Almost global existence of space periodic gravity-capillary water waves

Transcription

1 Almost global existence of space periodic gravity-capillary water waves Massimiliano Berti SISSA, Trieste Mini-course. St Etienne, 29 January-3 February 2017 Joint work with Jean-Marc Delort

2 Euler equations of fluids u t + (u )u = p W 1 u(t, x) = velocity field 2 x Ω(t) = variable domain occupied by the fluid 3 p(t, x) = internal pressure 4 W = external conservative forces 5 constant density ρ(t, x) = 1 + suitable conditions and forces acting at the boundary Ω(t) Continuity equation: uncompressible fluid divu = 0

3 Irrotational velocity fields rotu = 0 Remark: By Helmotz equation t ω + (u )ω = (ω )u, ω = rotu, the vorticity ω(t) 0, t, is a solution. Thus Velocity potential u = Φ (defined up to a constant C(t)) Bernoulli theorem t Φ + Φ W + p = 0

4 Water Waves equations for a fluid in S η (t) = { h < y < η(t, x)} under gravity and with capillarity forces at the free surface ( ) t Φ Φ 2 η + gη = κ x x at y = η(t, x) 1+η 2 x Φ = 0 in h < y < η(t, x) y Φ = 0 at y = h t η = y Φ x η x Φ at y = η(t, x) Unknowns: u = Φ = velocity field, rotu = 0 (irrotational), divu = Φ = 0 (uncompressible) g = gravity, κ = surface tension ( coefficient ) η Mean curvature = x x 1+η 2 x free surface y = η(t, x) and the velocity potential Φ(t, x, y)

5 Zakharov formulation Infinite dimensional Hamiltonian system: t u = J u H(u), u := canonical Darboux coordinates: ( ) η, J := ψ ( ) 0 Id, Id 0 η(x) and ψ(x) = Φ(x, η(x)) trace of velocity potential at y = η(x) (η, ψ) uniquely determines Φ in the whole { h < y < η(x)} solving the elliptic problem: Φ = 0 in { h < y < η(x)}, Φ y=η = ψ, y Φ = 0 at y = h

6 Hamiltonian: total energy on S η = T { h < y < η(x)} H := 1 Φ 2 dxdy + gy dxdy + κ 2 S η S η T 1 + η 2 x dx kinetic energy + potential energy + area surface integral Hamiltonian expressed in terms of (η, ψ) H(η, ψ) = 1 2 (ψ, G(η)ψ) L 2 (T x ) + T g η2 2 dx + κ T 1 + η 2 x dx Dirichlet Neumann operator (Craig-Sulem 93) G(η)ψ(x) := 1 + ηx 2 n Φ y=η(x) = (Φ y η x Φ x )(x, η(x))

7 Zakharov-Craig-Sulem formulation t η =G(η)ψ = L2 ψ H(η, ψ) ( ) 2 t ψ = gη ψ2 x G(η)ψ ηx ψ x κη xx 2(1 + ηx) 2 + (1 + ηx) 2 = L2 3/2 η H(η, ψ) Dirichlet Neumann operator G(η)ψ(x) := 1 G(η) is linear in ψ, 2 self-adjoint with respect to L 2 (T x ) 3 G(η) 0, G(1) = 0 4 η G(η) nonlinear, smooth 1 + η 2 x n Φ y=η(x) 5 G(η) is pseudo-differential, G(η) = D x tanh(hd x ) + OPS Alazard, Burq, Craig, Delort, Lannes, Metivier, Zuily,...

8 Symmetries Reversibility H(η, ψ) = H(η, ψ) Involution H S = H, S : (η, ψ) (η, ψ), S = [ ] S 2 = Id, Reversible vector field X H = J H X H S = S X H Φ t H S = S Φ t H Equivariance under the Z/(2Z)-action of the group {Id, S}

9 Standing Waves Invariant subspace: functions even in x η( x) = η(x), ψ( x) = ψ(x) Thus the velocity potential Φ( x, y) = Φ(x, y) and, using also 2π periodicity, Φ x (0, y) = Φ x (π, y) = 0 = no flux of fluid outside the walls {x = 0} and {x = π}.

10 Prime integral: mass η H s 0(T) := T η(x)dx { η H s (T) : T } η(x)dx = 0 The variable ψ is defined modulo constants: only the velocity field x,y Φ has physical meaning. ψ Ḣ s (T) = H s (T)/ u(x) v(x) u(x) v(x) = c

11 Linear water waves theory Linearized system at (η, ψ) = (0, 0) { t η = G(0)ψ, t ψ = gη + κη xx Dirichlet-Neumann operator at the flat surface η = 0 is G(0) = D tanh(hd), D = x i = Op(ξ) ξ R Fourier multiplier notation: given m : Z C m(d)h = Op(m)h = j Z m(j)h je ijx, h(x) = j Z h je ijx

12 Exercise: The solution of the elliptic problem: Φ = 0 in { h < y < 0}, Φ y=0 = ψ, y Φ = 0 at y = h where ψ(x) = j Z ψ je ijx is Thus Φ(x, y) = ψ 0 + j 0 ψ j cosh(j(y + h))eijx cosh(hj) G(0)ψ := ( y Φ)(x, 0) = j tanh(hj)ψ j e ijx =: D tanh(hd)ψ j Z

13 Linear water waves system [ [ η ] t ψ = 0 D tanh(hd) g+κ xx 0 ] [ η ] ψ Change of variable New linear system [ [ η ] t ψ = η Λη, ψ Λ 1 ψ, Λ = 0 ω(d) ω(d) 0 ( D tanh(hd) g+κd 2 ) 1/4 ] [ η ] ψ, ω(d) = D tanh(hd)(g + κd 2 ) Dispersion relation ω(ξ) = ξ tanh(hξ)(g + κξ 2 ) κ ξ 3/2

14 Complex representation u = η + iψ, u t + iω(d)u = 0 -decoupled Harmonic oscillators u(t, x) = e iω(j)t u j (0)e ijx j Z Linear frequencies of oscillations ω(j) = j tanh(hj)(g + κj 2 ), j Z, 1 All solutions are periodic, quasi-periodic, almost periodic in time: the solutions are defined for all times, and the Sobolev norm is constant u(t, ) H s = u(0, ) H s

15 Linear frequencies of oscillations ω j = j tanh(hj)(g + κj 2 ), j Z, 1 even dispersion relation. On the subspace of even functions ω j are simple 2 For κ > 0 -superlinerω j κj 3 2 as j +, 3 For κ = 0 (no surface tension) -sublinearω j = j tanh(hj)g jg as j +,

16 Nonlinear Water Waves equations t η = G(η)ψ ( ) 2 t ψ = gη ψ2 x G(η)ψ ηx ψ x κη xx 2(1 + ηx) 2 + (1 + ηx) 2 3/2 1 For which time interval solutions exist? Are global or not? 2 Which is their behavior for large times? 3 Are there periodic, quasi-periodic, almost periodic solutions? We are considering x T periodic boundary conditions, thus there are no "dispersive" effects of the linear PDE with x R 2, x R (Germain-Masmoudi-Shatah, Wu, Ionescu-Pusateri, Alazard-Delort, Ifrim-Tataru, Alazard-Burq-Zuily,... )

17 Water waves are a quasi-linear system 1 Linear part: ω(d) = u t + iω(d)u = N(u) D tanh(hd)(g + κd 2 ) 2 N = quadratic nonlinearity with derivatives the same order N = N( D 3/2 u)

18 Main result: Long time existence Birkhoff normal form result: M. Berti- J-M. Delort, 16, For any small initial condition the solutions are defined for long times For the same system recent KAM results: periodic solutions: Alazard-Baldi, 14 quasi-periodic solutions, Berti-Montalto, 15, Existence of solutions defined for all times, for "most" initial conditions

19 Almost global existence Theorem (B., J-M.Delort, 2016) There is a zero measure subset N in ]0, + [ 2 such that, for any (g, κ) in ]0, + [ 2 \N, for any N in N, there is s 0 > 0 and, for any s s 0, there are ε 0 > 0, c > 0, C > 0 such that, for any ε ]0, ε 0 [, any even function (η 0, ψ 0 ) in H s (T, R) Ḣ s 1 4 (T, R) with η 0 H s ψ 0 Ḣs 1 4 < ε the gravity-capillary water waves equation has a unique classical solution, even in space, (η, ψ) C 0( ] T ε, T ε [, H s (T, R) Ḣ s 1 4 (T, R) ) with T ε cε N, satisfying the initial condition η t=0 = η 0, ψ t=0 = ψ 0

20 Remark 1) Time of existence 1 N = 1, time of existence T ε = O(ε 1 ), local existence theory, Beyer-Gunther, Coutand-Shkroller, Alazard-Burq-Zuily. 2 N = 2, time of existence T ε = O(ε 2 ), Ifrim-Taturu, Ionescu-Pusateri for all the parameters, also for data that do not decay at infinity, 3 For N 3, to get time of existence T ε = O(ε N ), we have to erase parameters (g, κ): Birkhoff normal form. The assumption (g, κ) / N " is not technical. Used to avoid wave interactions" which may produce blow up, growth of Sobolev norms, instabilities: there are Wilton ripples" yet at 3-th order Birkhoff resonant system (ex. Craig-Sulem)

21 Remark 2) Parameters 1 We can also think to fix (κ, g, h) and the result holds for most space "wave-length": periodic boundary conditions x λt, λ R. The linear frequencies depend non-trivially w.r.t λ 2 We fixed h = 1. We can not use h as a parameter: h j tanh(hj)(g + κj 2 ) is not sub-analytic. The parameter h moves very little the frequency: tanh(hj) = 1 + O(e hj )

22 Remark 3) Reversible and Hamiltonian structure Algebraic property to exclude growth of Sobolev norms" 1 Hamiltonian 2 Reversibility (Moser) Dynamical systems heuristic explanation:

23 Water waves u t = iω(d)u + N 2 (u, ū), N 2 (u, ū) = O(u 2 ) Fourier and Action-Angle variables (θ, I) u(x) = j Z u je ijx, u j = I j e iθ j Sobolev norm u 2 H s = j Z (1 + j2 ) s I j Small amplitude solutions Rescaling u εu u t = iω(d)u + εo(u 2 ) in action-variables reads İ j = εf j (ε, θ, I), θ j = ω(j) + εg j (ε, θ, I) angles θ j = ω(j)t rotate fast", actions I j (t) slow" variables

24 Averaging - principle": The dynamics of the actions is expected to be İ j = ε f j (ε, I), f j (ε, I) = f (ε, θ, I)dθ T the average with respect to the angles θ = (θ) j Z If f j (ε, I) 0 = I j (t) diverges ("secular terms" of Celestial mechanics) The condition f j (I) = 0 is necessary to have quasi-periodic solutions and long time stability

25 The condition f j (I) = 0 is implied by Hamiltonian case: f (θ, I) = ( θ H)(θ, I) = ( θ H)(θ, I)dθ = 0 T Reversible vector field (Moser) θ = g(i, θ), İ = f (I, θ), f (I, θ) odd in θ, g(i, θ) even in θ = f (θ, I)dθ = 0 T Reversible vector field X(θ, I) = (g, f )(θ, I), X S = SX, S : (θ, I) ( θ, I)

26 The water waves equations (written in complex variables) are reversible with respect to the involution S : u ū that, on the subspace of functions u( x) = u(x), u(x) = j Z u j e ijx, u j = u j is Moser reversibility Indeed: (θ, I) ( θ, I) u(x) = j Z u j e ijx, u j = I j e iθ j Su = ū = ū j e ijx = ū j e ijx = j Z j Z j Z I j e iθ j e ijx

27 Alinhac good unknown" which has to be introduced to get energy estimates (local existence theory) preserves the reversible structure, not the Hamiltonian one

28 Remark 4) Global existence? Do these solutions exist for all times? We do not know. I expect typical scenario of quasi-integrable systems (Dyanchenko-ZakharoV, Craig-Workfolk, Craig-Sulem,... ) However there are many solutions defined for all times periodic solutions: Alazard-Baldi, 14 quasi-periodic solutions, B.-R.Montalto, 16, selection of initial conditions" which give rise to smooth solutions defined for all times

29 Quasi-periodic solution with n frequencies of u t = J H(u) Definition u(t, x) = U(ωt, x) where U(ϕ, x) : T n T R, ω R n (= frequency vector) is irrational ω k 0, k Z n \ {0} = the linear flow {ωt} t R is dense on T n The torus-manifold T n ϕ U(ϕ, x) phase space is invariant under the flow Φ t H of the PDE For n = 1 are time periodic Φ t H U = U Ψ t ω Ψ t ω : T n ϕ ϕ + ωt T n

30 Theorem (KAM for capillary-gravity water waves, B.-Montalto 16) For every choice of tangential" sites S N \ {0}, there exists s > S +1 2, ε 0 (0, 1) such that: for all ξ j (0, ε 0 ), j S, a Cantor like set G ξ [κ 1, κ 2 ] with asymptotically full measure as ξ 0, i.e. lim ξ 0 G ξ = κ 2 κ 1, such that, for any surface tension coefficient κ G ξ, the capillary-gravity water waves equation has a quasi-periodic standing wave solution (η, ψ) H s of the form η(t, x) = j S ξj cos( ω j t) cos(jx) + o( ξ) ψ(t, x) = j S ξj j 1 ω j sin( ω j t) cos(jx) + o( ξ) with frequencies ω j satisfying ω j ω j (κ) 0 as ξ 0. The solutions are linearly stable.

31 Linear stability -reducibility There exist coordinates (around the torus) (φ, y, v) T ν R ν (H s x L 2 S c ) in which the quasi-periodic solution reads t ( ωt, 0, 0) and the linearized equation reads φ = by ẏ = 0 v t = id v, v = j / S v je ijx, D := Op(µ j ), µ j R y(t) = y 0, v j (t) = v j (0)e iµ jt = v(t) H s x = v(0) H s x : stability 0, {iµ j } j S c = Floquet exponents

32 1 Sharp asymptotic expansion of the Floquet exponents µ j = λ 3 j 1 2 (1 + κj 2 ) λ1 j rj (ξ, κ) where λ 3, λ 1 R are constants satisfying λ λ 1 + sup j S c r j Cε a, a > 0, 2 The change of variables Φ(ϕ) satisfies tame estimates in Sobolev spaces: Φh s, Φ 1 h s h s + u s+σ h s0, s s 0

33 Gravity water Waves Periodic solutions: Plotnikov-Toland: 01 Gravity Water Waves with Finite depth Iooss-Plotnikov-Toland 04, Iooss-Plotnikov Gravity Water Waves with Infinite depth Completely resonant, infinite dimensional bifurcation equation Quasi-periodic solutions: Baldi, Berti, Haus, Montalto, in finite depth h Proved by Nash-Moser theorem

34 1 Main point of Nash-Moser iterative scheme: 1 Linearized PDE at an approximate solution. Reduce it to constant coefficients up to smoothing operators 2 KAM (Kolmogorov-Arnold-Moser) reducibility scheme to diminish the size of the perturbation 2 The proof of the long time existence result is a nonlinear analogue: 1 Paralinearization of the equations and paradifferential transformations to diagonalize and reduce it to constant coefficients up to smoothing remainders 2 Birkhoff normal form analysis for a reversible system

35 Major difficulty: water waves are quasi-linear PDEs Birkhoff normal form for Hamiltonian semilinear PDEs u t = Lu + P 2 (u), P 2 (u) = O(u 2 ) (1) 1 Bambusi, Grebert, Delort, Szeftel Goal: transform (1) into u t = Lu + P 3 (u), P 3 = O(u 3 ) (2) = the time of existence of a solution with u(0) = εu 0 becomes T = O(ε 2 )

36 Model semilinear PDE Reversible nonlinearity (D t m κ (D x ))u = P(u, ū), D t = 1 i m κ (ξ) = (ξ tanh ξ) 1 2 (1 + κξ 2 ) 1/2 t P(u, ū) = P(ū, u) P(u, ū) = a p,l u l ū p l a p,l is real Transforms under a map v = u + Q(u, ū), Q(u, ū) = M(u,..., u, ū,..., ū) }{{}}{{} l p l where M is p-linear map

37 Goal (D t m κ (D x ))u = a p,l u l ū p l (D t m κ (D x ))[u + Q(u, ū)] = (terms of order q > p) + (terms of order p that do not contribute to the H s -norm) (D t m κ (D x ))[u + M(u,..., u, ū,..., ū)] = a }{{}}{{} p,l u l ū p l + l p l l M(u,..., m κ (D x )u,..., u, ū,..., ū) j=1 p j=l+1 M(u,..., u, ū,..., m κ (D x )ū,..., ū) m κ (D x )M(u,..., ū) + higher order terms

38 Π n = spectral projector associated to the nth mode of on T m κ (D x )Π n = m κ (n)π n, u = n N Π nu To eliminate the terms homogeneous of degree p choose M so that Choice of M D κ,l (n 1,..., n p+1 )Π np+1 M(Π n1 u 1,..., Π np u p )= Π np+1 ( ap,l p j=1 Π n j u j ) small divisors D κ,l (n 1,..., n p+1 ) = l j=1 m κ (n j ) p+1 j=l+1 m κ(n j ) non-resonance condition D κ,l (n 1,..., n p+1 ) c(third largest among n 1,..., n p+1 ) N 0 for any n 1,..., n p+1 except in the case when p is odd, l = p + 1, and {n 1,..., n l } = {n l+1,..., n p+1 }, 2

39 Resonant terms The resonant terms that [ can not be eliminated are ] Π nl a l 1 2l 1,l j=1 Π n j u 2 Π nl u but it does not contribute to the energy, since a 2l 1,l is real. The actions Π n u 2 L are prime integrals of the resonant 2 system Above small divisor estimate and P semilinear imply that M is a bounded multilinear map on H s and u u + Q(u, ū) is bounded and invertible The main problem for quasi-linear PDEs is that u + Q(u, ū) is unbounded so it could not define a transformation of the same Sobolev space into itself and new vector field would loses more derivatives (formal expansions ex. Craig-Worfolk 94, Dyachenko-Zakharov 95)

40 Main idea: reduce the water waves system to a diagonal, constant coefficients paradifferential system up to bounded remainders before normal form procedure which reduces the size of the nonlinear terms. Steps: 1 Alinhac good unknown, Alazard, Burq, Metivier, Zuily 2 paracomposition operator 3 paradifferential operators steps (2)-(3) inspired to Alazard-Baldi and Berti-Montalto

41 Bony-Weyl quantization of a symbol with limited smoothness in x Op BW (a(u; x, ξ)) = Op W (a χ (u; x, ξ)) Weyl quantization of the regularized symbol a χ (x, ξ) = χ(d ξ 1 )a(, ξ) where χ is a cut-off function equal to 1 close to zero Weyl quantization Op W (b(x, ξ)) = 1 2π R R ( x + y e i(x y)ξ a 2 u(x) = einx n Z ûn 2π Op W (b(x, ξ)) = 1 ( 2π k n ), ξ u(y) dydξ ˆb(k n, n)û n ) eikx 2π

42 Symbols a Γ m p. m = order of symbol, p = size in O( u p ) a(u; x, ξ) = N 1 q=p a q(u; x, ξ) + a N (u; x, ξ) are polynomials in u up to a given order N and a N = O( u N ) 1 Homogeneous symbols: a q (U 1,..., U q ; x, ξ), U = [ ū u ], Π n = spectral projector on cos(nx) α, β N, n = (n 1,..., n q ) in (N ) q, α x β ξ a q(π n1 U 1,..., Π nq U q ; x, ξ) C n µ+α ξ m β Π n1 U L 2... Π nq U L 2 2 Non-homogeneous symbols : α, β in N, with α σ σ 0 x α β ξ a(u; x, ξ) C ξ m β U N 1 σ 0 U σ = a(u; x, ξ) is a symbol with limited smoothness if U H σ

43 The above definition is well posed up to Smoothing operators R ρ p R(V )U = N 1 q=p R q(v,..., V )U + R N (V )U 1 Homogeneous smoothing operators R q (V 1,..., V q ) 2 Non-homogeneous smoothing operators R ρ, ρ > 0, σ(ρ): U, V H s, s > σ, R(V )[U] H s+ρ s V N H σ U H s + V N 1 H V σ H σ U H σ

44 Action on Sobolev spaces of a para-differential operator Let a Γ m p. Then, there is σ > 0 such that for any U Ḣσ r, for any s R, Op BW (a(u; ))V Ḣs m C U ṗ H σ V Ḣ s Op BW (a) : Ḣs Ḣs m acts on homogeneous spaces modulo constants

45 Symbolic calculus Composition of paradifferential operators Let a Γ m p, b Γ m q. Then, for any U Ḣ σ, where R R m+m ρ p+q (a#b) ρ = Op BW (a) Op BW (b) = Op BW ((a b) ρ ) + R 0 l<ρ where 1 ( i l [ ] l! 2 σ(d x, D ξ, D y, D η )) a(x, ξ)b(y, η) x=y, ξ=η = ab + 1 {a, b} i σ(d x, D ξ, D y, D η ) = D ξ D y D x D η

46 Commutator Let a Γ m p, b Γ m q. [ Op BW (a), Op BW (b) ] = Op BW ( 1 i {a, b} + r 1) + R where the Poisson bracket: {a, b} := ξ a x b x a ξ b and R R ρ+m+m p+q, r 1 Γ m+m 3 p+q

47 Paralinearization of water waves First Change of variables (η, ω) Good-unknown of Alinhac ω = ψ Op BW (B(η, ψ))η Alazard-Metivier, Alazard-Burq-Zuily 1 We need multilinear expansion of the system. Thus the solution of Φ = 0 in { h < y < η(x)}, Φ y=η = ψ, y Φ = 0 at y = h not found as a variational solution, but via a parametrix a la Boutet de Monvel 2 This change of variable is reversibility preserving

48 u t = F (u), u = [ η ] ψ Reversible vector field F S = S F, S = [ ] Reversibility preserving change of variable Φ S = S Φ [ ηω ] [ 1 0 ][ η ] [ η ] = Op BW (B(η,ψ)) 1 ψ = Φ ψ is reversibility preserving B(η, ψ) is odd in ψ B is linear in ψ New system after the change of variable v = Φ(u) is reversible v t = X(v)

49 Complex variable u = Λ(D)ω + iλ 1 (D)η, Λ(D) = Water waves system ( D tanh(d) 1+κD 2 ) 1/4 D t U = Op BW (A(U; x, ξ))u + R(U)U, U = [ ū ] u, Dt = t ( i [ A(U; x, ξ) = m κ (ξ)(1 + ζ(u; x)) + λ 1/2 (U; x, ξ)) 1 0 ] 0 1 ( [ + m κ (ξ)ζ(u; x) + λ 1/2 (U; x, ξ)) 0 1 ] λ 1 (U; x, ξ) [ ] λ0 (U; x, ξ) [ ] m κ (ξ) = ( ξ tanh ξ(1+κξ 2 ) ) 1/2, ς(u, x) R, λj Γ j, Imλ j Γ j 1 1 the eigenvalues of the matrix A(U; x, ξ) are symbols with imaginary part or order 0 (actually - 1/2) 2 in (η, ψ) variables the eigenvalues would have imaginary part of positive order

50 Transformation rules Original PDE in paradifferential form: D t U = Op BW (A(U; x, ξ))u + R(U)U Paradifferential change of variable: W = Op BW (Q(U; x, ξ))u W H s U H s, s, U H σ r New PDE (it is still in paradifferential form) D t W = Op BW (A 1 (U, x, ξ)) ) W + R 1 (U)U A 1 (U, x, ξ) = D t Q(U) ( Q(U) ) 1 + Q(U) (A(U)) ( Q(U) ) 1 Choose Q(U) in order to have A 1 (U; x, ξ) which is diagonal and constant coefficients in x

51 Step 2. Diagonalization There exists a 2 2 matrix Q(U) of symbols of order 0, such that if W = Op BW (Q(U))U New system is diagonal, up to smoothing operators ( Dt Op BW (A (1) (W ; )) ) W = R(W )W where R R ρ and ( A (1) (U; x, ξ)= m κ (ξ)(1 + ζ (1) (U; x)) + λ (1) ) [ 1/2 (U; x, ξ) 1 0 ] λ (1) 1 (U; x, ξ)[ ] 1 0 where ζ (1) R, λ (1) j Γ j, Imλ (1) j Γ j 1 0 1

52 Goal: Reduce to constant coefficients in x the highest order part: iu t = Op BW ((1 + ζ(u; x))m κ (ξ))u Idea: use a change of variable x x + β(u; x) a diffeomorphism of T 1 so that the equation transforms into iu t = Op BW ((1 + β x (U; x)) 3/2 (1 + ζ(u; x)) κ ξ 3/2 +...)u Choice of β(u; x) β x (U; x) = (1 + β x (U; x)) 3/2 (1 + ζ(u; x)) = c(u) ( c(u) ) = determines c(u) and β(u; x) 1 + ζ(u; x)

53 Paracomposition Let x x + β(x) a diffeomorphism of T 1 Composition operator Φ β : u(x) u(x + β(x)) Remark: it mixes the low-high frequencies: b H s 0 < 1/2 u(x + β(x)) H s s u H s + b H s u H s 0 We want to associate to Φ β a para-composition operator Φ β which is the responsible above of just the contribution u H s

54 A definition of paracomposition operator We regard the change of variable u(x) u(x + β(x)) as a flow Homotopy This path is the flow of linear transport equation u(x) u(x + τβ(x)), τ [0, 1] τ u = b(τ, x) x u, b(τ, x) = τ u = Op(ib(τ, x)ξ)u Paracomposition operator Φ β : time one flow of τ u = iop BW (b(τ, x)ξ)u β(x) 1 + τβ x (x)

55 Paradifferential analogue of Egorov theorem. Using composition properties of paradifferential operators Proposition 1 Assuming β H s 0 < 1/2 then Φ β : H s H s, s, Φ βu H s C u H s 2 where (Φ β) 1( Op BW a(x, ξ) ) Φ β = ( Op BW α(x, ξ) ) + R α(x, ξ) = a ( x + β(x), ξ(1 + β y (y)) y=x+β(x) ) +... and R R ρ. y = x + β(x) x = y + β(y) Weyl quantization is convenient

56 Conjugation of t Proposition Let U be a solution of D t U = M(U)[U]. Then for any ρ large enough, we have Φ β t (Φ β) 1 = t + Φ β(φ β) 1 = t + Op BW (e(u; )) + R(U) where e is a symbol in Γ 1 with Re(e) in ΣΓ 1 and R(U; t) is a smoothing operator in R ρ 1. The conjugation with t gives a lower order term, order 1, All the transformations are determined by the spatial operator since ω(ξ) ξ 3/2 is superlinear

57 iu t = Op BW ((1 + ζ(u; x))m κ (ξ))u +... Under conjugation with the paracomposition operator we have iu t = Op BW ((1 + β x (U; x)) 3/2 (1 + ζ(u; x)) κ ξ 3/2 +...)u +... Choice of β(u; x) Reduction at highest order (1 + β x (U; x)) 3/2 (1 + ζ(u; x)) = c(u) ( D t Op BW( [ (1 + ζ(u))mκ (ξ) + λ(u; x, ξ) ][ ] )) V λ Γ 1/2 with Im λ Γ 1/2, = R(V ; t)v

58 Performing suitable paradifferential bounded non-linear changes of variables we transform (WW) into Reduction to constant coefficients up to smoothing remainders ( D t Op BW( (1 + ζ(u))m κ (ξ) [ ] ) ) H(U; ξ) U = R(U)U where R is a ρ-smoothing operator and H is a diagonal matrix of constant coefficients symbols (in x) of order one, such that ImH is of order zero. We deduce local existence of solutions with u(0) = ε with T ε = O(ε 1 ) = we are-back to a semilinear PDE situation

59 Normal form ( D t Op BW( (1 + ζ(u))m κ (ξ) [ ] H(U; ξ) ) ) U = R(U)U Transformation V = exp(op BW B(U; ξ))u, where B(U; ξ) = N 1 p=1 B p(u; ξ) and B p are homogenous in U Goal: new system ( D t Op BW( (1 + ζ(u))m κ (ξ) [ ] H1 (U; ξ) )) V = R 1 (V )V with Im(H 1 (U, ξ)) = O( U N ), Im(H 1 (U, ξ)) is of order 0

60 Using the equation D t U = m κ (D)U +... one finds that the coefficients of the multilinear form are determined by ( l p ) m κ (n j ) m κ (n j ) j=1 j=l+1 B p (Π + n 1 U 1,..., Π + n l U l, Π n l+1 U l+1,..., Π n p U p ; ξ) = iimh p (Π + n 1 U 1,..., Π + n l U l, Π n l+1 U l+1,..., Π n p U p ; ξ)

61 Small divisors Non-resonance condition l p+1 m κ (n j ) m κ (n j ) c (n 0,..., n p+1 ) τ (1) j=0 j=l+1 for any (n 0,..., n p+1 ) (N ) p+2 if p is odd or p is even and l p/2, and and for any (n 0,..., n p+1 ) (N ) p+2 such that {n 0,..., n l } {n l+1,..., n p+1 } if p is even and l = p/2. Frequencies: m κ (n) = n tanh(hn)(g + κn 2 ) It is fulfilled for most (g, κ). Tool: estimate of sublevels of sub-analytic functions, Delort-Szeftel 03

62 1 By reversibility the contribution when the divisor vanishes is not present 2 The small divisors are compensated by the para-differential regularization of Op BW (B p (u; ξ)) taking H s large enough 3 A similar normal form construction removes the smoothing remainder R(u) up to O( u N ) 4 The small divisors are compensated by the smoothing character of the remainder R(u) R ρ, s σ ρ Nτ

63 ( D t Op BW( (1 + ζ(u))m κ (ξ) [ ] H(U; ξ) ) ) U = R(U)U with Im(H(U, ξ)) is of order 0, Im(H(U, ξ)) = O( U N ), R(U) = O( U N ) is a regularizing operator = the solution can extended up to t O(ε N )