Long time existence of space periodic water waves

Transcription

1 Long time existence of space periodic water waves Massimiliano Berti SISSA, Trieste IperPV2017, Pavia, 7 Settembre 2017 XVII Italian Meeting on Hyperbolic Equations

2 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)

3 Zakharov formulation 68 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

4 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))

5 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,...

6 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}

7 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}.

8 Prime integral: mass T η(x)dx η H s 0(T) := { 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

9 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

10 Linear water waves system [ [ η ] t ψ = 0 G(0) g+κ xx 0 ] [ η ] ψ Complex variables Linear WW u = Λ(D)η + iλ 1 (D)ψ, Λ(D) = u t + iω(d)u = 0, ω(d) = ( g+κd 2 D tanh(hd) ) 1/4 D tanh(hd)(g + κd 2 ) Dispersion relation ω(ξ) = ξ tanh(hξ)(g + κξ 2 ) κ ξ 3/2

11 -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

12 Nonlinear water waves Major difficulty: Nonlinear Water Waves are a quasi-linear system u t + iω(d)u = N(u, ū) N = quadratic nonlinearity with derivatives of the same order N = N( D 3/2 u)

13 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,...

14 Main result: Joint work with Jean-Marc Delort Long time existence Birkhoff normal form result: M. Berti- J-M. Delort, 17, For any small initial condition the solutions are defined for long times For the same system recent KAM results: Existence of periodic solutions: Alazard-Baldi, 15, Arch. Rational Mech. quasi-periodic solutions, B.-Montalto, 16, Memoires AMS solutions defined for all times, for "most" initial conditions

15 Almost global existence Theorem (M.B., J-M.Delort, 2017) 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

16 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-Tataru, 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)

17 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 ω j = λj tanh(hλj)(g + κλ 2 j 2 ) 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 ) Maybe analytic initial data?

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

19 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-angle 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

20 Averaging principle": The dynamics of the actions is expected to be İ j = ε f j (ε, I), f j (ε, I) := f j (ε, θ, I)dθ T the average with respect to θ = (θ j ) 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

21 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 = S X, S : (θ, I) ( θ, I)

22 The water waves equations (written in complex variables) are reversible with respect to the involution S : u ū that, on the subspace of functions is Moser reversibility u( x) = u(x), (θ, I) ( θ, I) Alinhac good unknown" which has to be introduced to get energy estimates (local existence theory) preserves the reversible structure, not the Hamiltonian one

23 Remark 4) Global existence? Do these solutions exist for all times? We do not know. I would 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, 15 quasi-periodic solutions, Berti-Montalto, 16, selection of initial conditions" which give rise to smooth solutions defined for all times

24 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

25 Ideas of Proof 1 Quadratic nonlinearity u t = iω(d)u + P 2 (u), P 2 (u) = O(u 2 ) Time of existence of solution with u(0) = εu 0 is T ε = O(ε 1 ) 2 Cubic nonlinearity u t = iω(d)u + P 3 (u), P 3 = O(u 3 ) Time of existence of solution with u(0) = εu 0 is T ε = O(ε 2 ) Normal form idea, ex. Shatah 84 for PDEs Look for a change of variable s.t. the nonlinearity becomes smaller 1 For Hamiltonian semilinear PDEs Birkhoff normal form Bambusi, Grebert, Delort, Szeftel 02-07, 2 For quasi-linear PDEs such transformation is unbounded

26 New procedure: 1 Reduce the water waves system to a diagonal, constant coefficients paradifferential system up to smoothing remainders 2 Implement a normal form procedure which reduces the size of the nonlinear terms that could produce a growth of the Sobolev norm

27 1) Introducing Alinhac good unknown, paracomposition, and paradifferential non-linear changes of variables, bounded in H s, we transform (WW) into u t = iop BW ( (1 + ζ 3 (u))ω κ (ξ) + ζ 1 (u) ξ 1/2 + r 0 (u; ξ) ) u + R(u)u where Op BW denotes the Bony-Weil quantization and 1 ω κ (ξ) = (ξ tanh(ξ)(1 + κξ 2 )) 1/2, linear dispersion relation 2 ζ 3 (u), ζ 1 (u) are real valued, of size O(u), constant in x, 3 r 0 (u; ξ) is a symbol of order 0 constant in x 4 R(u) is a regularizing operator: for any ρ it maps H s H s+ρ, for s large, R(u)[u] H s+ρ C u H ρ u H s = we are back to a semilinear PDE situation

28 Correct notion" of normal form for quasi-linear systems: expansion in constant coefficient (nonlinear) operators with decreasing order: u t = iop BW ( (1 + ζ 3 (u))ω κ (ξ) + ζ 1 (u) ξ 1/2 + r 0 (u; ξ) ) u + R(u)u one could deduce 1 Local existence of solutions 2 Long time existence: semilinear Birkhoff normal form 3 Quasi-periodic solutions via semilinear KAM theory 4... etc The above reduction uses that the dispersion relation ω κ (ξ) ξ 3/2, superlinear, not for quasi-linear Klein-Gordon, Delort 10-14

29 The PDE u t = iop BW ( (1 + ζ 3 (u))ω κ (ξ) + ζ 1 (u) ξ 1/2 + Re(r 0 (u; ξ) ) u preserves for all times t R the L 2 x and H s x norms because the symbol (1 + ζ 3 (u))ω κ (ξ) + ζ 1 (u) ξ 1/2 + Re(r 0 (u; ξ)) is real (self-adjointness) and has constant coefficients in x Goal Perform Birkhoff normal form transformations to eliminate the size of Im(r 0 (u; ξ)) and R(u) up to O( u N ) = the u(t) H s norm of the solution with u(0) = O(ε) remains bounded up to times t O(ε N )

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

31 1 To eliminate R(u,..., u)[u], we look for a transformation }{{} p close to the identity of the form Id + Ψ(u,..., u) uniquely }{{} p determined by R(u,..., u) dividing by the divisors l p ω κ (n j ) ω κ (n j ) c (n 1,..., n p ) τ j=1 j=l+1 2 There is one term that we do not eliminate because the divisor vanishes. By reversibility it does not contribute to the growth of the Sobolev norm. Example: u j = iω j u j + ia j u j 2 u j, j Z, a j R. 3 Ψ(u,..., u) is (ρ τ) regularizing Ψ(u,..., u) H s+ρ τ 4 To perform N steps: we require s ρ Nτ C u p 1 H u ρ H s