Lecture 4: Birkhoff normal forms

Transcription

1 Lecture 4: Birkhoff normal forms Walter Craig Department of Mathematics & Statistics Waves in Flows - Lecture 4 Prague Summer School 2018 Czech Academy of Science August

2 Outline Two ODEs Water waves with and without surface tension Birkhoff normal forms Resonances Formal Birkhoff normal forms Mapping properties of normal forms transformations Critique

3 Contrast two ODEs Quadratic case ż = z 2, z(0) = ε z(t) = ε 1 εt, T = 1 ε Cubic case ẇ = w 3, w(0) = ε w(t) = ε 2 1 2ε 2 t, T = 1 2ε 2 The general time of existence does not change when these ODE are replaced by ż = iωz + z 2, ẇ = iωw + w 3

4 Free surface water waves Incompressible and irrotational flow u = 0, u = 0 which is a potential flow in the fluid domain S(η) u = ϕ, ϕ = 0 Fluid domain S(η): h < y < η(x, t), Bottom boundary conditions N ϕ = 0 x R d 1 Free surface conditions on y = η(x, t) t η = y ϕ x η x ϕ kinetic BC t ϕ = gη 1 2 ϕ 2 Bernoulli condition

5 Free surface water waves with surface tension Incompressible and irrotational flow u = 0, u = 0 which is a potential flow in the fluid domain S(η) u = ϕ, ϕ = 0 Fluid domain S(η): h < y < η(x, t), Bottom boundary conditions N ϕ = 0 x R d 1 Free surface conditions on y = η(x, t) with surface tension t η = y ϕ x η x ϕ kinetic BC t ϕ = gη 1 2 ϕ 2 + σκ(η) Bernoulli condition where κ(η) is the mean curvature of the free surface

6 Zakharov s Hamiltonian The energy functional H = K + P = x η(x) y= h Zakharov s choice of variables 1 2 ϕ 2 g dydx + 2 η2 + σ ( x η ) dx x z := (η(x), ξ(x)), where ξ(x) := ϕ(x, η(x)) That is ϕ = ϕ[η, ξ](x, y) Expressed in terms of the Dirichlet Neumann operator G(η) 1 H(η, ξ) = 2 ξg(η)ξ + g 2 η2 + σ ( x η ) dx

7 Taylor expansion of G(η) Given by the expression G(η)ξ = j 0 G (j) (η)ξ where each G (j) (η) is homogeneous of degree j in η Explicitely, for D x := i x G (0) ξ(x) = D x tanh(h D x )ξ(x) G (1) (η)ξ(x) = D x ηd x G (0) ηg (0) ξ(x) G (2) (η)ξ(x) = 1 2 (G(0) η 2 D 2 x + D 2 xη 2 G (0) 2G (0) ηg (0) ηg (0) )ξ(x) Accordingly, the Hamiltonian is analytic, with expansion 1 H(η, ξ) = R d 1 2 ξg(0) ξ + g 2 η2 + σ 2 xη 2 dx + H (j) (η, ξ) j 3 H (j) (η, ξ) = 1 2 R ξg(j 2) (η)ξ + σs j x η j dx d 1

8 Taylor expansion of the Hamiltonian From the analyticity of G(η) and an explicit Taylor expansion of 1 + x η 2 H = H (2) + H (3) + H (4) +... = 1 2 ξg (0) ξ + gη 2 dx + m ξg (m 2) (η)ξ + σs m x η m dx The water wave equations linearized about (η, ξ) = 0 are ( ) η t = J grad ξ η,ξ H (2) (η, ξ) namely t η = D x tanh(h D x )ξ t ξ = gη + σ 2 x η A harmonic oscillator with frequencies ω(k) = (g + σ k 2 ) k tanh(h k )

9 Birkhoff normal form Normal form - transform the equations to retain only essential nonlinearities ( ) η τ : z = w ξ in a neighborhood B R (0) H r Conditions: 1. The transformation τ is canonical, so the new equations are t w = J grad H(w), H(w) = H(τ 1 (w)) 2. The transformed Hamiltonian is H(w) = H (2) (w) + ( Z (3) + + Z (M)) + TR (M+1) where each Z (m) retains only resonant terms {H (2), Z (m) } = 0 The new Hamiltonian H(w) is conserved by the flow ϕ t (w)

10 Significance of the normal form This transformation procedure and reduction to Birkhoff normal form is part of the theory of averaging for dynamical systems Consider x T d 1 with periodic boundary conditions on a torus Fourier transform variables and complex symplectic coordinates 1 (η k, ξ k ) := T d 1 1/2 T e ik x d 1 (η(x), ξ(x)) dx z k := 1 ( Qk η k + iq 1 ) k ξ k, Qk = 4 2 g + σ k 2 k tanh(h k ) Action angle variables z k = I k e iθ k I k = z k 2 A Hamiltonian h(i) in action variables alone is integrable t Θ = I h(i), t I = Θ h(i) = 0, Θ(t) = Θ(0) + t I h(i) I(t) = I(0) Such flows conserve each I k, hence every Sobolev norm z(t) 2 r = k k 2r z k (t) 2 = k k 2r I k = z(0) 2 r

11 Reasons to study the normal form Long - time existence If Z (3) + + Z (M) are integrable, then the existence time for small initial data w(t) B ε (0) H s is T ε O(ε (M 1) ) Zakharov s theory of wave turbulence. The reduction of a Hamiltonian PDE to its resonant manifold is a normal forms transformation. In Zakharov s notation, a(x) := 1 ) (Q(D x )η(x) + iq 1 (D x )ξ(x) b(x) 2 In KAM theory the Arnold condition depends upon the normal form special solutions. Resonant terms Z (3) +... Z (M) describe an averaged system, which often has particular solutions of interest. Example: Wilton ripples and three wave resonances

12 Resonant terms Cubic resonances: It is well known in the folklore of fluid dynamics that when surface tension σ = 0 there are no three wave interactions. Namely ω k1 ± ω k2 ± ω k3 = 0, k 1 + k 2 + k 3 = 0 implies that at least one of k 1, k 2, k 3 = 0. This means that formally Z (3) = 0 Quartic resonances: Four wave interactions are nontrivial In deep water h = + with d = 2 and frequencies ω k = g k ω k1 ± ω k2 ± ω k3 ± ω k4 = 0, k 1 + k 2 + k 3 + k 4 = 0 has integrable solutions ω k1 + ω k2 ω k3 ω k4 = 0, {k 1, k 2 } = {k 3, k 4 } and nonintegrable Benjamin - Feir resonant interactions k 1 : k 2 : k 3 : k 4 = n 2 : (n + 1) 2 : n 2 (n + 1) 2 : (n 2 + n + 1) 2 ω 1 : ω 2 : ω 3 : ω 4 = n : (n + 1) : n(n + 1) : (n 2 + n + 1)

13 Formal fourth order Birkhoff normal form Theorem (Dyachenko, Lvov & Zakharov (1994), WC & Worfolk (1995)) The Benjamin - Feir resonant interaction terms vanish, specifically c k1 k 2 k 3 k 4 = 0 whenever k 1 : k 2 : k 3 : k 4 = n 2 : (n + 1) 2 : n 2 (n + 1) 2 : (n 2 + n + 1) 2 ) ω 1 : ω 2 : ω 3 : ω 4 = n : (n + 1) : n(n + 1) : (n 2 + n + 1) Set I 1 (k) = 1 2 (z k z k + z k z k ) and I 2 (k) = 1 2 (z k z k z k z k ). The formal Birkhoff normal form up to quintic residual is integrable H + = ω k I 1 (k) 1 k 3( I 1 (k) 2 3I 2 (k) 2) 2π k k + 4 I 2 (k 1 )I 2 (k 4 ) + R (5) (w) π k 4 < k 1 =H (2) (I) + H (4) (I) + R (5) (w)

14 Mapping properties of normal form transformations A PDE question: mapping properties of the transformation τ: is the transformation well defined, on which Banach spaces Theorem (Properties of τ (3) ) C. Sulem & WC (2016) Fix d = 2, h = + and s > 3/2. There exists R 0 > 0 such that for any R < R 0, the transformation τ (3) is defined on B R (0) H s η H s ξ τ (3) : B R (0) B 2R (0) (τ (3) ) 1 : B R/2 (0) B R (0) Other results on the transformation of three-wave interactions for water waves WC (1996) Canonical normal forms on scales of analytic spaces S. Wu (2009) Transformations in Lagrangian coordinates P. Germain N. Masmoudi & J. Shatah (2012) geometric coordinates J. Hunter, M. Ifrim & D. Tataru (2014) ad hoc methods in holomorphic cordinates A. Ionescu & F. Pusateri (2015) global solutions for x R 1

15 Fourth order resonance relations Define the energy space E r := H r η H r+1/2 ξ Quartet interactions are indexed by {(k 1, k 2, k 3, k 4 ) Z 4 : Σ 4 j=1k j = 0} Due to Theorem 1 the resonant set is R = {k 1 k 4, k 2 k 3 > 0 : k 1 +k 2 = 0 = k 3 +k 4 or k 1 +k 3 = 0 = k 2 +k 4 } A quasihomogeneous neighborhood of R is a set of near-resonant modes C + R :={(k 1, k 2, k 3, k 4 ) Z 4 : Σ 4 j=1k j = 0 satisfying k 1 + k 2 < ( k 1 + k 2 ) 1/4 and k 3 + k 4 < ( k 3 + k 4 ) 1/4 } The neighborhood C R Z4 is similar with k 2 k 3

16 Fourth order partial Birkhoff normal form Theorem (WC & Sulem (2016)) Let Q Z 4 be a set of quartet interactions, such that Q\B ρ (0) C ± R = ρ < + is symmetric under (k k), (k 2 k 3 ) and (k 1 k 4 ). Then for r > 3/2 there exists a canonical transformation τ (4) Q on B R (0) E r such that τ (4) Q : H(2) + H (4) + R (5) H = H (2) + Z (4) + R (5) such that supp Z (4) C ± R. For (k 1, k 2, k 3, k 4 ) R then Z (4) k 1,k 2,k 3,k 4 = Z (4) k 1,k 2,k 3,k 4 (I)

17 Surface tension σ > 0 Theorem (C. Sulem & WC (2015)) In the case of positive surface tension, with 0 < h +, a similar statement holds for r > 1, namely z τ (3) : B R (0) Hη r+1 H r+1/2 ξ Hη r+1 possible that Z (3) is nonzero (Wilton ripples). H r+1/2 ξ. However it is NB Furthermore τ (3) is smooth on a scale of Hilbert spaces. That is, in the case with surface tension the Jacobian maps energy spaces z τ (3) : H r+1/2 η H r ξ Hr+1/2 η H r ξ In the case w/o surface tension the Jacobian maps z τ (3) : H s 1 η H s 1 ξ H s 1 η H s 1 ξ NB: The transformation mixes the domain η and the potential ξ.

18 Critique This talk in on work in progress for several reasons Existence theorems depend upon solving the initial value problem in special variables. Nalimov (1971) shows that this is possible in Lagrangian coordinates, as in the work of S. Wu In Eulerian coordinates such proofs depend upon Alinhac s good variables, as in the work of D. Lannes Question: is there a systematic symmetrization for the water wave equations, that is independent of coordinates Mapping properties of normal forms in the case σ = 0 and 0 < h < + are not included In the case of a variable bathymetry h(x), periodic for example, the dispersion relation is replaced by the Bragg frequencies ω h (k) Properties of the normal forms transformations τ (M) on energy spaces H s η H s+1/2 ξ

19 Thank you