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The natural question was then to ask if one can see the effect of the dispersive term by analyzing more precisely the regularity of the flow map. For this we can start by noticing that independently of α the flow map is Lipschitz from bounded sets of Hs(D) to C0([0, T ],Hs−1(D)) and ask: can the space Hs−1(D) be replaced by Hs−µ(D) with µ< 1 depending on α? Again in [21] we proved that the best µ one can hope for is µ = 1 (α 1)+, more precisely we showed that: − − the flow map cannot be Lipschitz from bounded sets of Hs(D) to • + C0([0, T ],Hs−1+(α−1) +ǫ(D)) for ǫ> 0. Looking to the literature to assess the optimality of the result, first in [23] the equation (1.1) is actually shown to be quasi-linear for α [0, 3[ and becomes semi- linear for α = 3, i.e the Korteweg-de Vries equation, when∈ D = R suggesting that our results are sub-optimal. Then when D = T, in [19], for the case α = 2 and the Benjamin-Ono equation, the flow map is shown to be Lipschitz (and even has s analytic regularity) on bounded sets of H0 the (Sobolev spaces of functions with mean value 0). Which suggests that our results could be optimal but with a subtlety in the low frequencies. The aim of the current paper is to prove that the results obtained in [21] are optimal on the torus and for the full periodic Water Waves system with surface tension, i.e the Gravity Capillary equation, while clarifying in all of those cases the effect brought on by the low frequencies. Remark 1.1. In Appendix B, we look to the problem on R and use the same Gauge transform to show that the lack of regularity obtained in [23] for α 2 is essentially due to the lack of control of the L1 norm in Sobolev spaces. ≥ 1.1. On the torus. We show that the flow map associated to (1.1) is Lipschitz + s T 0 s−1+(α−1) T from bounded sets of H0( ) to C ([0, T ],H0 ( )). We begin by recalling a classical result. 1 Theorem 1.1. Consider three real numbers α [0, + [, s ]1+ 2 , + [, r> 0 and s ∈ ∞ ∈ ∞ s u0 H (T; R). Then there exists T > 0 such that for all v0 B(u0, r) H (T; R), there∈ exists a unique v C([0, T ],Hs(T; R)) solving the Cauchy∈ problem:⊂ ∈ α−1 ∂tv + v∂xv + D ∂xv = 0 | | . (1.2) v(0, )= v ( ) ( · 0 · Moreover, for all δ > 0 and all of µ [0,s], there is C : R+ R+ non decreasing such that: ∈ → v(t) µ T C v0 3 v0 µ T . (1.3) k kH ( ) ≤ k kH 2 +δ(T) k kH ( ) Taking two different solutions u, v, and assuming moreover
that u Hs+1(T) then: 0 ∈ (u v)(t) s T Cs( (u, v) s T ; t u s+1 T ) u v s T . (1.4) k − kH ( ) ≤ k kH ( ) k kH ( ) k 0 − 0kH ( ) In [21] we studied the regularity of the flow map and proved the following. 1 Theorem 1.2 ([21]). Consider three real numbers α [0, 2[, s ]2 + 2 , + [, r> 0 and u Hs(T; R). ∈ ∈ ∞ 0 ∈ Then the flow map associated to the Cauchy problem (1.2): • B(u , r) C([0, T ],Hs(T; R)) 0 → v v 0 7→ 2 is continuous but not uniformly continuous. Moreover for all ǫ> 0 the flow map: • + B(u , r) C([0, T ],Hs−1+(α−1) +ǫ(T; R)) 0 → v v 0 7→ is not C1. Here we prove that those results are essentially optimal on the Torus more pre- cisely we prove the following. Theorem 1.3 ([21]). Consider three real numbers α [1, + [, s ]1+ 2−α + 1 , + [, ∈ ∞ ∈ α−1 2 ∞ r> 0 and u Hs(T; R). Then the flow map associated to the Cauchy problem (1.2): 0 ∈ 0 + B(u , r) Hs(T; R) C([0, T ],Hs−(2−α) (T; R)) 0 ∩ 0 → 0 v v 0 7→ is Lipschitz. Several remarks are in order. Remark 1.2. (1) As a corollary of Theorem 1.3 we prove in Section 2.2 the following. 2−α Corollary 1.1. Consider three real numbers α [0, + [, s ]1 + α−1 + 1 s T R ∈ ∞ ∈ 2 , + [, r> 0 and u0 H ( ; ). ∞Then the flow map∈ associated to the Cauchy problem (1.2): • B(u , r) C([0, T ],Hs(T; R)) 0 → v v 0 7→ is continuous but not uniformly continuous. For all ǫ> 0 the flow map: • B(u , r) C([0, T ],Hs−1+ǫ(T; R)) 0 → v v 0 7→ is not C1. 3 (2) The case α = 2 is closely related to the system obtained after reduction and para-linearization of the periodic Water Waves system in dimension 1 obtained in [4] Proposition 3.3 by T. Alazard, N. Burq and C. Zuily, which we will treat in the second part of this paper. (3) The case α = 2 and the Benjamin-Ono equation on the circle was obtained by Molinet in [19]. Though Molinet’s result extends to the Cauchy problem on L2(T) and only studied the flow map regularity for data with 0 mean value. 1.2. The periodic Gravity Capillary equation. We follow here the presentation in [4], [3] and [2].

1.2.1. Assumptions on the domain. We consider a domain with free boundary, of the form: (t,x,y) [0, T ] R R : (x,y) Ωt , { ∈ × × ∈ } where Ωt is the domain located between a free surface:

Σt = (x,y) R R : y = η(t,x) { ∈ × } and a given (general) bottom denoted by Γ = ∂Ωt Σt. More precisely we assume \ that initially (t = 0) we have the hypothesis (Ht) given by: 3 The domain Ωt is the intersection of the half space, denoted by Ω ,t, located • 1 below the free surface Σt,

Ω ,t = (x,y) R R : y < η(t,x) (Ht) 1 { ∈ × } 1+1 and an open set Ω2 R such that Ω2 contains a fixed strip around Σt, which means that there⊂ exists h> 0 such that,

(x,y) R R : η(t,x) h y η(t,x) Ω . (Ht) { ∈ × − ≤ ≤ }⊂ 2 We shall assume that the domain Ω2 (and hence the domain Ωt =Ω1,t Ω2) is connected. ∩

1.2.2. The equations. We consider an incompressible inviscid liquid, having unit density. The equations of motion are given by the Euler system on the velocity field v: ∂tv + v v + P = gey ·∇ ∇ − in Ωt, (1.5) (div v = 0 where gey 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 is then coupled with the boundary conditions: v n =0 onΓ, · 2 ∂tη = 1+ η v ν on Σt, (1.6)  |∇ | · P = κH(η) onΣt, −p where n and ν are the exterior normals to the bottom Γ and the free surface Σt, κ is the surface tension and H(η) is the mean curvature of the free surface: η H(η) = div ∇ . 1+ η 2  |∇ |  We are interested in the case with surfacep tension and take κ = 1. The first con- dition in (1.6) expresses in fact that the particles in contact with the rigid bottom remain in contact with it. As no hypothesis is made on the regularity of Γ, this con- dition is shown to make sense in a weak variational meaning due to the hypothesis (Ht), for more details on this we refer to Section 2 in [4].

The fluid motion is supposed to be irrotational and Ωt is supposed to be simply connected thus the velocity v field derives from some potential φ i.e v = φ and: ∇ ∆φ =0inΩ, (∂nφ =0onΓ. The boundary condition on φ becomes:

∂nφ =0 onΓ, ∂tη = ∂yφ η φ on Σt, (1.7) −∇ ·∇ 1 2 ∂tφ = gη + H(η) x,yφ on Σt. − − 2 |∇ | Following Zakharov [27] and Craig-Sulem [12] we reduce the analysis to a system on the free surface Σt. If ψ is defined by ψ(t,x)= φ(t,x,η(t,x)), then φ is the unique variational solution of

∆φ =0inΩt, φ|y=η = ψ, ∂nφ =0onΓ. 4 Define the Dirichlet-Neumann operator by

2 (G(η)ψ)(t,x) = 1+ η ∂nφ |∇ | |y=η = (∂yφ)(t,x,η(t,x)) η(t,x) ( φ)(t,x,η(t,x)). p −∇ · ∇ For the case with rough bottom we refer to [6], [4] and [3] for the well-posedness of the variational problem and the Dirichlet-Neumann operator. Now (η, ψ) (see for example [12]) solves:

∂tη = G(η)ψ, (1.8)

1 2 1 η ψ + G(η)ψ ∂tψ = gη + H(η)+ ψ + ∇ ·∇ . − 2 |∇ | 2 1+ η 2 |∇ | The system is completed with initial data

η(0, )= ηin, ψ(0, )= ψin. · · We consider the case when η, ψ are 2π-periodic in the space variable x.

1.2.3. Flow map regularity. In [4] and [3], Alazard, Burq, and Zuily perform a par- alinearization and symmetrization of the the water waves system that takes the form:

∂tu + TV . u + iTγ u = f, ∇ 3 where γ is elliptic of order 2 which closely resemble the model problem we presented on T but with an extra non linearity in γ. The paralinearization and symmetrization of the system was used to prove the well-posedness of the Cauchy problem in the 1 optimal threshold s > 2 + 2 in which the velocity field v is Lipschitz. We will complete this and our result in [21] by giving the precise regularity of the flow map. First we recall some previously known results on the Cauchy problem from [3, 4].

1 Theorem 1.4 (From [3, 4]). Consider two real numbers r> 0, s ]2 + 2 , + [ and s+ 1 s ∈ ∞ (η , ψ ) H 2 (T) H (T) such that, 0 0 ∈ × ′ ′ s+ 1 s (η , ψ ) B((η , ψ ), r) H 2 (T) H (T) ∀ 0 0 ∈ 0 0 ⊂ × the assumption (Ht)t=0 is satisfied. Then there exists T > 0 such that the Cauchy problem (1.8) with initial data (η′ , ψ′ ) B((η , ψ ), r) has a unique solution 0 0 ∈ 0 0 ′ ′ 0 s+ 1 s (η , ψ ) C ([0, T ]; H 2 (T) H (T)) ∈ × and such that the assumption (Ht) is satisfied for t [0, T ]. Moreover the flow map (η′ , ψ′ ) (η′, ψ′) is continuous. ∈ 0 0 7→ In [21] we completed this by the following.

1 Theorem 1.5. Consider two real numbers r > 0, s ]2 + 2 , + [ and (η0, ψ0) s+ 1 s ∈ ∞ ∈ H 2 (T) H (T) such that, × ′ ′ s+ 1 s (η , ψ ) B((η , ψ ), r) H 2 (T) H (T) ∀ 0 0 ∈ 0 0 ⊂ × the assumption (Ht)t=0 is satisfied. Then for all R> 0 the flow map associated to the Cauchy problem (1.8):

s+ 1 s B(0,R) C([0, T ],H 2 (T) H (T)) → × (η′ , ψ′ ) (η′, ψ′) 0 0 7→ is not uniformly continuous. 5 1 And at least a loss of 2 derivative is necessary to have Lipschitz control over the flow map, i.e for all ǫ′ > 0 the flow map s+ǫ′ s− 1 +ǫ′ B(0,R) C([0, T ],H (T) H 2 (T)) → × (η′ , ψ′ ) (η′, ψ′) 0 0 7→ is not Lipschitz. Here it is shown that those results are sufficient after suitable re-normalization of the flow map. 1 Theorem 1.6. Consider two real numbers r > 0, s ]3 + 2 , + [ and (η0, ψ0) s+ 1 s ∈ ∞ ∈ H 2 (T) H (T) such that, × ′ ′ s+ 1 s (η , ψ ) B((η , ψ ), r) H 2 (T) H (T) ∀ 0 0 ∈ 0 0 ⊂ × ′ ′ the assumption (Ht)t=0 is satisfied. Define (η, ψ) and (η , ψ ) as the solutions to the Cauchy problem (1.8) on [0, T ], T> 0. Define the following change of variables: x 1 t 1 χ(t,x)= dy 2 + ∂xφ dΣ (1.9) 1 + (∂ η(t,y))2 − 1 + (∂xη(t,y)) Z0 x Z0 ZΣ   x t 2π p 2 1 2 = 1 + (∂xη(t,y)) dy + ∂xφ 1 + (∂xη) dy, − 1 + (∂ η(t,y))2 0 0 0  x  Z p Z Z p and χ′ is defined analogously from (η′, ψ′). Then for r sufficiently small and t [0, T ] we have: ∈ ∗ ′ ′ ∗′ (η, ψ) (η , ψ ) (t, ) 1 − · Hs×Hs− 2 ′ ′ ∗ ′ ′ ∗′ 1 C( (η0, ψ0, η0, ψ0) s+ s ) (η0, ψ0) (η0, ψ0) 1 , (1.10) ≤ H 2 ×H − Hs×Hs− 2

∗ ∗′ ′ where and are the paracomposition by χ and χ , which we recall it’s definition in A.3. Remark 1.3. The time integral in the re-normalization 1.9 is to insure that the mean value of the transport term vanishes. This re-normalization is needed here to 3 compensate the non-linearity in the dispersive term of order 2 here. s 1.3. Strategy of the proof. For Theorem 1.3, we first work on H0 and the main idea is to conjugate (1.1) to a semi-linear dispersive equation of the form: α−1 ∂tw + D ∂xw = Ru, | | where R is continuous from Hs to itself. For the viscous Burgers equation such a result is obtained by the Cole-Hopf transformation that reduces the problem to a one dimensional heat equation. In [24], T.Tao used a complex version of the Cole- Hopf transformation to reduce the problem on the Benjamin-Ono equation to a one dimensional Schr¨odinger type equation, this idea was extensively used to lower the regularity needed for the well-posedness of the Cauchy problem as in Molinet’s work in [19]. A generalized pseudodifferential form of this transformation was used in [1] to reduce the one dimensional water waves system to a a one dimensional semi-linear Schr¨odinger type system.

Formally if we follow the same lines of those previous papers, the transformation we will have to use is a pseudodifferential transformation of the form: w = Op(a)u, 1 ξ|ξ|1−αU (1.11) (a = e iα , 6 where U is a real valued periodic primitive of u that exists because u has mean value 0. The main problem is that such an operator belongs to a H¨ormander symbol class 0 3 0 of the form Sα−1,2−α, which for α = 2 becomes S 1 1 which is a ”bad” symbol class 2 , 2 with no general symbolic calculus rules. Thus we have to treat this transformation with care. The idea here is inspired by the particular form of the formal computation, we express the desired operator as the time one of a flow map associated to a hyper- bolic equation, i.e a = A1 where (Aτ )τ∈R is defined as the group generated by the paradifferential operator iTp where p is a real valued symbol of order smaller than 1. This is inspired by previous results of Alazard, Baldi and P.G´erard [7]. Take a different operator Tb. The main new idea is to apply a Baker-Campbell- Hausdorff formula. Formally this allows one to express Aτ TbA−τ as a series of successive Lie derivatives [iTp, , [iTp, Tb]. The same kind of computations go for · · · [Aτ , Tb]. The convergence of such a series is a non trivial problem, equivalent to m solving a linear ODE in the Fr´echet space of paradifferential symbol classes Γ+∞ defined in Appendix A.2. Such an ODE is not generally well posed and to solve such a problem one usually has to look at a Nash-Moser type scheme. Though in our case we have an explicit ODE that can be solved locally with loss of derivative thus inspired by H¨ormander’s [15] and Beals in [9], we prove the existence of a symbol bτ such τ that Aτ TbA−τ = Tbτ , moreover b is shown to have the asymptotic expansion given by the Baker-Campbell-Hausdorff formula. The use of paradifferential operators is key here, as in H¨ormander’s [15], because the continuity of paradifferential operators given by Theorem A.2 insures that we do not need to control an infinite number of semi-norms as would have been the case for pseudodifferential operators. Finally the transformation defined in this way helps us reduce the transport term of order 1 to a term of order 2 α which is enough for our problem. s s − Passing from H0 to H we use the following gauge transform: 1 u˜(t,x)= u(t,x t u0) u0, where u0, − − −− T T Z Z | | Z which we prove is continuous on Hs but not uniformly continuous and C1 only from Hs to Hs−1. For the Gravity-Capillary equation the problem is more delicate. Indeed the model problems we study are for the paralinearized and symmetrized system, though the change of variable from the original system to the paralinearized and symmetrized s 1 one is known to be Lipschitz on H for s> 2+ 2 . Thus the problem is reduced to the study of the flow map regularity of an equation of the form

∂tu + TV . u + iTγ u = f. ∇ In the same spirit as [1, 2] we preform a para-change of variable, i.e we para-compose with χ defined by (1.9), to get:

∗ ∗ ∗ ∂t[u ]+ TW . u + iT 3 u = f, with W = 0. ∇ |ξ| 2 T Z We then proceed exactly as for Equation (1.1) (with the 0 mean value hypothesis insured by the choice of χ). Remark 1.4. Transformation (1.11), in which we use a primitive of the solution is called• a gauge transform in the literature. As for the Cole-Hopf transformation, this gauge transform (1.11) is essen- • tially one dimensional. It is interesting to know that the same type of transformation can be iterated • and get at the step of order k a remainder of order k + 1 kα which is − 1 acceptable for k sufficiently large as α> 1 but the price to pay is s> 1+ α−1 . 7 In [22] we use this iteration to prove that for possible for 2 <α< 3, the paradifferential version of (1.1) can be transformed to a semi-linear equation with a regularizing remainder, i.e: α−1 α−1 ∂tu + Tu∂xu + ∂x D u = 0 ∂tAu + ∂x D Au = R (u), | | ⇒ | | ∞ where the operator norm of R∞(u) is controlled by u ∞ 2−α,∞ . k kL ([0,T ],C∗ ) 1.4. Acknowledgement. I would like to express my sincere gratitude to my thesis advisor Thomas Alazard.

2. Study of the model problems s 2.1. Proof of Theorem 1.3, the estimates on H0. We keep the notations of Theorem 1.3, fixing u Hs(T; R) and r> 0 and taking: 0 ∈ 0 v , w B(u , r) Hs(T; R). 0 0 ∈ 0 ⊂ 0 As the mean value is conserved by the flow of (1.2) we consider the solutions 0 s T R u, v, w C ([0, T ]; H0 ( ; )) to (1.2) with initial data u0, v0, w0 and on a uniform small interval∈ [0, T ]. The main goal of the proof is to show the following estimate:

v(t, ) w(t, ) + C( (v , w ) s ) v w + , (2.1) k · − · kHs−(2−α) ≤ k 0 0 kH k 0 − 0kHs−(2−α) with the following tame control, ′ C( (v , w ) s ) C ( (v , w ) + )[ (v , w ) s + 1], (2.2) k 0 0 kH ≤ k 0 0 kHs−(2−α) k 0 0 kH where C and C′ are non decreasing positive functions. The final simplification we make in this paragraph is that given the well-posedness of the Cauchy problem in Hs, and the density of H+∞ in Hs, it suffice to prove (2.1) for v , w H+∞, which henceforth we will suppose. 0 0 ∈ We start by applying the paralinearization Theorem A.3 to the term u∂xv to get:

∂tv + Tviξv + T α−1 v = R (v)v, i|ξ| ξ 1 (2.3) v(0, )= v ( ), ( · 0 · 1 where R1 verifies as s> 1+ 2 :

R (v) + + C( v 1,∞ ) C( v s ), k 1 kHs−(2−α) →Hs−(2−α) ≤ k kW ≤ k kH [R (v) R (w)]v + + C v w + v s , k 1 − 1 kHs−(2−α) →Hs−(2−α) ≤ k − kHs−(2−α) k kH + where C verifies 2.2. Now we reduce Hs−(2−α) estimates to L2 ones by defining s−(2−α)+ s−(2−α)+ f1 = D v. Commuting D with (2.3), using the symbolic calculus rulesh ofi Theorem A.3, we get that:h i

∂tf1 + Tviξf1 + Ti|ξ|α−1ξf1 = R1(v)f1 s−(2−α)+ (2.4) (f1(0, )= D v0( ), · h i · where R1 was modified to include terms verifying the same estimate i.e:

R (v) 2 2 C( v s ), k 1 kL →L ≤ k kH [R (v) R (w)]f 2 C v w + f + . k 1 − 1 1kL ≤ k − kHs−(2−α) k 1kH(2−α) + We define analogously g = D s−(2−α) w and notice that by definition: 1 h i f g 2 = v w + , k 1 − 1kL k − kHs−(2−α) 2 thus the problem is reduced to getting L estimates on f1 g1. Here we explain the scheme of the proof, using some− estimates which will be proved in Section 4. 8 2.1.1. Gauge transform and Energy estimate. The goal of this section is to find an operator Av such that

∂t[Avf1]+ AvTi|ξ|α−1ξf1 + AvTviξf1 + [Av, Ti|ξ|α−1ξ]f1 = (∂tAv)f1 + AvR1(f1)f1, + and AvTviξ + [Av, Ti|ξ|α−1ξ] is a hyperbolic operator of order (2 α) < 1. −1 − If we define V = ∂x v which is the periodic zero mean value primitive of v, then vˆ(ξ) Vˆ (0) = 0 and Vˆ (ξ)= , for ξ Z∗, iξ ∈ and we define analogously W from w. Then a formal computation shows that one 0 can choose Av = T 1 1−α S (T Z) which is a symbol class with no e iα ξ|ξ| V α−1,2−α ∈ × 1 general symbolic calculus rules. Here we will define Av differently . Av is defined as the time one of the flow map generated by:

1 1−α 2−α pv = ξ ξ V Γ (T), α | | ∈ 2 which is well defined by Proposition 4.1. We define analogously Aw and pw from w. Now introduce:

f2 = Avf1, g2 = Awg1. (2.5) The study of the symbolic calculus associated to this very specific form of symbols is given by Proposition 4.1 and the change of variable (2.5) is Lipschitz from L2 to 2 (2−α)+ L but under H control on (f2, g2). Indeed we write:

f g 2 = Avf Awg 2 k 2 − 2kL k 1 − 1kL Av[f g ] 2 + (Av Aw)g 2 . ≤ k 1 − 1 kL k − 2kL Applying estimate (1) of Proposition 4.1 and estimate (4.4):

f g 2 C( v s ) f g 2 + V W ∞ g + k 2 − 2kL ≤ k kH k 1 − 1kL k − kL k 2kH(2−α) C( v s ) f g 2 + v w + g + ≤ k kH k 1 − 1kL k − kHs−(2−α) k 2kH(2−α) C( (v, w) s ) f g 2 , ≤ k kH k 1 − 1kL where C verifies the estimate 2.2. As A−1 and A−1 are the time 1 generated by v w − the flow map pv,pw respectively which is well defined by Proposition 4.1. We get by symmetry:

f g 2 C( (v, w) s ) f g 2 , k 1 − 1kL ≤ k kH k 2 − 2kL thus, −1 C ( (v, w) s ) f g 2 f g 2 C( (v, w) s ) f g 2 , k kH k 1 − 1kL ≤ k 2 − 2kL ≤ k kH k 1 − 1kL and the problem is reduced to getting L2 estimates on f g . 2 − 2 To get the equations on f2 and g2 we commute Av and Aw with (2.4), we make the computations for f2, those for g2 are obtained by symmetry:

Av∂tf1 + AvTviξf1 + AvTi|ξ|α−1ξf1 = AvR1(v)f1, thus,

∂t Avf + T α−1 Avf + (AvTviξ [T α−1 , Av])f + [Av, ∂t]f = AvR (v)f . 1 i|ξ| ξ 1 − i|ξ| ξ 1 1 1 1 By definition
of pv and Proposition 4.4 we have: 1 α−1 p p ∂ξ(ξ ξ )∂xpv = vξ and [Av, ∂t]= Av A Ti∂ p A dr. | | − −r t v r Z0 1Similar ideas were used in Appendix C of [1] to get estimates on a change of variable operator m which are still in the usual symbol classes S1,0, the difficulty here being that we are no longer in those symbol classes. 9 Thus by Corollary 4.2 we have:

−1 ∂tf2 + Ti|ξ|α−1ξf2 = R2(v)f2 + AvR1(v)Av f2, (2.6)

−1 where R2 and AvR1(v)Av verify:

Re(R (v)) 2 2 C( v s ), k 2 kL →L ≤ k kH [R (v) R (w)]g 2 C v w + g + , k 2 − 2 2kL ≤ k − kHs−(2−α) k 2kH(2−α) −1 −1 [AvR (v)A AwR (w)A ]g C v w s−(2−α)+ g (2−α)+ . 1 v − 1 w 2 L2 ≤ k − kH k 2kH We get analogously on g , 2 −1 ∂tg2 + Ti|ξ|α−1ξg2 = R2(w)g2 + AwR1(w)Aw g2. (2.7) Taking the difference between (2.6) and (2.7) yields:

−1 −1 ∂t(f g )+T α−1 (f g ) = [R (w) R (v) AvR (v)A +AwR (w)A ]g . 2 − 2 i|ξ| ξ 2 − 2 2 − 2 − 1 v 1 w 2 Thus the usual energy estimate combined with the Gronwall lemma on f2 g2 gives for 0 t T : − ≤ ≤

f g 2 C( (u , v ) s , (f , g )(0, ) + ) (f g )(0, ) 2 k 2 − 2kL ≤ k 0 0 kH k 2 2 · kH(2−α) k 2 − 2 · kL C( (u , v ) s ) (f g )(0, ) 2 , ≤ k 0 0 kH k 2 − 2 · kL with C verifying (2.2), which concludes the proof.

2.2. Proof of Corollary 1.1, the estimates on Hs. The starting point is noticing that the mean value is preserved by (1.2) and by doing the change of unknowns:

u˜(t,x)= u(t,x t− u ) − u − 0 − 0 , (2.8) v˜(t,x)= v(t,x t− v ) − v ( − R 0 − R 0 − 1 R R where u0 = 2π T u0 is the mean value. We can reduce the Cauchy problem for s general data to ones with 0 mean value by verifying thatu, ˜ v˜ H0 still solve (1.2). Thus theR main goalR is to prove that the change of variable (2.8)∈ is not regular. More precisely we will show that there exists a positive constant C and two sequences λ λ 0 s T (uǫ ) and (vǫ ) solutions of 1.2 in C ([0, 1],H ( )) such that for every t T , where T is a uniform small time, ≤

λ λ sup uǫ + vǫ (t, ) C, λ,ǫ Hs(T)(t,·) · Hs(T) ≤

λ λ (uǫ,τ ) and (vǫ,τ ) satisfy initially:

λ λ lim uǫ (0, ) vǫ (0, ) = 0, λ→+∞ · − · Hs(T) ǫ→0 but, λ λ lim inf uǫ (t, ) vǫ (t, ) c> 0. λ→+∞ · − · Hs(T) ≥ ǫ→0

Which proves the non uniform continuity. Considering a weaker control norm we want to get, for all δ > 0 and for t> 0: uλ(t, ) vλ(t, ) ǫ · − ǫ · Hs−1+δ(T) lim inf λ λ =+ . λ→+∞ uǫ (0, ) vǫ (0, ) Hs T ∞ ǫ→0 k · − · k ( ) 10 2.2.1. Definition of the Ansatz. Take ω C∞(T) such that for x [0, 2π]: ∈ 0 ∈ 1 ω(x)=1if x , ω(x)=0if x 1. | |≤ 2 | |≥ Let (λ, ǫ) be two positive real sequences such that: λ + , ǫ 0, λǫ + . (2.9) → ∞ → → ∞ Put for x [0, 2π], ∈ 0 1 −s 0 0 u (x)= λ 2 ω(λx), v (x)= u (x)+ ǫω(x), and extend u0 and v0 periodically. The main trick here will be to use the time reversibility of equation (1.2) by definingu, ˜ v˜ as the solution of (1.2) with data fixed at time t> 0 given by u˜(t,x)= u − u 0 − 0 , (2.10) v˜(t,x)= v − v ( 0 − R 0 where t t0 is chosen small enough for theR equations to be well-posed. Finally, define u ≤and v by (2.8).

2.2.2. Main estimates. First the estimates at time 0, for 0 ν s: ≤ ≤

u(0,x) v(0,x) Hν = u˜(0,x) v˜(0,x)+ u0 v0 k − k − − −− ν Z Z H

By the estimate (2.1), the tame control (2.2) and the Cauchy-Schwartz inequality,

u(0,x) v(0,x) ν C( (u , v ) + ) u v ν k − kH ≤ k 0 0 kHν+(2−α) k 0 − 0kH + C[1 + λν−s+(2−α) ]ǫ. (2.11) ≤ Now the estimates at a fixed time t> 0, by construction:

u(t,x) v(t,x) Hν = u0(x + t u0) v0(x + t v0) k − k − − − ν Z Z H

= u0(x + t u0) u0(x + t v0) + OHν (ǫ) − − − ν Z Z H

− − − Now by hypothesis λǫ + and t ω > 0, thus u0( + t u0) and u0( + t v0) have disjoint supports, thus→ ∞ · · R R R

u(t,x) v(t,x) Hν = u0(x + t u0) + u0(x + t v0) + OHν (ǫ) k − k − ν − ν Z H Z H ν−s = Cλ + OHν (ǫ ). (2.12)

Now to conclude the proof we differentiate the cases: + in the case of non uniform continuity we take ǫ such that ǫλ(2−α) 0 and • apply the previous estimates with ν = s. → In the case of non Lipschitz control we take ǫ such that λ−1+δǫ−1 + • and apply the previous estimates with ν = s 1+ δ. → ∞ −

3. Flow map regularity for the periodic Gravity Capillary equation 3.1. Prerequisites from the Cauchy problem. We start by recalling the apriori estimates given by Proposition 5.2 of [4] combined with the results of [3]. We keep the notations of Theorem 1.6. 11 1 Proposition 3.1. (From [4] and [3]) Consider a real number s> 2+ 2 . Then there exists a non decreasing function C such that, for all T ]0, 1] and all solution (η, ψ) of (1.8) such that: ∈

0 s+ 1 s (η, ψ) C ([0, T ]; H 2 (T) H (T)) and (Ht) is verified for t [0, T ], ∈ × ∈ we have:

(η, ψ) 1 C((η0, ψ0) 1 )+ TC( (η, ψ) 1 ). k kL∞(0,T ;Hs+ 2 ×Hs) ≤ Hs+ 2 ×Hs k kL∞(0,T ;Hs+ 2 ×Hs) The proof will rely on the para-linearised and symmetrized version of (1.8) given by Proposition 4.8 and corollary 4.9 of [4] which are valid on T as shown in [3]. Before we recall this, for clarity as in [4], we introduce a special class of operators Σm Γm given by: ⊂ 0 Definition 3.1. (From [4]) Given m R, Σm denotes the class of symbols a of the form ∈ a = a(m) + a(m−1), with, (m) a = F (∂xη(t,x),ξ),

(m−1) k a = Gα(∂xη(t,x),ξ)∂x η(t,x), |Xk|=2 such that

(1) Ta maps real valued functions to real-valued functions; (2) F is of class C∞ real valued function of (ζ,ξ) R (Z 0), homogeneous of order m in ξ; and such that there exists a continuous∈ × function\ K = K(ζ) > 0 such that F (ζ,ξ) K(ζ) ξ m , ≥ | | for all (ζ,ξ) R (Z 0); ∞ ∈ × \ (3) Gα is a C complex valued function of (ζ,ξ) R (Z 0), homogeneous of order m 1 in ξ. ∈ × \ − Σm enjoys all the usual symbolic calculus properties modulo acceptable remain- ders that we define by the following:

Definition-Notation 3.1. (From [4]) Let m R and consider two families of operators of order m, ∈

A(t) : t [0, T ] , B(t) : t [0, T ] . { ∈ } { ∈ } 3 We shall say that A B if A B is of order m 2 and satisfies the following estimate: for all µ ∼R, there− exists a continuous− function C such that for all t [0, T ], ∈ ∈ A(t) B(t) 3 C( η(t) 1 ). k − kHµ→Hµ−m+ 2 ≤ k kHs+ 2 In the next Proposition we recall the different symbols that appear in the para- linearization and symmetrization of the equations.

Proposition 3.2. (From [4])We work under the hypothesis of Proposition 3.1. Put

λ = λ(1) + λ(0), l = l(2) + l(1) with, 12 λ(1) = ξ , | | 2 (0) 1+|∂xη| (1) ξ (1) λ = ∂x α ∂xη + i ∂xα ,  2|ξ| |ξ| (3.1)       (1) 1 α = 2 ξ + i∂xηξ . √1+|∂xη| | |     (2) 2 − 3 2 l = (1+ ∂xη ) 2 ξ , (1) i | | (2) (3.2) l = (∂x ∂ξ)l . ( − 2 · 0 1 3 Now let q Σ ,p Σ 2 , γ Σ 2 be defined by ∈ ∈ ∈ 2 − 1 q = (1+ ∂xη ) 2 , | | 2 − 5 1 (− 1 ) p = (1+ ∂xη ) 4 ξ 2 + p 2 , | | | | (2) (0) (2) (1) l Re λ i (2) (1) γ = l λ + (∂ξ ∂x) l λ , sλ(1) 2 − 2 · p p (− 1 ) 1 (1) ( 1 ) ( 1 ) ( 3 ) ( 1 ) p 2 = ql γ 2 p 2 + i∂ξγ 2 ∂xp 2 . ( 3 ) γ 2 − · n o Then ∗ TqTλ TγTq, TqTl TγTp, Tγ (Tγ ) , ∼ ∼ ∼ ∗ where (Tγ ) is the adjoint of Tγ . Now we can write the paralinearization and symmetrization of the equations (1.8) after a change of variable: Corollary 3.1. (From [4])Under the hypothesis of Proposition 3.1, introduce the unknowns2

U = ψ TBη, Φ = Tpη and Φ = TqU, − 1 2 where we recall, ∂xη·∂xψ+G(η)ψ B = (∂ φ) = 2 , y |y=η 1+(∂xη) (V = (∂xφ)|y=η = ∂xψ B∂xη. − Then Φ , Φ C0([0, T ]; Hs(T)) and 1 2 ∈

∂tΦ + TV ∂xΦ Tγ Φ = f , 1 × 1 − 2 1 (3.3) ∂tΦ + TV ∂xΦ + Tγ Φ = f , ( 2 × 2 1 2 with f ,f L∞(0, T ; Hs(T)), and f ,f have C1 dependence on (Φ , Φ ) verifying: 1 2 ∈ 1 2 1 2 (f1,f2) ∞ s T C( (η, ψ) 1 ). k kL (0,T ;H ( )) ≤ k kL∞(0,T ;Hs+ 2 ×Hs(T)) 3.2. Proof of Theorem 1.6. Corollary 3.1 shows that the paralinearization and symmetrization of the equations (1.8) are of the form of the equations treated in Theorem 1.3, so the proof will follow the same lines but with more care in treating the non linearity in the dispersive term. s+ 1 s We keep the notations of Theorem 1.6, fixing (η0, ψ0) H 2 (T) H (T) and s+ 1∈ s × r> 0. We begin by taking (˜η0, ψ˜0) B((η0, ψ0), r) H 2 (T) H (T) and consider 0 ∈ s+ 1 ⊂ s × the solutions (η, ψ), (˜η, ψ˜) C ([0, T ]; H 2 (T) H (T)) to (1.2) with initial data ∈ ×

2U is commonly called the ”good” unknown of Alinhac. Introduced by Alazard-Metivier in [6], following earlier works by Lannes in [16]. 13 (η0, ψ0), (˜η0, ψ˜0), on a uniform small interval [0, T ] where the hypothesis (Ht) is also supposed to be verified. Define the following change of variables: x 1 t 1 χ(t,x)= dy 2 + ∂xφ dΣ 1 + (∂ η(t,y))2 − 1 + (∂xη(t,y)) Z0 x Z0 ZΣ   x t 2π p 2 1 2 = 1 + (∂xη(t,y)) dy + ∂xφ 1 + (∂xη) dy, − 1 + (∂ η(t,y))2 0 0 0  x  Z p Z Z p (3.4) andχ ˜ is defined analogously from (˜η, ψ˜). The main goal of the proof is to show the following estimate: ∗ ˜ ∗˜ (η, ψ) (t, ) (˜η, ψ) (t, ) 1 · − · Hs×Hs− 2

˜ ∗ ˜ ∗˜ C (η0, ψ0, η˜0, ψ0) 1 (η0, ψ0) (˜η0, ψ0) 1 , (3.5) ≤ Hs+ 2 ×Hs − Hs×Hs− 2   ∗ ∗˜ where and are the paracomposition operators defined by χ andχ ˜ respectively. We recall that the definition of the paracomposition operator is given in Section A.3. Put Φ = (Φ1, Φ2) the unknowns obtained from (η, ψ) after paralinearization and symmetrization of the equations as in Corollary 3.1. Define analogously Φ˜ = (φ˜1, φ˜2) from (˜η, ψ˜). Let us notice that, in order to prove 3.5, it suffice to get estimates on Φ Φ.˜ Indeed by the ellipticity of the symbols p and q combined with the immediate 2− 1 L estimates( as s> 2+ 2 ) we have: ∗ ˜ ∗˜ (η, ψ) (t, ) (˜η, ψ) (t, ) s 1 · − · Hs×H − 2

˜ ∗ ˜ ∗˜ C (η0, ψ0, η˜0, ψ0) 1 Φ (t, ) Φ (t, ) 1 1 , (3.6) ≤ Hs+ 2 ×Hs · − · Hs− 2 ×Hs− 2   ∗ ˜ ∗˜ Φ (t, ) Φ (t, ) s− 1 s− 1 · − · H 2 ×H 2

˜ ∗ ˜ ∗˜ C (η0, ψ0, η˜0, ψ0) 1 (η, ψ) (t, ) (˜η, ψ) (t, ) 1 . (3.7) ≤ Hs+ 2 ×Hs · − · Hs×Hs− 2  

1 2 3.2.1. Gauge transform. Again, as s> 2+ 2 we have an immediate L estimates on s− 1 s− 1 Φ Φ,˜ thus we only need to get H˙ 2 H˙ 2 estimates. Let us start by by writing − × Φ=Φ1 + iΦ2 in equation (3.3):

∂tΦ+ TV ∂xΦ+ iTγ Φ= R (Φ)Φ, (3.8) · 1 Where R1 verifies ˜ R1(Φ) s− 1 s− 1 C (η0, ψ0, η˜0, ψ0) 1 , k kH 2 →H 2 ≤ Hs+ 2 ×Hs     ˜ ˜ ˜ ˜  [R1(Φ) R1(Φ)]Φ s 1 C (η0, ψ0, η˜0, ψ0) s 1 Φ Φ s 1 .  − H − 2 ≤ H + 2 ×Hs − H − 2    The next step is to preform the change of variable by χ, by Theorem A.6 we get: ∗ ∗ ′ ∗ ∗ ∂tΦ + TW ∂xΦ+ iT 3 Φ = R1(Φ )Φ , with (3.9) · |ξ| 2

Where R1 verifies

′ ∗ ˜ ∗˜ ˜ ∗˜ ˜ ∗ ˜ ∗˜ [R1(Φ ) R1(Φ )]Φ 1 C (η0, ψ0, η˜0, ψ0) 1 Φ Φ 1 . − Hs− 2 ≤ Hs+ 2 ×Hs − Hs− 2   ˜ We get the same equation on Φ˜ ∗ by symmetry. 14 Introduce the following gauge transform AΦ as the time one of the flow map defined by Propositions 4.1 with

2 1 −1 2−α p = ξ 2 ∂ W Γ (T), Φ 3 | | x ∈ 2 and put, ∗ θ = AΦΦ . (3.10) ˜ ˜ ∗˜ We define analogously AΦ˜ and θ from Φ . From Theorem Propositions 4.1 the s− 1 s− 1 s change of variable (3.10) is Lipschitz from H 2 to H 2 but under H control on ˜ ˜ (Φ, Φ) which is equivalent by Theorem A.3 to a control on (η0, ψ0, η˜0, ψ0) 1 . Hs+ 2 ×Hs We have:

∗ ˜ ∗˜ ˜ ˜ Φ (t, ) Φ (t, ) 1 C (η0, ψ0, η˜0, ψ0) 1 θ(t, ) θ(t, ) 1 , · − · H˙ s− 2 ≤ Hs+ 2 ×Hs · − · H˙ s− 2   (3.11)

˜ ˜ ∗ ˜ ∗˜ θ(t, ) θ(t, ) 1 C (η0, ψ0, η˜0, ψ0) 1 Φ (t, ) Φ (t, ) 1 . · − · H˙ s− 2 ≤ Hs+ 2 ×Hs · − · H˙ s− 2   (3.12)

˜ To get the equations on θ and θ we commute AΦ and AΦ˜ with (3.9), we make the computations for θ, those for θ˜ are obtained by symmetry:

3 ∗ ∗ 2 ′ ∗ ∗ A ∂tΦ + A TW ∂xΦ + iA T Φ= A R (Φ )Φ Φ Φ · Φ |ξ| Φ 1

3 3 ∗ 2 ∗ 2 ∗ ∗ ′ ∗ ∗ ∂tA Φ + iT A Φ + (A TW ∂x [iT , A ])Φ (∂tA )Φ = A R (Φ )Φ Φ |ξ| Φ Φ − |ξ| Φ − Φ Φ 1 By definition of pΦ and Proposition 4.4 we have:

3 1 pΦ pΦ ∂ξ( ξ 2 )∂xp = Wξ and ∂tA = A A Ti∂ p A dr. | | Φ Φ Φ −r t Φ r Z0 thus by Corollary 4.2 we get:

3 2 ∗ −1 ∗ ∂tθ + iT|ξ|θ = R2(θ)θ + AΦR1(Φ )AΦ Φ , (3.13)

−1 1 where R2 and AΦR1(Φ)AΦ , as s> 3+ 2 , verify:

˜ Re(R2(θ)) s− 1 s− 1 C (η0, ψ0, η˜0, ψ0) 1 , k kH 2 →H 2 ≤ Hs+ 2 ×Hs   ˜ ˜ ˜ ˜ [R2(θ) R2(θ)]θ 1 C (η0, ψ0, η˜0, ψ0) 1 θ(t, ) θ(t, ) 1 , − Hs− 2 ≤ Hs+ 2 ×Hs · − · Hs− 2   and,

−1 ˜ −1 ˜ [AΦR1(Φ)AΦ AΦ˜ R1(Φ)A˜ ]θ s− 1 − Φ H 2

˜ ˜ C (η0, ψ0, η˜0, ψ0) 1 θ(t, ) θ(t, ) 1 . ≤ Hs+ 2 ×Hs · − · Hs− 2  

Thus we have succeeded to eliminate the term T V ∂x of order 1 in (3.9) and got 1 · a term of order 2 . The result then follows with a standard energy estimate. 15 4. Baker-Campbell-Hausdorff formula: composition and commutator estimates We will start by giving the propositions defining the operators used in the gauge transforms and the symbolic calculus associated to them. From those propositions we will deduce the direct estimates used in in Sections 2.1.1 and 3.2.1. Notation 4.1. To compute the conjugation and commutation of operators with a flow map, we introduce Lie derivatives, i.e commutators. More precisely we introduce the following notations for commutation between operators: 0 2 k L b = b, Lab = [a, b]= a b b a, L b = [a, [a, b]], L b = [a, [ , [a, b]] ]. a ◦ − ◦ a a · · · · · · k times

In the following proposition the variable t [0, T ] is the generic| time{z variable} that appeared in the previous section and a new∈ variable τ R will be used and they should not be confused. ∈ We start with the proposition defining the Flow map and its standard properties. Proposition 4.1. Consider two real numbers δ < 1, s R and a real valued symbol p Γδ(D). The following linear hyperbolic equation is globally∈ well-posed: ∈ 1 ∂τ h iTph = 0, − (4.1) h(0, )= h ( ) Hs(D). ( · 0 · ∈ p For τ R, define Aτ as the flow map associated to 4.1 i.e: ∈ Ap :Hs(D) Hs(D) τ → h h(τ, ). (4.2) 0 7→ · Then for τ R we have, p ∈ s (1) Aτ L (H (D)) and ∈ p C|τ|M δ(p) A s s e 1 . k τ kH →H ≤ (2) p p p p p iTp Aτ = Aτ iTp, Aτ+τ ′ = Aτ Aτ ′ . p ◦ ◦ (3) Aτ is invertible and, p −1 p (Aτ ) = A−τ . Moreover, ∗ p ∗ (Tp) p (Aτ ) = A−τ = A−τ + R, ∗ (Tp) where R is a δ 1 regularizing operator and Aτ is the flow generated by the Cauchy problem:− ∗ ∂τ h i(Tp) h = 0, − (4.3) h(0, )= h ( ) Hs(D). ( · 0 · ∈ (4) Taking a real valued symbol p˜ Γδ(D) we have: ∈ 1 δ p p˜ C|τ|M1 (p,p˜) δ [A A ]h C τ e M (p p˜) h s+δ . (4.4) τ − τ 0 Hs ≤ | | 0 − k 0kH Proof. Points (1), (2), (3) are simple consequences of the hyperbolicity and well posedness of the Cauchy problem (4.1). Point (4) comes by writing: p p˜ p p˜ p˜ ∂τ [A A ]h iTp[A A ]h = iTp pA h , τ − τ 0 − τ − τ 0 −˜ τ 0 and making the usual energy estimate.  δ δ Remark 4.1. The hypothesis p Γ1 can be relaxed to p Γδ, which is the minimal hypothesis to ensure well posedness∈ in Sobolev spaces of ∈(4.1). 16 p At the moment the only bounds we obtained on Aτ is the continuity bounds on Sobolev spaces, in order to study it’s symbol and the symbol of conjugated operators we need to transfer those continuity bounds to estimates of symbols seminorms. This m was first done by Beals in [9] for pseudodifferential operators in the class Sρ,ρ with ρ< 1,m R. The following Lemma gives explicitly the key estimate adapted from [9] given∈ by (4.5), and we give one new estimate (4.6) that can then be directly applied to the paradifferential setting. Lemma 4.1. Consider an operator A continuous from S (D) to S ′(D) and let a S ′(D Dˆ) the unique symbol associated to A (cf [10] for the uniqueness), i.e, let∈K be the× kernel associated to A then:

u, v S (D), (Au, v)= K(u v), a(x,ξ)= Fy ξK(x,x y). ∈ ⊗ → − If A is continuous from Hm to L2, with m R, and [ 1 d , A] is continuous • ∈ i dx from Hm+δ to L2 with δ < 1, then (1 + ξ )−ma(x,ξ) L∞ (D Dˆ) and we | | ∈ x,ξ × have the estimate:

−m 1 d (1 + ξ ) a L∞ Cm A Hm→L2 + , A . (4.5) | | x,ξ ≤ k k i dx m+δ 2    H →L 

If A is continuous from Hm to L2, with m R, and [ix, A] is continuous • ∈ from from Hm−ρ to L2 with ρ 0, then (1 + ξ )−ma(x,ξ) L∞ (D Dˆ) ≥ | | ∈ x,ξ × and we have the estimate: −m (1 + ξ ) a L∞ Cm[ A Hm→L2 + [ix, A] Hm−ρ→L2 ]. (4.6) | | x,ξ ≤ k k k k

Proof. First without loss of generality through a standard mollification argument we work with a S (D Dˆ). We study the cases on T and R separately. Operators defined∈ on×T. The first key observation is the following: (x,ξ) T Z, e−ix·ξAeix·ξ = a(x,ξ), (4.7) ∈ × which one can right as eix·ξ L2 (T). Thus taking L2 norms in x we get: ∈ x ix·ξ m a( ,ξ) 2 A m 2 e Cm A m 2 (1 + ξ ) . (4.8) Lx H →L m H →L k · k ≤ k k Hx ≤ k k | |

Now to get the analogue of (4.8) but in the ξ variable we observe that the continuity hypothesis reads for (u, v) S : ∈ ix·ξ −mF F e a(x,ξ)(1 + ξ ) (v)(ξ)u(x)dxdξ A Hm→L2 (v) L2 u L2 , T Z | | ≤ k k k k ξ k k x Z × which we rewrite as:

F ix·ξ −m F (v)(ξ) e (1 + ξ ) a(x,ξ)u(x)dx dξ A Hm→L2 (v) L2 u L2 , Z T | | ≤ k k k k ξ k k x Z  Z 

Now we treat a as an operator with roles of (x,ξ) exchanged. Choosing,

u(x)= F −1(eix·ξ), we get analogously to (4.8): −m (1 + ξ ) a(x,ξ) L∞L2 Cm A Hm→L2 . (4.9) | | x ξ ≤ k k

The second key observation is: 1 d e−ix·ξ , A eix·ξ = ∂ a(x,ξ), e−ix·ξ[ix, A]eix·ξ = ∂ a(x,ξ). (4.10) i dx x ξ   Iterating the previous computations we get:

m+δ 1 d ∂xa( ,ξ) 2 Cm(1 + ξ ) , A , (4.11) Lx k · k ≤ | | i dx m+δ 2   H →L 17

and as ρ 0: ≥ −m ∂ξ[(1 + ξ ) a(x,ξ)] L∞L2 Cm[ A Hm→L2 + [ix, A] Hm−ρ→L2 ]. (4.12) | | x ξ ≤ k k k k

By the Sobolev embedding (4.9) and (4.12) give the desired result (4.6) on the torus. To get (4.5) we introduce: x b(x,ξ,ξ )= a , (1 + ξ )δξ . 0 (1 + ξ )δ | 0|  | 0|  As δ < 1 we have that (1 + ξ )δ (1 + ξ )δ for ξ ξ c(1 + ξ )δ for some | | ∼ | 0| | − 0|≤ | 0| fixed c > 0. Considering b as a function of (x,ξ) on T B(ξ0, c), inequalities (4.8) 2 1 × and (4.11) give uniform Lx and Hx estimates thus by the Sobolev embedding we get:

1 d m b(x,ξ) ∞ Cm A m 2 + , A (1 + ξ ) , (4.13) Lx H →L k k ≤ k k i dx m+δ 2 | |    H →L 

which transferred back to a give the desired result (4.5) on the torus. Operators defined on R. The main problem we face on R when adapting the previous proof is we can no longer use eix·ξ as a test function as it no longer belongs to L2(R). One way to get over this was given by Beals in [9], we choose g in S (R) such that g(0) = 1, F (g) is supported in ξ 1 and g(x) = g( x). Let {| |≤ } − gx(y)= g(y x) and compute for u S : − ∈ i(x−y)·ξ −iy·ξ ix u(x)= u(x)gx(x)= e gx(y)u(y)dydξ = e e gy(x)u(y)dydξ. R R R R Z × Z × We now compute an analogue of (4.7):

−iy·ξ ix Au(x)= e A e gy (x)u(y)dydξ R R Z × i(x−y)·ξ
= e a0(x,y,ξ)u(y)dydξ R R Z × = eix·ξa(x,ξ)F (u)(ξ)dξ, R Z where, −ix·ξ ix·ξ a0(x,y,ξ)= e A e gy (x), and,
i(x−y)·(η−ξ) a(x,ξ)= e a0(x,y,η)dηdy. R R Z × Applying the same arguments as in the periodic case we get:

−m k 1 d (1 + ξ ) ∂ a C A m 2 + , A , k N y 0 ∞ m,k H →L | | L ≤ k k i dx m+δ 2 ∈ x,y,ξ    H →L  (4.14) and,

−m k (1 + ξ ) ∂ a C [ A m 2 + [ix, A] m−ρ 2 ], k N. (4.15) y 0 ∞ m,k H →L H →L | | Lx,y,ξ ≤ k k k k ∈

Thus to conclude the proof we need to transfer the information on the amplitude a0 to the symbol a which is a simple application of Oscillatory Integrals. Indeed it 18 suffices to write: i(x−y)·(η−ξ) a(x,ξ)= e a0(x,y,η)dηdy R R Z × 1 i(x−y)·(η−ξ) = (I ∆y)e a0(x,y,η)dηdy R R ξ η 2 − Z × h − i 1 i(x−y)·(η−ξ) ∗ = e (I ∆y) a0(x,y,η)dηdy , R R ξ η 2 − Z × h − i  which gives the desired result as a0(x,y,η) and all of it’s y derivatives are y inte- grable.  Remark 4.2. It is worth noting that if instead of Sobolev estimates we had H¨older continuity estimates on A, then combined with (4.7) it gives directly the analogue of estimates (4.5) and (4.6). Analogously to Beals characterization of pseudodifferential through the continuity 1 d of the successive commutators Op(x), i dx with A on Sobolev spaces, we give the following characterization of Paradifferential operators through estimate (4.6). Corollary 4.1. Consider two real numbers m and ρ 0 and the spaces of paradiffer- m D ≥ m ential symbols Γρ ( ) equipped with the topology induced by the seminorms Mρ ( ; k) for k N defined in A.5 giving it a Fr´echet space structure. · For∈p Γm(D) we introduce the following family of seminorms: ∈ ρ k m Lj H0 (p; k)= ixTp , Hm→Hj j=0 X n k m Lj Ll N Hn (p; k)= ix 1 d Tp , n ,n ρ, i dx Hm→Hj ∈ ≤ j Xl=0 X=0 and if ρ / N: ∈ k m m n(ρ−⌊ρ⌋) Lj L⌊ρ⌋ Hρ (p; k)= H⌊ρ⌋(p; k)+sup 2 ix 1 d TpP≤n(D) . N i dx m j n∈ j=0 H →H X m m Then H (p; k)k N induces an equivalent Fr´echet topology to M ( ; k)k N on: ρ ∈ ρ · ∈ ψB,b Γm(D) = σB,b,p Γm(D) . ρ p ∈ ρ   n o In order to give the key symbolic calculus results in Propositions 4.2 and 4.3 we need to introduce the paradifferential analogue of the H¨ormander symbol class 0 S1−δ,δ. For this we follow [26] and introduce the space of non regular symbols: Definition-Proposition 4.1. Consider s R , for 0 δ,ρ < 1, we say: ∈ + ≤ k m−ρk Dξ p(x,ξ) Ck ξ s,∞ m D ≤ h i D D˜ p W Sρ,δ( ) , (x,ξ) , k 0. ∈ ⇐⇒  k m−ρk+sδ ∈ × ≥  Dξ p( ,ξ) Ck ξ  · W s,∞ ≤ h i (4.16)  ρ,δ m N The best constant in (4.16 ) defines a seminorm denoted by Ms ( ; k), k where k is the number of derivatives we make on the frequency variable ·ξ, we also∈ define the seminorm ρ,δM m = ρ,δM m( ; 1). We define analogously W s,∞Sm (D∗ D˜). s s · ρ,δ × Analogously to Corollary 4.1 we introduce the following family of seminorms: k ρ,δ m Lj H0 (p; k)= ixp , Hm→Hjρ j X=0 19

n k ρ,δ m Lj Ll N Hn (p; k)= ix 1 d p , n ,n ρ, i dx Hm→Hjρ−lδ ∈ ≤ l=0 j=0 X X and if s / N: ∈ k ρ,δ m m n(s−⌊s⌋) Lj L⌊s⌋ Hs (p; k)= H⌊s⌋(p; k)+sup 2 ix 1 d [Pn(D)p] , N i dx m jρ lδ n∈ j=0 H →H − X where Pn(D) is applied to p in the x variable. ρ,δ m ρ,δ m Then Hs (p; k)k∈N induces an equivalent Fr´echet topology to Ms ( ; k)k∈N on s,∞ m · W Sρ,δ. Proof. The L2 continuity of such operators and control by their symbol seminorms is given in Appendix B of [22] and the reciprocal is given by Lemma 4.1.  Remark 4.3. We note that for the standard paradifferential symbols classes we have: m ρ,∞ m Γρ = W S1,0. Now all of the ingredients are in place to give the key commutation and conjuga- tion result. Proposition 4.2. Consider three real numbers δ < 1, s R, ρ 1, a real valued δ D p R ∈ ≥ symbol p Γρ( ). Let Aτ ,τ be the flow map defined by Proposition 4.1 and ∈ β ∈ take a symbol b Γρ (D), β R then we have: ∈ ∈ p ρ,∞ β (5) There exists bτ W S (D) such that: ∈ 1−δ,δ p p A Tb A = T p . (4.17) τ ◦ ◦ −τ bτ Moreover we have the estimates:

⌈ρ−1⌉ k lim τ k β δ ⌈ρ⌉ T p L Tb CρM (b)M (p) , (4.18) bτ − k! iTp ≤ ρ ρ k=0 s s β ρ δ ρ X H →H − −⌈ ⌉ +

1−δ,δ β p δ β β δ H (b ; k) Ck(M (p ))[H (b; k)+ H (b; k)H (p; k)], k N. (4.19) ρ τ ≤ 1 ρ ρ ρ ∈ c p ρ−1,∞ β+δ−1 (6) There exists bτ W S (D) such that: ∈ 1−δ,δ p p lim lim lim [Aτ , Tb]= Aτ Tcbp Tcbp = Tb Tbp . (4.20) τ ⇐⇒ τ − −τ Moreover we have the estimates:

⌈ρ−1⌉ k lim k−1 τ k β δ ⌈ρ⌉ Tc p ( 1) L Tb CρM (b)M (p) , (4.21) bτ − − k! iTp ≤ ρ ρ k=1 s s β ρ δ ρ X H →H − −⌈ ⌉ +

1 −δ,δ β+δ−1 c p δ β β δ H ( b ; k) Ck(M (p))[H (b; k)+ H (b; k)H (p; k)], k N. (4.22) ρ−1 τ ≤ 1 ρ ρ ρ ∈ Remark 4.4. It is important to notice that the main result of this propo- • p sition is the factorization of the Aτ terms in (4.17) and (4.20) where the right hand sides contain symbols in the usual classes modulo a more regular remainder. This was not a priori the case of the left hand sides containing p m Aτ . In other words we prove the stability of Γρ under the conjugation by p Aτ . This is crucial when studying the regularity of the flow map for: 2 α 1 s> 1+ − + . α 1 2 20− p p Indeed if p depends on a parameter λ, DλAτ Tb A−τ is a priori an operator lim ◦ ◦ of order β + δ by (4.4), but DλT p is shown in proposition 4.5 to be an bτ operator of order β. p In the language of pseudodifferential operators, Tbτ is the asymptotic sum • k τ Lk of the series ( k! iTp Tb) i.e the Baker-Campbell-Hausdorff formal series. p Though Tbτ is not necessarily equal to this sum, for this sum need not con- verge. lim s By the right hand side we have the continuity of T p on H for s R and • bτ ∈ not only s> 0 which is an improvement on A.2. Proof. The structure of the proof is as follows: p (I) We will give a proof of the estimate (4.18) assuming bτ exists. p ρ,∞ β D (II) We will prove the existence of bτ W S1−δ,δ( ) which is the subtle part of the proof. (III) Finally we will deduce point (6) from point (5). Point (I). For point (5) we compute, p p p p p p ∂τ [A Tb A ]= iTp A Tb A A Tb iTp A τ ◦ ◦ −τ ◦ τ ◦ ◦ −τ − τ ◦ ◦ ◦ −τ Using (2), p p p p p p ∂τ [A Tb A ]= A iTp Tb A A Tb iTp A τ ◦ ◦ −τ τ ◦ ◦ ◦ −τ − τ ◦ ◦ ◦ −τ p p = Aτ [iTp, Tb]A−τ . (4.23)

As A0 = Id, integrating on [0,τ] we get: τ p p p p Aτ Tb A−τ = Tb + Ar[iTp, Tb]A−r dr. ◦ ◦ 0 Z ∗ Iterating the computation in we get for n N∗, ∗ ∈| {z } n k τ n p p τ k (τ r) p n+1 p A Tb A = L Tb + − A L TbA dr. (4.24) τ ◦ ◦ −τ k! iTp n! r iTp −r k 0 X=0 Z Now the key point is the continuity of paradifferential operators given by Theorem A.1 and the symbolic calculus rules given by Theorem A.3. By Lemma 4.2: L⌈ρ⌉ β δ ⌈ρ⌉ R iT Tb CρMρ (b)Mρ (p) , for ρ +, (4.25) p Hs→Hs−β−⌈ρ⌉δ+ρ ≤ ∈ where ρ is the upper integer part of ρ. Thus⌈ applying⌉ point (1) combined with (4.25) we get (4.18). ρ Point (II). The constant Cρ in (4.25) is estimated ”brutally” by Lemma 4.2: O(2 × ρ !), thus even though one has a 1 in (4.24) the convergence result is non trivial. ⌈ ⌉ ⌈ρ⌉! To get past this let us express explicitly the difficulty in the problem. Rearranging the terms in (4.23) we see that: p p p p p p ∂τ [A Tb A ]= A [iTp, Tb]A = [iTp, A TbA ], (4.26) τ ◦ ◦ −τ τ −τ τ −τ thus we have to solve the following the Cauchy problem in L (Hs(D),Hs−β(D)): β D ∂τ f(τ) = [iTp,f(τ)] Γρ−1( ), β ∈ (4.27) f(0) = Tb Γρ (D). ( ∈ This amounts to the non trivial problem of solving a linear ODE in the Fr´echet β D space Γ+∞( ), indeed such a problem need not have a solution in general, and even if it does, it need not be unique. To solve such a problem one usually has to look at a Nash-Moser type scheme, though in our case we have an explicit ODE that can be solved with a series and a loss of derivative. Thus inspired by H¨ormander’s [15], we 21 remark another key estimate given by the continuity of paradifferential operators given by Theorem A.1 and the symbolic calculus rules given by Theorem A.3(cf Lemma 4.2): L⌈ρ⌉ ⌈ρ⌉ β δ ⌈ρ⌉ R iT Tb C M0 (b)M0 (p) , for ρ +. (4.28) p Hs→Hs−β−⌈ρ⌉δ ≤ ∈

This means that if we can compensate the loss of β + ρ δ derivatives with a cost negligible in comparison to ρ !, we would have a convergent⌈ ⌉ series in (4.24). A first approach would be to interpolate⌈ ⌉ (4.25) and (4.28) which gives:

⌈ρ⌉ ρ−⌈ρ⌉δ ⌈ρ⌉ β ρ−⌈ρ⌉δ ρ−⌈ρ⌉δ ⌈ρ⌉ L ρ ρ δ ρ iT Tb C M0 (b) M0 (p) (4.29) p Hs→Hs−β ≤ (⌈ρ⌉+1) ⌈ρ⌉δ ⌈ρ⌉ ⌈ρ⌉δ ⌈ρ⌉δ ⌈ρ⌉δ ⌈ρ⌉ ⌈ρ⌉δ C ρ 2 ρ ρ ! ρ M β(b) ρ M δ(p) ρ , × ⌈ ⌉ ρ ρ This indeed solves the cost ρ ! of (4.25) but depends on M · norms of b and p. An ⌈ ⌉ ρ idea to control those norms in a cost negligible in comparison to ρ ! would be to mollify p and b using an analytic mollifier, this might work but we⌈ found⌉ it better to mollify differently. For this we introduce a mollification with the Gaussian function φǫ(D) with sym- bol: 2 1 − ξ φǫ(ξ)= e 2ǫ2 ,ǫ> 0. (4.30) ǫ√2π Other than the standard properties of mollifiers, we have the following properties: s For h H (D), φǫ(D)h is real analytic. • The moments∈ of the Gaussian can be explicitly computed by, for k n: • ∈ k k+1 ξ2 2 1 if k is even 1 k − 2 Γ( 2 ) ξ e 2 dξ = = (k 1)!! . √2π R | | √π − 2 if k is odd Z ( π From the moments of the Gaussian we deduce that,q for h Hs(D) and k n: • ∈ ∈ k −k ∂xφǫ(D)h Ckǫ h Hs , Hs ≤ k k

and C verifies for all K > 0, KkC = o(k!). k k Now by the symbolic calculus rules given by Theorem A.3, for ǫ> 0, there exists ǫ p 0 bτ Γ (D) such that: ∈ ρ +∞ k τ k lim L Tbφǫ(D)= Tǫ p , with, k! iTp bτ Xk=0 +∞ k 0 ǫ p k τ −kδ−β β δ k M ( b ) C Ck | | ǫ M (b)M (p) . (4.31) ρ τ ≤ k! ρ ρ k=0 X ǫ p In order to pass to the limit in ǫ we will express bτ differently, for all ǫ > 0, n τ k Lk T φ (D) converges in L (Hs(D)), thus by unicity of the limit: k! iTp b ǫ Xk=0 +∞ k p p τ k A Tb A φǫ(D)= L Tbφǫ(D). (4.32) τ ◦ ◦ −τ k! iTp k X=0 Thus, p p lim A Tb A φǫ(D)= Tǫ p . (4.33) τ ◦ ◦ −τ bτ δ−1 β ǫ p Now we estimate the Hρ ( ; k)k∈N norms of bτ . To do so we need, in the word of H¨ormander [14], a result which· interpolates between information on the norm a 22 of an operator and bounds for the derivatives of it’s symbol. This was exactly the goal of Lemma 4.1. 1 d By commuting i dx and ix with (4.33) we get:

1 d lim 1 d p p p 1 d p [ , Tǫ p ] =[ , A ] Tb A φǫ(D)+ A [ , Tb] A φǫ(D) i dx bτ i dx τ ◦ ◦ −τ τ ◦ i dx ◦ −τ p 1 d p p p 1 d + A Tb[ , A ]φǫ(D)+ A Tb A [ , φǫ(D)]. τ ◦ i dx ◦ −τ τ ◦ ◦ −τ i dx and, lim p p p p [ix, Tǫ p ] = [ix, A ] Tb A φǫ(D)+ A [ix, Tb] A φǫ(D) bτ τ ◦ ◦ −τ τ ◦ ◦ −τ p p p p + A Tb[ix, A ]φǫ(D)+ A Tb A [ix, φǫ(D)]. τ ◦ ◦ −τ τ ◦ ◦ −τ 1 d p p To estimate [ i dx , Aτ ] and [ix, Aτ ] we get back to (4.1) and see that: 1 d p τ p 1 d p [ , Aτ ]= A [ , T ]Ar , i dx 0 τ−r i dx ip (4.34) [ix, Ap ]= τ Ap [ix, T ]Ap. ( τ 0R τ−r ip r p Thus by iteration, the continuity ofRAτ and Lemma 4.1 we get: 1−δ,δ β p δ β β δ H (b ; k) Cn,k(M (p))[H (b; k)+ H (b; k)H (p; k)], (k,n) N. n τ ≤ 1 n n n ∈ p 0,∞ β Thus we can pass to the limit in ǫ in (4.33), there exist bτ W S such that: ∈ 1−δ,0 p p lim A Tb A = T p . (4.35) τ ◦ ◦ −τ bτ 1−δ,δ β p Now to get Hρ (bτ ; k), ρ / N estimates we will use the Littelwood Paley decom- ∈ position. Indeed recalling (P≤k) as the Littlewood Paley projectors defined in A.1 we have by Lemma A.1: T limP = P T lim P , a ≤k ≤k+1 (P≤k+1(D)a) ≤k where F (P≤k+1(D)a) = P≤k+1(η)F (a)(η,ξ). Thus going back to the sum (4.32) we find:

(P≤k+1(D)p (P≤k+1(D)p lim Aτ T(P (D)b A τ P≤k = P≤k+1T p P≤k ◦ ≤k+1 ◦ − (P≤k+1(D)bτ 1 d thus we get by commuting with ix, i dx and estimating as previously: 1−δ,δ β L⌊ρ⌋ p δ β L⌊ρ⌋ H (P≤k+1(D) 1 d bτ ; k) Cn,k(M1 (p))Hn (P≤k+1(D) 1 d b; k) i dx ≤ i dx δ β L⌊ρ⌋ δ L⌊ρ⌋ + Ck(M1 (p))Hn (P≤k+1(D) 1 d b; k)Hn( 1 d p; k) i dx i dx Thus by Proposition A.3 characterizing Zygmund spaces and Definition 4.1 we get: 1−δ,δ β p δ β β δ H (b ; k) Cρ,k(M (p))[H (b; k)+ H (b; k)H (p; k)], (k,n) N. ρ τ ≤ 1 ρ ρ ρ ∈ which gives (4.19) for ρ / N. Point (III). For point (6)∈ we compute: p p p p ∂τ [A , Tb] = [iTp A , Tb]= iTp[A , Tb] + [iTp, Tb]A . τ ◦ τ τ τ p Thus by definition of Aτ as the flow map we get the following Duhamel formula, τ p p p [Aτ , Tb]= Aτ−r[iTp, Tb]Ardr, 0 Z τ p p = Aτ A−r[iTp, Tb]Ar(u)dr . 0 Z ⋆ 23 | {z } Applying point (5) to ⋆ we get: n k p p k−1 τ k [A , Tb]= A ( 1) L Tb τ τ − k! iTp k=1 X τ n n p (τ r) p n+1 p + ( 1) A − A L TbA dr. − τ n! −r iTp r Z0 Again applying point (1) combined with (4.25) we get (4.21). To get (4.20) we inject (4.17) in ⋆, which concludes the proof.  δ D Lemma 4.2. Consider two real numbers δ, β, ρ 0, and two symbols p Γρ( ) β ≥ ∈ and b Γρ (D) then the exists a constant C > 0 such that: ∈ L⌈ρ⌉ ⌈ρ⌉+1 ⌈ρ⌉ β δ ⌈ρ⌉ R T Tb C 2 ρ !Mρ (b)Mρ (p) , for ρ +, (4.36) p Hs→Hs−β−⌈ρ⌉δ+ρ ≤ ⌈ ⌉ ∈

L⌈ρ⌉ ⌈ρ⌉ β δ ⌈ρ⌉ R T Tb C M0 (b)M0 (p) , for ρ +. (4.37) p Hs→Hs−β−⌈ρ⌉δ ≤ ∈

Proof. For (4.37 ), we notice that L⌈ρ⌉T contains 2⌈ρ⌉ terms of the form: Tp b ⌈ρ⌉ Tp Tb Tp, i [0, 2 ], ◦···◦ ◦···◦ ∈ position i Now by the continuity of paradifferential|{z} operators given in Theorem A.1 we have: ⌈ρ⌉ β δ ⌈ρ⌉ Tp Tb Tp s s−β−⌈ρ⌉δ K M (b)M (p) , k ◦···◦ ◦···◦ kH →H ≤ 0 0 which gives (4.37). For (4.36), we start by the case k N∗ is a an integer. We first notice that again by Theorem A.1 we have: ∈ L1 β+δ−1 Tp Tb Γk−1 , β+δ−∈1 1 β δ M (L Tb) CM (b)M (p). ( k−1 Tp ≤ k k Thus iterating this formula we get: Lk β+kδ−k Tp Tb Γ0 , ∈ k  β+δ−k k β δ M (L Tb) CkM (b) M (p),  0 Tp ≤ k i  i Y≥1 and Ck verifies:   k Ck = 2kCk Ck = C2 k!, −1 ⇒ Thus giving the result in the case k integer. For ρ 0, it suffice to see that for ρ 1 again by Theorem A.1 we have: ≥ ≤ L1 β+δ−ρ Tp Tb Γ0 , β+δ−∈ρ 1 β δ M (L Tb) CMρ (b)M (p), ( 0 Tp ≤ ρ which concludes the proof.  p We give a result on the symbol of Aτ Proposition 4.3. Consider two real numbers δ < 1, ρ 1 and a real valued symbol δ D ≥ p Γρ( ). ∈ p Let Aτ ,τ R be the flow map defined by Proposition 4.1, then there exists a iτp ∈ symbol e W ρ,∞S0 (D∗ D˜) such that: ⊗ ∈ 1−δ,δ × p lim p Aτ = T iτp + Aτ (Id T1). (4.38) e⊗ − 24 Moreover we have the identities: lim lim ∂τ [T iτp h0]= iTpT iτp h0, e⊗ e⊗ s for h0 H (D),s R. (4.39) T lim h = T h . ∈ ∈ eiτp 0 |τ=0 1 0  ⊗ τ lim p  T iτp = Teiτp + Aτ−s TipTeisp Tipeisp ds. (4.40) e⊗ − Z0 lim
Proof. The idea is that morally T iτp should be defined by the asymptotic series: e⊗ k k k k lim i τ k i τ T iτp (T ) = T k , e p ⊗ ∼ k! k! ⊗p k X X where p is defined by Theorem A.3. To make this series converge we again intro- ⊗ 1 duce the Gaussian multiplier φǫ(D) defined by (4.30), as we still have the factor k! as in Proposition 4.2. By the symbolic calculus rules in Theorem A.3 we then have that there exists a symbol ǫeiτp CρS0 (D∗ D˜) such that: ⊗ ∈ ∗ 1−δ,δ × +∞ k k +∞ k k i τ k i τ lim (Tp) φǫ(D)= T k φǫ(D)= Tǫ iτp , (4.41) k! k! ⊗p e⊗ Xk=0 Xk=0 +∞ k 1−δ,δ 0 ǫ iτp k τ −kδ δ k M ( e ) C Ck | | ǫ M (p) , (4.42) ρ ⊗ ≤ k! ρ k=0 k X where Ck verifies for all K > 0, K Ck = o(k!). Now in order to pass to the limit in ǫ we need to get uniform estimates on 1−δ,δ m ǫ iτp Hs ( e⊗ ; k)k∈N. To do so we see that: lim lim ∂τ [Tǫ iτp h0]= iTpTǫ iτp h0, e⊗ e⊗ s for h0 H (D),s R. (4.43) T lim h = φ (D)T h , ∈ ∈ eiτp 0 |τ=0 ǫ 1 0  ⊗ Thus a standard energy estimate combined with the commutation identities (4.34)  and Lemma 4.1 we get: 1−δ,δ m ǫ iτp δ δ H ( e ; k)k N Ck,n(M (p; k))M (p; k), (k,n) N. n ⊗ ∈ ≤ 1 n ∈ In order to get higher Zygmund estimates we see that by getting back to the sum (4.41), we have: lim lim ∂τ [T iτp h ]= iT T iτp h , P D ǫe 0 [P≤k(D)p] P D ǫe 0 ≤k( )[ ⊗ ] [ ≤k( ) ⊗ ] s D R lim for h0 H ( ),s . (4.44) T iτp h = φǫ(D)h . ∈ ∈ [P (D)e ] 0 |τ=0 0  ≤k ⊗ 1 d As previously commuting with ix and i dx and Proposition A.3 we get: 1−δ,δ m ǫ iτp δ δ H ( e ; k)k N Ck,n(M (p; k))M (p; k), (k,n) N. s ⊗ ∈ ≤ 1 s ∈ Thus we can pass to the limit in when ǫ 0 and get the desired result. Finally for identity (4.39) we pass to→ the limit in (4.43). Identity (4.40) comes s p from the following computation. Fix an h H ,s R, then [Aτ T iτp ]h solves: 0 ∈ ∈ − e 0 p p ∂τ [Aτ Teiτp ]h0 iTp [Aτ Teiτp ]h0 = TipTeiτp Tipeiτp h0, p − − − − (4.45) [Aτ T iτp ]h (0, ) = (Id T )h ( ), ( − e 0
· − 1 0 ·

p which by definition of
Aτ gives (4.40).  We will now compute the different Gateaux derivatives of the operators defined above. 25 Proposition 4.4. Consider two real numbers δ < 1, ρ 1, two real valued symbols ′ δ D p p′ R ≥ p,p Γρ( ). Let Aτ , Aτ ,τ be the flow maps defined by Proposition 4.1, then for τ∈ R we have: ∈ ∈ τ p p′ p p′ Aτ Aτ = Aτ−rTip′−pAr dr. (4.46) − 0 Z p Another way to see this is with the Gateaux derivative of p Aτ on the Fr´echet δ D 7→ space Γρ( ) is given by: τ p p p DpAτ (h)= Aτ−rTihArdr. (4.47) Z0 R 1 δ D Moreover consider an open interval I , and a real valued symbols p C (I, Γρ( )). p ⊂ ∈ Let Aτ ,τ R be the flow map defined by Proposition 4.1 then for τ R, z I we ∈ ∈ ∈ have: τ p p p ∂zAτ = Aτ−rTi∂zpArdr. (4.48) Z0 Proof. Fix h Hs,s R then: 0 ∈ ∈ p p p p p ∂τ [Aτ h0] iTp[Aτ h0] = 0 ∂τ [∂zAτ h0] iTp[∂zAτ h0] Ti∂zp[Aτ h0] = 0, − ⇒ p − − which gives (4.48) by the definition of Aτ and the Duhamel formula. The identities (4.46) and (4.47) are obtained in the same way.  Proposition 4.5. Consider two real numbers δ < 1, ρ 1, two real valued symbols ′ δ D p p′ R ≥ p,p Γρ( ). Let Aτ , Aτ ,τ be the flow maps defined by Proposition 4.1 and ∈ β ∈ take a symbol b Γρ (D) then for τ R we have: ∈ ∈ τ lim lim p lim p T p T ′ = A L T ′ A dr (4.49) bτ p τ−r iTp−p′ p r−τ − bτ 0 br Z τ lim = L lim T ′ dr. (4.50) iT p p p p−(p′) (br ) Z0 τ−r τ−r lim Another way to see this is with the Gateaux derivative of p T p on the Fr´echet 7→ bτ δ D space Γρ( ) is given by: τ lim lim lim D T p (h)= L lim T p dr = L τ lim T p . (4.51) p b iT p b i R T p dr b τ h τ 0 h τ Z0 τ−r τ−r lim lim lim lim Writing, Tc p = Tb T p , and, T ′ = Tb T ′ we get: bτ b c p p − −τ bτ − b−τ −τ lim lim p lim p Tc p T ′ = A LiT T ′ A dr (4.52) bτ cbp −τ−r p−p′ bp r+τ − τ − 0 r Z −τ lim = L lim T ′ dr. (4.53) iT p p p − p−(p′) (br )−τ−r Z0 −τ−r −τ lim lim lim DpTc p (h)= L lim T p dr = L −τ lim T p . (4.54) bτ iT p b i R T p dr b − h −τ − 0 h −τ Z0 −τ−r −τ−r Proof. From (4.26) and (4.27) we have: lim lim lim lim lim ∂ [T p T ′ ]= L (T p T ′ )+ L T ′ , τ b p iTp b p iTp−p′ p τ − bτ τ − bτ bτ lim lim (4.55) T p T p′ = 0. b0 b  − 0 Thus the Duhamel formula gives (4.49) and (4.50). For the Gateaux derivative  passing to the limit in (4.49) we have: τ lim p lim p D T p (h)= A L T p A dr, p bτ τ−r iTh br r−τ 0 Z 26 which gives (4.51). 

Corollary 4.2. Consider three real numbers α > 1, β < α and s R, two real α D β D ∈ valued symbols a Γ 1+β−α ( ) and b Γ 1+β−α ( ). Suppose that there exists a ∈ 2+ α−1 ∈ 1+ α−1 β+1−α D real valued symbol p Γ 1+β−α ( ) such that: ∈ 2+ α−1

b = ∂ξp∂xa + ∂xp∂ξa. (4.56) − p Define Aτ (u) as the flow map generated by iTp from Proposition 4.1. For τ R, Let, ∈ τ p p Rτ = τTib + A−s[Tip, Tia]Asds, (4.57) Z0 and, ˜ p p p p p p Rτ = Aτ Rτ A−τ = τAτ iTbA−τ + [Aτ , Tia]A−τ . (4.58)

s+(β+1−α)+ s Then Rτ , R˜τ L (H (D),H (D)) and ∈ ˜ α β β+1−α (Rτ , Rτ ) CM 1+β−α (a)M 1+β−α (b)M 1+β−α (p). s+(β+1−α)+ s 2+ H →H ≤ α−1 1+ α−1 2+ α−1

′ ′ ′ ′ ˜′ Moreover taking three different symbols a , b and p and defining analogously Rτ , Rτ , we have for h Hs: ∈ ′ [Rτ R ]h s − τ H α ′ ′ β ′ ′ β+1−α ′ ′ CM β α (a, a , a a )M (b, b , b b )M (p,p ,p p ) 2+ 1+ − 1+β−α 1+β−α ≤ α− 1 − 1+ α−1 − 2+ α−1 −

h + , × k kHs+(β+1−α) and,

˜ ˜′ [Rτ Rτ ]h − Hs α ′ ′ β ′ ′ β+1−α ′ ′ CM β α (a, a , a a )M (b, b , b b )M (p,p ,p p ) 2+ 1+ − 1+β−α 1+β−α ≤ α− 1 − 1+ α−1 − 2+ α−1 −

h + . × k kHs+(β+1−α)

Proof. First we notice that by definition Rτ , R˜τ are of order β and that we can write 2−α 2−α as p, a and b have a regularity of 2 + α−1 and 1+ α−1 respectively:

lim lim (α−1−β)− Rτ = T (β) + T (β+1−α) + Rτ , rτ rτ  ˜ lim lim ˜(α−1−β)− Rτ = T (β) + T (β+1−α) + Rτ ,  r˜τ r˜τ

(β) (β+1−α) (β) (β+1−α) where rτ , rτ , r˜τ , r˜τ are operators in the usual paradifferential classes and thus there differential with respect to p do not generate the undesired loss of 1+ β α derivative. Now− by Proposition 4.2:

lim lim T (β) = T (β) = Tib + [Tip, Tia], rτ r˜τ but the choice of p ensures by Theorem A.3 that Tib + [Tip, Tia] is of order β + 1 α, giving the desired result. −  27 Appendix A. Paradifferential Calculus In this paragraph we review classic notations and results about paradifferential and pseudodifferential calculus that we need in this paper. We follow the presenta- tions in [13], [14], [25], and [17] which give an accessible and complete presentation. Notation A.1. In the following presentation we will use the usual definitions and k k k standard notations for the regular functions C , Cb for bounded ones and C0 for those with compact support, the distribution space D ′,E ′ for those with compact support, D ′k,E ′k for distributions of order k, Lebesgue spaces (Lp), Sobolev spaces (Hs,W p,q) and the Schwartz class S and it’s dual S ′. All of those spaces are equipped with their standard topologies. We also use the Landau notation Ok k(X). For the definition of the periodic symbol classes we will need the following defini- tions and notations. Notation A.2. We will use D to denote T or R and Dˆ to denote their duals that is Z in the case of T and R in the case of R. For concision an integral on on Z i.e should be understood as . A function a is said to be in C∞(T Z) if for Z × Z Z ∞ X every ξ Z, a( ,ξ) C (T). For ξ Z, ∂ξ should be understood as the forward difference∈ operator,· i.e∈ ∈

∂ξa(ξ)= a(ξ + 1) a(ξ), ξ Z. − ∈ We recall the following simple identities for the Fourier transform on the Torus: F α αF Z T(∂x f)(ξ)= ξ T(f)(ξ),ξ , iπx α α ∈ FT((e−2 1) f)(ξ)= ξ FT(f)(ξ),ξ Z. ( − ∈ A.1. Littlewood-Paley Theory. Definition A.1 (Littlewood-Paley decomposition). Pick P C∞(Rd) so that: 0 ∈ 0 P (ξ) = 1 for ξ < 1 and 0 for ξ > 2. 0 | | | | We define a dyadic decomposition of unity by: −k for k 1, P k(ξ)=Φ (2 ξ), Pk(ξ)= P k(ξ) P k (ξ). ≥ ≤ 0 ≤ − ≤ −1 Thus, ∞ P≤k(ξ)= Pj(ξ) and 1= Pj(ξ). j k j 0≤X≤ X=0 Introduce the operator acting on S ′(Dd): −1 −1 P≤ku = F (P≤k(ξ)u) and uk = F (Pk(ξ)u). Thus,

u = uk. Xk Finally put k 1,Ck = supp Pk the set of rings associated to this decomposition. { ≥ } Remark A.1. An interesting property of the Littlewood-Paley decomposition is that even 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. 28 Proposition A.1 (Bernstein’s inequalities). Suppose that a Lp(Dd) has its spec- trum contained in the ball ξ λ . ∈ ∞ {| |≤ d} Then a C and for all α N and 1 p q + , there is Cα,p,q (indepen- dent of λ)∈ such that ∈ ≤ ≤ ≤ ∞ d d α |α|+ p − q ∂ a q Cα,p,qλ a p . k x kL ≤ k kL In particular, α |α| ∂ a q Cαλ a p , and for p = 2, p = k x kL ≤ k kL ∞ d a ∞ Cλ 2 a 2 . k kL ≤ k kL If moreover a has it’s spectrum is in 0 <µ ξ λ then: { ≤ | |≤ } −1 |α| α |α| C µ a q ∂ a q Cα,qλ a q . α,q k kL ≤ k x kL ≤ k kL Proposition A.2. For all µ > 0, there is a constant C such that for all λ > 0 and for all α W µ,∞ with spectrum contained in ξ λ . one has the following estimate: ∈ {| |≥ } −µ a ∞ Cλ a µ,∞ . k kL ≤ k kW Definition A.2 (Zygmund spaces on Dd). For r R we define the space: ∈ r Dd S ′ Dd r Dd S ′ Dd qr C∗ ( ) ( ), C∗ ( )= u ( ), u r = sup 2 uq ∞ < ⊂ ∈ k k q k k ∞   equipped with its canonical topology giving it a Banach space structure. It’s a classical result that for r / N, Cr(Dd)= W r,∞(Dd) the classic H¨older spaces. ∈ ∗ Proposition A.3. Let B be a ball with center 0. There exists a constant C such ′ d that for all r> 0 and for all (uq)q N S (D ) verifying: ∈ ∈ q qr q, suppu ˆq 2 B and (2 uq )q N is bounded, ∀ ⊂ k k∞ ∈ r d C qr then, u = u C (D ) and u sup N2 u . q ∗ r 1 2−r q∈ q ∞ q ∈ k k ≤ k k X − Definition A.3 (Sobolev spaces on Dd). It is also a classical result that for s R : ∈ 1 2 s Dd S ′ Dd 2qs 2 H ( )= u ( ), u s = 2 uq L2 < ( ∈ | | q k k ∞)  X  with the right hand side equipped with its canonical topology giving it a Hilbert space structure and is equivalent to the usual norm on s . | |s k kH Proposition A.4. Let B be a ball with center 0. There exists a constant C such ′ d that for all s> 0 and for all (uq) N S (D ) verifying: ∈ ∈ q qs 2 q, suppu ˆq 2 B and (2 uq 2 )q N is in L (N), ∀ ⊂ k kL ∈ 1 C 2 then, u = u Hs(Dd) and u 22qs u 2 . q s 1 2−s q L2 q ∈ | | ≤ q k k X −  X  We recall the usual nonlinear estimates in Sobolev spaces: s d s d If uj H j (D ), j = 1, 2, and s + s > 0 then u u H 0 (D ) and if • ∈ 1 2 1 2 ∈ d s sj, j = 1, 2 and s s + s , 0 ≤ 0 ≤ 1 2 − 2

then u u s K u s u s , k 1 2kH 0 ≤ k 1kH 1 k 2kH 2 d where the last inequality is strict if s1 or s2 or s0 is equal to 2 . 29 − ∞ s Dd d For all C function F vanishing at the origin, if u H ( ) with s > 2 , • then ∈

F (u) s C( u s ), k kH ≤ k kH for some non decreasing function C depending only on F. A.2. Paradifferential operators. We start by the definition of symbols with lim- ited spatial regularity. Let W S ′ be a Banach space. ⊂ Definition A.4. Given m R, Γm (D) denotes the space of locally bounded func- ∈ W tions a(x,ξ) on D (Dˆ 0), which are C∞ with respect to ξ for ξ = 0 and such that, for all α Nd and× for all\ ξ = 0, the function x ∂αa(x,ξ) belongs6 to W and there ∈ 6 7→ ξ exists a constant Cα such that, for all ǫ> 0: α m−|α| ξ > ǫ, ∂ a(., ξ) Cα,ǫ(1 + ξ ) . (A.1) ∀ | | ξ W ≤ | | m The spaces ΓW (D) are equipped with their natural Fr´echet topology induced by the semi-norms defined by the best constants in (A.1). For quantitative estimates we introduce as in [17]: Definition A.5. For m R and a Γm (D), we set ∈ ∈ W m m−|α| α MW (a; n)= sup sup (1 + ξ ) ∂ a(., ξ) , for n N. ξ W |α|≤n |ξ|≥ 1 | | ∈ 2 ρ, For W = W ∞, ρ 0, we write: ≥ m D m D m m ΓW ρ,∞ ( )=Γρ ( ) and Mρ (a)= MW ρ,∞ (a; 1). Moreover we introduce the following spaces equipped with their natural Fr´echet space structure: ∞ ρ,∞ m m −∞ m C (D)= ρ W , Γ (D)= ρ Γ (D), Γ (D)= m RΓ (D) and, b ∩ ≥0 ∞ ∩ ≥0 ρ ρ ∩ ∈ ρ −∞ m Γ (D)= ρ m R Γ (D). ∞ ∩ ≥0 ∩ ∈ ρ m Remark A.2. In higher dimension the 1 in the definition of Mρ should be replaced by 1+ d . ⌊ 2 ⌋ Definition A.6. Define an admissible cutoff function as a function ψB,b C∞, B > 1,b> 0 that verifies: ∈ (1) ψB,b(η,ξ) = 0 when ξ B η + b + 1. | | | | | | | | d d (2) for all (α, β) N N , there is Cα , with C , 1, such that: ∈ × β 0 0 ≤ α β −|α|−|β| (ξ, η) : ∂ ∂ ψ(ξ, η) Cα,β(1 + ξ ) . (A.2) ∀ ξ η ≤ | | 1,b (ψ )B>1,b>0 will be called limit cutoff functions.

m Definition A.7. Consider a real numbers m R, a symbol a ΓW and an admis- B,b ∈ ∈ sible cutoff function ψ define the paradifferential operator Ta by:

B,b Tau(ξ) = (2π) ψ (ξ η, η)ˆa(ξ η, η)ˆu(η)dη, Dˆ − − Z ix.η where aˆ(η,ξ) = de− a(x,ξ)dx is the Fourier transform of a with respect to the first variable. In the language of pseudodifferential operators: R B,b Tau = op(σa)u, where Fxσa(ξ, η)= ψ (ξ, η)Fxa(ξ, η). For a limit cutoff ψ1,b we define:

\lim 1,b Ta u(ξ) = (2π) ψ (ξ η, η)ˆa(ξ η, η)ˆu(η)dη, Dˆ − − Z 30 lim and define analogously σa . An important property of paradifferential operators is their action on functions with localized spectrum. Lemma A.1. Consider two real numbers m R, ρ 0, a symbol a Γm(D), an ∈ ≥ ∈ 0 admissible cutoff function ψB,b, a limit cutoff function ψ1,b and u S (Dd). ∈ For R >> b, if supp F u ξ R , then: • ⊂ {| |≤ } 1 b supp F Tau ξ (1 + )R , (A.3) ⊂ | |≤ B − B   b and supp F T limu ξ 2R . (A.4) a ⊂ | |≤ − B   For R >> b, if supp F u ξ R , then: • ⊂ {| |≥ } 1 b supp F Tau ξ (1 )R + , (A.5) ⊂ | |≥ − B B   b and supp F T limu ξ . (A.6) a ⊂ | |≥ B   The main features of symbolic calculus for paradifferential operators are given by the following Theorems taken from [17] and [20]. R m D Theorem A.1. Let m . if a Γ0 ( ), then Ta is of order m. Moreover, for all µ R there exists a constant∈ K such∈ that: ∈ m Ta µ µ−m KM (a), and, k kH →H ≤ 0 m Ta µ,∞ µ−m,∞ KM (a), µ / N. k kW →W ≤ 0 ∈ R m Rd Theorem A.2. Take m and a Γ0 ( ), then for all µ > 0 there exists a constant K such that: ∈ ∈ lim m Ta KM0 (a). Hµ→Hµ−m ≤

lim m N Ta KM0 (a), µ / . W µ,∞ →W µ−m,∞ ≤ ∈ ′ Theorem A.3. Let m,m ′ R, and ρ> 0, a Γm(D)and b Γm (D). ∈ ∈ ρ ∈ ρ Composition: Then TaTb is a paradifferential operator with symbol: • ′ a b Γm+m (D), more precisely, ⊗ ∈ ρ ′ ′ BB ,b ψB,b ψB ,b ψ B+B′−1 Ta Tb = Ta⊗b . ′ Moreover TaTb Ta b is of order m + m ρ where a#b is defined by: − # − 1 a#b = ∂αa∂αb, i|α|α! ξ x |αX|<ρ ′ and there exists r Γm+m −ρ(D) such that: ∈ 0 ′ ′ ′ M m+m −ρ(r) K(M m(a)M m (b)+ M m(a)M m (b)), 0 ≤ ρ 0 ρ 0 and we have BB′ ′ ′ ,b BB ,b B,b B ,b B+B′−1 ′ T ψ T ψ T ψ = T ψ B+B −1 , a b − a#b r lim lim lim lim Ta Tb Ta#b = Tr . −31 ∗ Adjoint: The adjoint operator of Ta, Ta is a paradifferential operator of order • m with symbol a∗ defined by: 1 a∗ = ∂α∂αa.¯ i|α|α! ξ x |αX|<ρ Moreover, for all µ R there exists a constant K such that ∈ ∗ m T Ta∗ µ µ−m+ρ KM (a). k a − kH →H ≤ ρ If a = a(x) is a function of x only, the paradifferential operator Ta is called a paraproduct. It follows from Theorem A.3 and the Sobolev embedding that: If a Hα(D) and b Hβ(D) with α,β > d , then • ∈ ∈ 2 d TaTb Tab is of order min α, β . − − { }− 2   If a Hα(D) with α> d , then • ∈ 2 ∗ d T Ta∗ is of order α . a − − − 2   If a W r,∞(D), r N then: • ∈ ∈ au Tau r C a r,∞ u 2 . k − kH ≤ k kW k kL An important feature of paraproducts is that they are well defined for function ∞ r d a = a(x) which are not L but merely in some Sobolev spaces H with r< 2 . d −m µ Proposition A.5. Let m > 0. If a H 2 (D) and u H (D) then Tau Hµ−m(D). Moreover, ∈ ∈ ∈

Tau Hµ−m K a d −m u Hµ k k ≤ k kH 2 k k A main feature of paraproducts is the existence of paralinearisation Theorems which allow us to replace nonlinear expressions by paradifferential expressions, at the price of error terms which are smoother than the main terms. Theorem A.4. Let α, β R be such that α,β > d , then ∈ 2 Bony’s Linearization Theorem: For all C∞ function F, if a Hα(D) then; • ∈ 2α− d F (a) F (0) T ′ a H 2 (D). − − F (a) ∈ α β α+β− d If a H (D) and b H (D), then ab Tab Tba H 2 (D). Moreover • there∈ exists a positive∈ constant K independent− − of a∈ and b such that:

ab Tab Tba α+β− d K a Hα b Hβ . k − − kH 2 ≤ k k k k A.3. 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]. Dd Dd 1+r,∞ r,∞ Theorem A.5. Let χ : be a Wloc diffeomorphism with Dχ W , r> 0, r / N and take s R then→ the following maps are continuous: ∈ ∈ ∈ Hs(Dd) Hs(Dd) → ∗ u χ u = Pl(D)uk χ, 7→ ◦ k≥0 l≥0 X k−NX≤l≤k+N N N −1 N where N is chosen such that 2 > supk,Dd ΦkDχ and 2 > supk,Dd ΦkDχ . ∈ 32 | | | | Taking χ˜ : Dd Dd a C1+˜r diffeomorphism with Dχ W r,˜ ∞ map with r˜ > 0, then the previous→ operation has the natural fonctorial prop∈erty: u Hs(Dd),χ∗χ˜∗u = (χ χ˜)∗u + Ru, ∀ ∈ ◦ with, R : Hs(Rd) Hs+min(r,r˜)(Rd) continous. → We now give the key paralinearization theorem taking into account the paracom- position operator. 1,∞ Dd Dd Dd 1+r,∞ Theorem A.6. Let u be a W ( ) map and χ : be a Wloc diffeo- morphism with Dχ W r,∞, r> 0, r / N. Then: → ∈ ∈ ∗ u χ(x)= χ u(x)+ TDu χχ(x)+ R (x)+ R (x)+ R (x) ◦ ◦ 0 1 2 where the paracomposition given in the previous Theorem verifies the estimates: ∗ s R, χ u(x) s C( Dχ ) u(x) s , ∀ ∈ k kH ≤ k k∞ k kH ′ 0 d u χ Γ , d (D ) for u Lipchitz, ◦ ∈ W 0 ∞(D ) and the remainders verify the estimates:

R0 1+r+min(1+ρ,s− d ) C Dχ r u H1+s k kH 2 ≤ k k k k R 1+r+s C( Dχ ) Dχ u 1+s . k 1kH ≤ k k∞ k kr k kH −1 R 1+r+s C( Dχ , Dχ ) Dχ u 1+s . k 2kH ≤ k k∞ ∞ k kr k kH Finally the commutation between a paradifferential operator a Γm(Dd) and a ∈ β paracomposition operator χ∗ is given by the following ∗ ∗ ∗ m−β d χ Tau = Ta∗ χ u + Tq∗ χ u with q Γ (D ), ∈ 0 where a∗ has the local expansion: ∗ 1 α −1 ⊤ m Dd a (x,ξ) ∂ a(χ(x),Dχ (χ(x)) ξ)Pα(χ(x),ξ) Γmin(r,β)( ), ∼ α α! ∈ |α|≤⌊Xmin(r,ρ)⌋ (A.7) where, ′ α i(χ−1(y′)−χ−1(x′)−Dχ−1(x′)(y′−x′)).ξ Pα(x ,ξ)= Dy′ (e )|y′=x′ |α| and Pα is polynomial in ξ of degree , with P = 1, P = 0. ≤ 2 0 1 Remark A.3. 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 large in the definition: ∗ u(Ax) = (Ax) u, and Tu′(Ax)Ax = 0.

Appendix B. Gauge transform on R Closely inspecting the two problems in [23] and [19], we saw that the lack of regularity obtained in [23] for α 2 is essentially due to the lack of control of the L1 norm in Sobolev spaces. To show≥ this we start give a simple, albeit an artificial example: 1 s R Theorem B.1. Consider two real numbers s ]1 + 2 , + [, r> 0 and u0 H ( ). ∈ ∞ s ∈ Then there exists T > 0 such that for all v0 in the ball B(u0, r) H (R) there exists a unique v C([0, T ],Hs(R)) solving the Cauchy problem: ⊂ ∈ ∂ v + Re(v)∂ v + i∂2v = 0, t x x (B.1) v(0, )= v ( ), ( · 0 · 33 Moreover we have the estimates:

0 µ s, v(t) µ R Cµ v µ R . (B.2) ∀ ≤ ≤ k kH ( ) ≤ k 0kH ( ) Taking two different solutions u, v such that u v L1(R), then: − ∈ (u v)(t) s R C( u s R ) u v s R k − kH ( ) ≤ k 0kH ( ) k 0 − 0kH ( ) + C( u s R )[ u v 1 R + (u v)(t) 1 R ]. (B.3) k 0kH ( ) k 0 − 0kL ( ) k − kL ( ) Proof. This is the simplest theorem to prove as the transformation is straightforward m and the symbols used are in the usual H¨ormander symbol classes S1,0. Given the well posedness of the Cauchy problem in Hs, and the density of S in Hs, it suffice to prove the result for u , v S (R) which henceforth we suppose. Define u, v as 0 0 ∈ the solution to (B.1) with initial data u0, v0 on [0, T ]. s 2 s The first step we reduce H estimates to L ones by defining f1 = D u. Com- muting D s with (B.1), by the symbolic calculus rules in Appendix hA iwe get the h i PDE on f1: ∂ f + Re(u)∂ f + i∂2f = R (f )f , t 1 x 1 x 1 1 1 1 (B.4) f (0, )= D s u ( ), ( 1 · h i 0 · where R1 verifies

R (f ) 2 2 C( u s ), ∂f R (f ) 2 2 C( u s ), k 1 1 kL →L ≤ k kH k 1 1 1 kL →L ≤ k kH We define analogously g1 from v and notice that by definition:

f g 2 = u v s . k 1 − 1kL k − kH 2 thus the problem is reduced to getting L estimates on f1 g1. x ∞ R − Then we introduce F (t,x) = 0 Re(u)(t,y)dy C ( ) and make the following change of variable: ∈ R − i F f2 = e 2 f1.

Analogously define G and g2 from v. As remarked in [24], F, G do not necessarily − i F − i G m decay at infinity but we still have e 2 , e 2 S1,0. Indeed because ∂xF = Re(u) +∞ +∞ ∈ ∈ H and ∂xG = Re(v) H . Now to get Lipschitz control we have ∈ − i G − i F [e 2 e 2 ]f1 G F L∞ f1 L2 v u L1 f1 L2 , − L2 ≤ k − k k k ≤ k − k k k i G i F [e 2 e 2 ]f2 G F L∞ f2 L2 v u L1 f2 L2 , − L2 ≤ k − k k k ≤ k − k k k thus,

f g 2 C[ f g 2 + v u 1 f 2 ], k 2 − 2kL ≤ k 1 − 1kL k − kL k 1kL f g 2 C[ f g 2 + v u 1 f 2 ], (k 1 − 1kL ≤ k 2 − 2kL k − kL k 2kL 2 i F and the problem is reduced to getting L estimates on f2 g2. Commuting e 2 with (B.4) yields: − 2 ∂tf2 + i∂xf2 = R2(f2)f2, i F s (B.5) f (0, )= e 2 0 D u ( ), ( 2 · h i 0 · where R2 verifies

R (f ) 2 2 C( u s ), ∂f R (f ) 2 2 C( u s ). k 2 2 kL →L ≤ k kH k 2 2 2 kL →L ≤ k kH Analogously we get on g2 2 ∂tg2 + i∂xg2 = R2(g2)g2 i G s (B.6) g (0, )= e 2 0 D v ( ). ( 2 · h i 0 · 34 Now the usual energy estimate on f2 g2 combined the Gronwall lemma on f2 g2 gives for 0 t T : − − ≤ ≤ f g 2 C( (u, v) s ) (f g )(0, ) 2 k 2 − 2kL ≤ k kH k 2 − 2 · kL 1 As s> 1+ 2 , by the Sobolev embedding Theorem:

i i F0 s G0 s f2 g2 L2 C( (u0, v0) Hs ) e 2 D u0 e 2 D v0 k − k ≤ k k h i − h i L2

C( (u0, v0) Hs )[ u0 v0 Hs + u0 v0 L1 ], ≤ k k k − k k − k which concludes the proof. 

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