S m ( D d × ˆ D d ) , Op( a ) has a kernel K defined by K ( x, y ) = (2 π ) − d Z ˆ D d e i ( x − y ) .ξ a ( x, ξ ) dξ = (2 π ) − n F ξ a ( x, y − x ) . (A.1) Which can be inverted to give: a ( x, ξ ) = F y → ξ K ( x, x − y ) = Z D d e − iy.ξ K ( x, x − y ) dy = ( − d e − ix.ξ Z D d e iy.ξ K ( x, y ) dy (A.2) Definition A.3.
Let m ∈ R , an operator T is said to be of order m if, and only if,for all µ ∈ R , it is bounded from H µ ( R d ) to H µ − m ( R d ) . Theorem A.1. If a ∈ S m ( D d × ˆ D d ) , then a ( x, D ) is an operator of order m. More-over we have the norm estimate: k a ( x, D ) k H µ → H µ − m ≤ CM mµ,m + d/ ( a )We will now present the main results in symbolic calculus associated to pseudo-differential operators. Theorem A.2.
Let m, m ′ ∈ R , a ∈ S m ( D d × ˆ D d ) and b ∈ S m ′ ( D d × ˆ D d ) . • Composition: Then
Op( a ) ◦ Op( b ) is a pseudodifferential operator of order m + m ′ with symbol a b defined by: a b ( x, ξ ) = (2 π ) − d Z D d × ˆ D d e i ( x − y ) . ( ξ − η ) a ( x, η ) b ( y, ξ ) dydη Moreover,
Op( a ) ◦ Op( b )( x, ξ ) − Op( X | α | Op( a ) , Op( a ) ⊤ is a pseudodifferential op-erator of order m with symbol a ⊤ defined by: a ⊤ ( x, ξ ) = (2 π ) − d Z D d × ˆ D d e − iy.ξ a ( x − y, ξ − η ) dydη Moreover, Op( a ⊤ )( x, ξ ) − Op( X | α | Let ( a j ) ∈ S m j ( D d × ˆ D d ) be a series of symbols with ( m j ) ∈ R decreasing to −∞ . We say that a ∈ S m ( D d × ˆ D d ) is the asymptotic sum of ( a j ) if ∀ k ∈ N , a − k X j =0 a j ∈ S m k +1 ( D d × ˆ D d ) . We denote a ∼ P a j Remark A.3. We can now write simply: a b ∼ X | α | i | α | α ! ( ∂ αξ a ( x, ξ ))( ∂ αx b ( x, ξ )) nd a ⊤ ∼ X | α | i | α | α ! ( ∂ αξ ∂ αx ¯ a ( x, ξ )) . Now we present the classic results of change of variables in pseudodifferentialoperators. Theorem A.3. Let χ : D d → D d be a C ∞ diffeomorphism with Dχ ∈ C ∞ b and A = a ( x, D ) ∈ S mloc (Ω ′ × R d ) a properly supported pseudodifferential operator withkernel K.Then the operator A ∗ defined by K ∗ i.e: ∀ u ∈ C ∞ (Ω) , A ∗ u = Z D d K ( χ ( x ) , χ ( y )) u ( y ) | detDχ ( y ) | dy is a properly supported pseudodifferential operator with symbol a ∗ ( x, ξ ) = ( − d e − ix.ξ Z D d × ˆ D d a ( χ ( x ) , η ) e i ( χ ( x ) − χ ( y ) .η + iy.ξ | detDχ ( y ) | dydη ∈ S m ( D d × ˆ D d ) , and verifies (Op( a ) u ) ◦ χ = Op( a ∗ )( u ◦ χ ) . An expansion of a ∗ is given by: a ∗ ( x, ξ ) ∼ X α α ∂ α a ( χ ( x ) , Dχ − ( χ ( x )) ⊤ ξ ) P α ( χ ( x ) , ξ ) , (A.3) where, P α ( x ′ , ξ ) = D αy ′ ( e i ( χ − ( y ′ ) − χ − ( x ′ ) − Dχ − ( x ′ )( y ′ − x ′ )) .ξ ) | y ′ = x ′ and P α is polynomial in ξ of degree ≤ | α | , with P = 1 , P = 0 . A.3. Paradifferential operators. We start by the definition of symbols with lim-ited spatial regularity. Let W ⊂ S ′ ( D d ) be a Banach space. Definition A.5. Given m ∈ R , Γ m W ( D d ) denotes the space of locally bounded func-tions a ( x, ξ ) on D d × ( ˆ D d \ , which are C ∞ with respect to ξ for ξ = 0 and suchthat, for all α ∈ N d and for all ξ = 0 , the function x ∂ αξ a ( x, ξ ) belongs to W andthere exists a constant C α such that, ∀ | ξ | > , (cid:13)(cid:13) ∂ αξ a ( ., ξ ) (cid:13)(cid:13) W ≤ C α (1 + | ξ | ) m −| α | Given a symbol a , define the paradifferential operator T a by d T a u ( ξ ) = (2 π ) − d Z ˆ D d θ ( ξ − η, η )ˆ a ( ξ − η, η ) ψ ( η )ˆ u ( η ) dη, where ˆ a ( η, ξ ) = R e − ix.η a ( x, ξ ) dx is the Fourier transform of a with respect to thefirst variable; θ and ψ are two fixed C ∞ functions such that: ψ ( η ) = 0 for | η | ≤ , ψ ( η ) = 1 for | η | ≥ , and θ ( ξ, η ) is homogeneous of degree 0 and satisfies for 0 < ǫ < ǫ small enough, θ ( ξ, η ) = 1 if | ξ | ≤ ǫ | η | , θ ( ξ, η ) = 0 if | ξ | ≥ ǫ | η | . For quantitative estimates we introduce as in [17]: Definition A.6. For m ∈ R , ρ ≥ and a ∈ Γ m W ( D d ) , we set M m W ( a ) = sup | α |≤ d +1+ c sup | ξ |≥ (cid:13)(cid:13)(cid:13) (1 + | ξ | ) m −| α | ∂ αξ a ( ., ξ ) (cid:13)(cid:13)(cid:13) W , with c > . We will essentially work with W = W ρ, ∞ and write M mW ρ, ∞ ( a ) = M mρ ( a ) with c = ρ . he main features of symbolic calculus for paradifferential operators are given bythe following Theorems. Theorem A.4. Let m ∈ R . if a ∈ Γ m ( D d ) , then T a is of order m. Moreover, forall µ ∈ R there exists a constant K such that k T a k H µ → H µ − m ≤ KM m ( a ) . Theorem A.5. Let m, m ′ ∈ R , and ρ > , a ∈ Γ mρ ( D d ) and b ∈ Γ m ′ ρ ( D d ) . • Composition: Then T a T b is a paradifferential operator of order m + m ′ and T a T b − T a b is of order m + m ′ − ρ where a b is defined by: a b = X | α | <ρ i | α | α ! ∂ αξ a∂ αx b Moreover, for all µ ∈ R there exists a constant K such that k T a T b − T a b k H µ → H µ − m − m ′ + ρ ≤ KM mρ ( a ) M m ′ ρ ( b ) . • Adjoint: The adjoint operator of T a , T ⊤ a is a paradifferential operator oforder m with symbol a ⊤ defined by: a ⊤ = X | α | <ρ i | α | α ! ∂ αξ ∂ αx ¯ a Moreover, for all µ ∈ R there exists a constant K such that (cid:13)(cid:13)(cid:13) T ⊤ a − T a ⊤ (cid:13)(cid:13)(cid:13) H µ → H µ − m + ρ ≤ KM mρ ( a ) . If a = a ( x ) is a function of x only, the paradifferential operator T a is called aparaproduct. It follows from Theorem A.5 and the Sobolev embeddings that: • If a ∈ H α ( D d ) and b ∈ H β ( D d ) with α, β > d , then T a T b − T ab is of order − (cid:18) min { α, β } − d (cid:19) . • If a ∈ H α ( D d ) with α > d , then T ⊤ a − T a ⊤ is of order − (cid:18) α − d (cid:19) . • If a ∈ W r, ∞ ( D d ), r ∈ N then: k au − T a u k H r ≤ C k a k W r, ∞ k u k L . An important feature of paraproducts is that they are well defined for function a = a ( x ) which are not L ∞ but merely in some Sobolev spaces H r with r < d . Proposition A.6. Let m > . If a ∈ H d − m ( D d ) and u ∈ H µ ( D d ) then T a u ∈ H µ − m ( D d ) . Moreover, k T a u k H µ − m ≤ K k a k H d − m k u k H µ A main feature of paraproducts is the existence of paralinearisation Theoremswhich allow us to replace nonlinear expressions by paradifferential expressions, atthe price of error terms which are smoother than the main terms. Theorem A.6. Let α, β ∈ R be such that α, β > d , then • Bony’s Linearization Theorem For all C ∞ function F, if a ∈ H α ( D d ) then F ( a ) − F (0) − T DF ( a ) a ∈ H α − d ( D d ) . In our recent work [20] we give a generalization to this Theorem. If a ∈ H α ( D d ) and b ∈ H β ( D d ) , then ab − T a b − T b a ∈ H α + β − d ( D d ) . More-over there exists a positive constant K independent of a and b such that: k ab − T a b − T b a k H α + β − d ≤ K k a k H α k b k H β . A.4. Paracomposition. We recall the main properties of the paracomposition op-erator first introduced by S. Alinhac in [8] to treat low regularity change of variables.Here we present the results we reviewed and generalized in some cases in [20]. Theorem A.7. Let χ : D d → D d be a W r, ∞ loc diffeomorphism with Dχ ∈ W r, ∞ , r > , r / ∈ N and take s ∈ R then the following maps are continuous: H s ( D d ) → H s ( D d ) u χ ∗ u = X k ≥ X l ≥ k − N ≤ l ≤ k + N P l ( D ) u k ◦ χ, where N ∈ N is chosen such that N > sup k, D d | Φ k Dχ | − and N > sup k, D d | Φ k Dχ | .Taking ˜ χ : D d → D d a C r diffeomorphism with Dχ ∈ W ˜ r, ∞ map with ˜ r > ,then the previous operation has the natural fonctorial property: ∀ u ∈ H s ( D d ) , χ ∗ ˜ χ ∗ u = ( χ ◦ ˜ χ ) ∗ u + Ru, with, R : H s ( R d ) → H s + min ( r, ˜ r ) ( R d ) continous . We now give the key paralinearization theorem taking into account the paracom-position operator. Theorem A.8. Let u be a W , ∞ ( D d ) map and χ : D d → D d be a W r, ∞ loc diffeo-morphism with Dχ ∈ W r, ∞ , r > , r / ∈ N . Then: u ◦ χ ( x ) = χ ∗ u ( x ) + T Du ◦ χ χ ( x ) + R ( x ) + R ( x ) + R ( x ) where the paracomposition given in the previous Theorem verifies the estimates: ∀ s ∈ R , k χ ∗ u ( x ) k H s ≤ C ( k Dχ k ∞ ) k u ( x ) k H s ,u ′ ◦ χ ∈ Γ W , ∞ ( D d ) ( D d ) for u Lipchitz,and the remainders verify the estimates: k R k H r + min (1+ ρ,s − d ≤ C k Dχ k r k u k H s k R k H r + s ≤ C ( k Dχ k ∞ ) k Dχ k r k u k H s . k R k H r + s ≤ C ( k Dχ k ∞ , (cid:13)(cid:13) Dχ − (cid:13)(cid:13) ∞ ) k Dχ k r k u k H s . Finally the commutation between a paradifferential operator a ∈ Γ mβ ( D d ) and aparacomposition operator χ ∗ is given by the following χ ∗ T a u = T a ∗ χ ∗ u + T q ∗ χ ∗ u with q ∈ Γ m − β ( D d ) , where a ∗ has the local expansion: a ∗ ( x, ξ ) ∼ X α | α |≤⌊ min ( r,ρ ) ⌋ α ∂ α a ( χ ( x ) , Dχ − ( χ ( x )) ⊤ ξ ) P α ( χ ( x ) , ξ ) ∈ Γ m min( r,β ) ( D d ) , (A.4) where, P α ( x ′ , ξ ) = D αy ′ ( e i ( χ − ( y ′ ) − χ − ( x ′ ) − Dχ − ( x ′ )( y ′ − x ′ )) .ξ ) | y ′ = x ′ and P α is polynomial in ξ of degree ≤ | α | , with P = 1 , P = 0 . emark A.4. The simplest example for the paracomposition operator is when χ ( x ) = Ax is a linear operator and in that case we see that if N is chosen sufficiently largein the definition: u ( Ax ) = ( Ax ) ∗ u, and T u ′ ( Ax ) Ax = 0 . Appendix B. Energy estimates and well-posedness of some pulledback hyperbolic equations Theorem B.1. Let T > , χ ∈ W , ∞ ([0 , T ] , W , ∞ loc ( D d )) with D x χ ∈ L , ∞ ([0 , T ] , L ∞ ( D d )) and consider ( a t ) t ∈ R a family of symbols in Γ β ( D d ) with β ∈ R , such that t a t iscontinuous and bounded from R to Γ β ( D d ) and such that Re ( a t ) = a t + a ⊤ t is boundedin Γ ( D d ) . Suppose moreover that χ ( t, · ) is a diffeomorphism between open sets of D d and that we have the bounds: ∃ C > , ∀ t ≤ T, ∀ x, C − ≤ | D x χ ( t, x ) | ≤ C. (B.1) Put ( · ) ∗ is the change of variables by χ as presented in Theorem A.8.Then for all initial data u ∈ H s ( D d ) and f ∈ C ([0 , T ]; H s ( D d )) the Cauchy prob-lem: ( ∂ t u + T a ∗ u = f ∀ x ∈ D d , u (0 , x ) = u ( x ) (B.2) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimates: k u ( t ) k H s ≤ e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ,M ( Re ( a ))) t k u k H s (B.3)+ 2 Z t e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ,M ( Re ( a )))( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ . Again fixing the initial data at 0 is an arbitrary choice. More precisely, ∀ ≤ t ≤ T and all data u ∈ H s ( D d ) the Cauchy problem: ( ∂ t u + T a ∗ u = f ∀ x ∈ D d , u ( t , x ) = u ( x ) (B.4) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimate: k u ( t ) k H s ≤ e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ,M ( Re ( a ))) | t − t | k u k H s + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ,M ( Re ( a )))( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ (cid:12)(cid:12)(cid:12)(cid:12) . Proof. The existence of a solution follows from standard compacity arguments afterregularization given the priory estimates (B.3). Also, the equation being linear thoseestimates give the unicity immediately. Thus we will only show the desired prioryestimates.Put Γ s = h D i s , we will compute ddt (Γ s ∗ u, Γ s ∗ u ) L ( D d , | D x χ ( t,x ) | dx ) in two different ways. • Method 1. First notice that by Theorem A.8Γ s ∗ ( x, ξ ) ∼ ([ Dχ − ( t, χ ( t, x ))] t ξ ) s + R Where R is of order s − | Dχ ( t, x ) | combined with upper ound on ddt | Dχ ( t, x ) | we have: C ( (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) ddt [(Γ s u, Γ s u ) L ] − C ( k D x χ k L ∞ L ∞ ) k Γ s u k L ≤ ddt (Γ s ∗ u, Γ s ∗ u ) L ( | D x χ ( t,x ) | dx ) . • Method 2. Now we use the PDE, ddt (Γ s ∗ u, Γ s ∗ u ) L ( | D x χ ( t,x ) | dx ) = 2 Re (( ∂ t Γ s ∗ u, Γ s ∗ u )) L ( | Dχ ( t,x ) | dx ) ) + (Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) = − Re ((Γ s ∗ T a ∗ u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ) + 2 Re ((Γ s ∗ f, Γ s ∗ u )) L ( | Dχ ( t,x ) | dx ) + 2 Re (([ ∂ t Γ s ∗ ] u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ) + (Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) by change of variables, ddt (Γ s ∗ u, Γ s ∗ u ) L ( | D x χ ( t,x ) | dx ) = − Re ((Γ s ∗ T a ∗ u ◦ χ − , Γ s ∗ u ◦ χ − ) L ) + 2 Re ((Γ s ∗ f, Γ s ∗ u )) L ( | Dχ ( t,x ) | dx ) + 2 Re (([ ∂ t Γ s ∗ ] u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ) + (Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) . Now notice that,Re (cid:18) Z T D [Γ s ∗ T a ∗ u Γ s ∗ u ] ◦ χ − χ − dx (cid:19) = Z T D [Γ s T Re( a ) u Γ s u ] ◦ χ − χ − dx + R, where R verifies by Theorem A.8: | R | ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L . (B.5)Thus by Theorem A.8: ddt (Γ s ∗ u, Γ s ∗ u ) L ( | D x χ ( t,x ) | dx ) (B.6)= − s T Re ( a ) [( χ − ) ∗ u ] , Γ s [( χ − ) ∗ u ]) L + Re (cid:18) Z T D [Γ s ∗ T a ∗ u Γ s ∗ u ] ◦ χ − χ − dx (cid:19) + R + 2 Re ((Γ s ∗ f, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ) + 2 Re (([ ∂ t Γ s ∗ ] u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) )+ (Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) , = − s T Re ( a ) [( χ − ) ∗ u ] , Γ s [( χ − ) ∗ u ]) L + Z T D [Γ s T Re( a ) u Γ s u ] ◦ χ − χ − dx + R + 2 Re ((Γ s ∗ f, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ) + 2 Re (([ ∂ t Γ s ∗ ] u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) )+ (Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) , Now we have (cid:12)(cid:12)(cid:12)(cid:12)Z T D [Γ s T Re( a ) u Γ s u ] ◦ χ − χ − dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( (cid:13)(cid:13) Dχ − (cid:13)(cid:13) L ∞ ) k Γ s u k L . (B.7)By the upper bound on (cid:12)(cid:12) Dχ − ( t, x ) (cid:12)(cid:12) :(Γ s T Re ( a ) [( χ − ) ∗ u ] , Γ s [( χ − ) ∗ u ]) L ( D d ) ≤ M ( Re ( a )) C ( (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L . (B.8)Now by the upper bound on ddt | Dχ ( t, x ) | and ddt (cid:12)(cid:12) Dχ − ( t, x ) (cid:12)(cid:12) we have:(Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) ≤ C ( k D x χ k L ∞ L ∞ ) k Γ s ∗ u k L ow using the upper bound on | Dχ ( t, x ) | :(Γ s ∗ u, Γ s ∗ u ) L ( ddt | Dχ ( t,x ) | dx ) ≤ C ( k D x χ k L ∞ L ∞ , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L . (B.9)Analogously we get:(Γ s ∗ f, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ≤ C ( k D x χ k L ∞ L ∞ ( D d ) , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L k Γ s f k L , (B.10)([ ∂ t Γ s ∗ ] u, Γ s ∗ u ) L ( | Dχ ( t,x ) | dx ) ≤ C ( k D x χ k L ∞ L ∞ , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L . (B.11)Thus finally we get by combining (B.6), (B.5), (B.7), (B.8), (B.9), (B.10)and(B.11):(Γ s T Re ( a ) [( χ − ) ∗ u ] , Γ s [( χ − ) ∗ u ]) L ( D d ) ≤ C ( k D x χ k L ∞ L ∞ , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L (B.12)To conclude we combine the computations from both methods and get: ddt [(Γ s u, Γ s u ) L ] ≤ C ( k D x χ k L ∞ L ∞ , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ , M ( Re ( a ))) k Γ s u k L + C ( k D x χ k L ∞ L ∞ , (cid:13)(cid:13) D x χ − (cid:13)(cid:13) L ∞ L ∞ ) k Γ s u k L k Γ s f k L . The result then follows from the Gronwall Lemma. (cid:3) We see that the proof depends essentially on symbolic calculus rules and thosestill clearly hold in the case of pseudodifferential operators as presented in AppendixA. Theorem B.2. Let T > , χ ∈ W , ∞ ([0 , T ] , C ∞ ( D d )) such that D x χ ∈ C ∞ b ( D d ) and consider ( a t ) t ∈ R a family of symbols in S β ( D d ) with β ∈ R , such that t a t iscontinuous and bounded from R to S β ( D d ) and such that Re ( a t ) = a t + a ⊤ t is boundedin S ( D d ) . Suppose moreover that χ ( t, · ) is a diffeomorphism between open sets of D d and that we have the bounds: ∃ C > , ∀ t ≤ T, ∀ x, C − ≤ | D x χ ( t, x ) | ≤ C. (B.13) Put ( · ) ∗ is the change of variables by χ as presented in Theorem A.3.Then for all initial data u ∈ H s ( D d ) and f ∈ C ([0 , T ]; H s ( D d )) the Cauchy prob-lem: ( ∂ t u + Op( a ∗ ) u = f ∀ x ∈ D d , u (0 , x ) = u ( x ) (B.14) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimates: k u ( t ) k H s ≤ e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ) t k u k H s (B.15)+ 2 Z t e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ )( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ , where C depends also on a finite symbol semi-norm of Re ( a t ) . Again fixing the initialdata at 0 is an arbitrary choice. More precisely, ∀ ≤ t ≤ T and all data u ∈ H s ( D d ) the Cauchy problem: ( ∂ t u + Op( a ∗ ) u = f ∀ x ∈ D d , u ( t , x ) = u ( x ) (B.16) as a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimate: k u ( t ) k H s ≤ e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ ) | t − t | k u k H s + 2 (cid:12)(cid:12)(cid:12)(cid:12)Z tt e C ( k D x χ k L ∞ L ∞ , k D x χ − k L ∞ L ∞ )( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s dt ′ (cid:12)(cid:12)(cid:12)(cid:12) . We finally show a general regularizing effect due to integration in time. Theorem B.3. Consider ( a t ) t ∈ R a family of symbols in S β ( D d ) with β ∈ R , such that t a t is continuous and bounded from R to S β ( D d ) and such that Re( a t ) = a t + a ⊤ t is bounded in S ( D d ) , and take T > . Then for all initial data u ∈ H s ( D d ) , and f ∈ C ([0 , T ]; H s ( D d )) the Cauchy problem: ( ∂ t u + op ( a ) u = f ∀ x ∈ D d , u (0 , x ) = u ( x ) (B.17) has a unique solution u ∈ C ([0 , T ]; H s ( D d )) ∩ C ([0 , T ]; H s − β ( D d )) which verifiesthe estimates: k u ( t ) k H s ( D d ) ≤ e Ct k u k H s ( D d ) + 2 Z t e C ( t − t ′ ) (cid:13)(cid:13) f ( t ′ ) (cid:13)(cid:13) H s ( D d ) dt ′ , where C depends on a finite symbol semi-norm M (Re( a t )) .Suppose moreover that a is elliptic that is: ∀ ( x, ξ ) ∈ R d , | a ( x, ξ ) | ≥ C h ξ i β . Then ∀ t ∈ [0 , T ] : (cid:13)(cid:13)(cid:13)(cid:13)Z t u ( s, · ) ds (cid:13)(cid:13)(cid:13)(cid:13) H s ≤ C ( k u k H s − + k u k H s − β + k f k L ∞ ([0 ,T ] ,H s − ) + k f k L ∞ ([0 ,T ] ,H s − β ) ) . Proof. 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