Lecture 1: Water waves and Hamiltonian partial differential equations

Transcription

1 Lecture : Water waves and Hamiltonian partial differential equations Walter Craig Department of Mathematics & Statistics Waves in Flows Prague Summer School 208 Czech Academy of Science August

2 Outline... the neverchanging everchanging water... Ulysses James Joyce Free surface water waves Symplectic forms and Hamiltonian PDEs Hamiltonian PDE - examples Quasilinear wave equations Nonlinear Schrödinger equations Shallow water equations Boussinesq equations Korteweg devries equations A Hamiltonian for overturning waves

3 Euler s equations and free surface water waves Fluid domain S(η) := {x R d, y ( h, η(x))}, d = 2, 3 Incompressibility and irrotationality therefore u = ϕ where u = 0, u = 0 ϕ = 0 On the solid bottom boundary of S(η) N u = 0 On the free surface Γ(η) := {y = η(x)} t η = y ϕ x η x ϕ t ϕ = gη 2 ϕ 2,

4 Figure : Great waves off the Oregon coast

5 kinetic and potential energy The energy H of the system of equations is η(x) η(x) H = K + P := R d 2 u 2 dydx + gy dydx h R d h η(x) = R d 2 ϕ 2 g dydx + h R d 2 η2 dx C, Rewrite the kinetic energy by integrating by parts η(x) K = R d 2 ϕ 2 dydx = h R d h + R d 2 ϕn ϕ ds bottom + η(x) 2ϕ ϕ dydx R d 2 ϕn ϕ ds free surface Define ξ(x) := ϕ(x, η(x)), the kinetic energy is therefore K = R d 2 ξn ϕ ds free surface = R d 2ξG(η)ξ dx where G(η) is the Dirichlet Neumann operator.

6 Dirichlet Neumann operator The Dirichlet Neumann operator G(η)ξ(x) = ( y x η(x) x )ϕ(x, η(x)) = R(N ϕ)(x, η(x)) where R = + x η 2 is a normalization factor so that G(η) is self-adjoint on L 2 (dx). The Hamiltonian H = R d 2 ξg(η)ξ + g 2 η2 dx Theorem (Zakharov (968)) The pair of functions (η(x), ξ(x)) are canonical variables for the water waves problem in which it can be written in Darboux coordinates, with Hamiltonian H(η, ξ).

7 Hamiltonian formulation Therefore the equations for water waves can be rewritten in Darboux coordinates; η = grad ξ H = G(η)ξ () ξ = grad η H = gη grad η K The expressions for K and grad η K involve derivatives of G(η) with respect to perturbations of the domain S(η). Proposition Let η C. Then G(η) satisfies:. G(η) is positive semidefinite. 2. It is self-adjoint (on an appropriately chosen domain). 3. G(η) maps H (R d ) to L 2 (R d ) continuously. 4. As an operator G(η) : H (R d ) L 2 (R d ) it is analytic in η B R (0) C (R d ).

8 ZCS system Rewriting the system for water waves in these coordinates t η = G(η)ξ t ξ = gη grad η K(η, ξ) Variations of the kinetic energy with respect to the free surface η(x) (the shape derivative) are given by grad η K(η, ξ) = ( 2( + x η 2 x ξ 2 (G(η)ξ) 2 ) 2( x η x ξ)g(η)ξ + ( x η 2 x ξ 2 ( x η x ξ) 2)) Variational formula of Hadamard (90), (96), related to the derivative of the Green s function with respect to the domain. Among other things, this is a useful formulation for numerical simulations

9 Simulations using the Dirichlet Neumann formulation 0.4 η/h I t/(h/g)/ x/h Figure : Head-on collision of two solitary waves, case S/h = 0.4 W. Craig, J. Hammack, D. Henderson, P. Guyenne & C. Sulem, Phys. Fluids 8, (2006)

10 proof by first principles of mechanics The Lagragian for surface water waves L := K P expressed in the tangent space variables (η, η). Use the kinematic condition η = y ϕ x η x ϕ = G(η)ξ The Lagrangian is thus L(η, η) = 2 R d ηg (η) η g 2 η2 dx The canonical conjugate variables are precisely those given by Zakharov. (η, η L) = (η, G (η) η) = (η, ξ)

11 Taylor expansion of G(η) The operator G(η)ξ(x) is analytic, 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 () (η)ξ(x) = D x ηd x G (0) ηg (0) ξ(x) G (2) (η)ξ(x) = 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 H(η, ξ) = R d 2 ξg(0) ξ + g 2 η2 dx + 2 j 3 R ξg(j 2) (η)ξ dx d = j 2 H (j) (η, ξ)

12 Hamiltonian PDE Phase space taken to be v M a Hilbert space with inner product X, Y M, for X, Y T(M) Symplectic form given by a two form ω(x, Y) = X, J Y M, J T = J for tangent vectors X, Y T(M). Antisymmetry J T = J Hamiltonian vector field X H defined through the relation dh(y) = ω(y, X H ) = grad v H, Y M for all Y T(M) t v = Jgrad v H(v), v(x, 0) = v 0 (x) The flow v(x, t) = ϕ t (v(x)), defined for v M 0 M Poisson brackets between H and other functions F are given by the expressions {H, F}(v) := d F(ϕ t (v)) = grad dt v H(v), Jgrad v F(v) M t=0

13 Poisson brackets and conservation laws Express a conservation law using the Poisson bracket t K(η(t, ), ξ(t, )) = {H, K} := grad η K grad ξ H grad ξ K grad η H dx Conservation of mass M(η) = η dx {H, M} = grad η M grad ξ H grad ξ M grad η H dx = G(η)ξ dx = G(η) ξ dx = 0 Momentum I(η, ξ) = η x ξ dx t I = {H, I} = 0 Energy H(η, ξ) t H = {H, H} = 0

14 example 2: the quasilinear wave equation Quasilinear wave equations 2 t u u + t G 0 ( t u, x u) + Σ j xj G j ( t u, x u) = 0 (2) with initial data u(0, x) = f (x) and t u(0, x) = g(x) This equation can be written as a Hamiltonian PDE. The Lagrangian is L = 2 u2 2 xu 2 + G( u, x u) dx Use the Legendre transform to change to canonical conjugate variables p := u L = p( u, x u) The Hamiltonian is now H(u, p) := 2 p2 + 2 xu 2 + R(p, x u) dx where R(p, x u) = G( u, x u) for u = u(p, x u)

15 quasilinear wave equation (cont.) Equation (2) can be rewritten as t u = grad p H(u, p) = p + p R t p = grad u H(u, p) = u Σ j xj uxj R(p, x u) which is in Darboux coordinates. Theorem (local existence theorem) Systems of equations of this form are symmetrizable hyperbolic systems. Therefore for Sobolev data of sufficient smoothness r > 0 a solution exists locally in time (u 0 (x), p 0 (x)) H r (R d ) (u(t, x), p(t, x)) = ϕ t (u 0, p 0 ) C([ T, +T] : H r (R d )) := M 0 L 2 (R d )

16 Example 3: NLS Nonlinear Schrödinger equation On R d or on a domain T d = R d /Γ, for period lattice Γ i t u 2 xu + Q(x, u, u) = 0 (3) with Hamiltonian H NLS (u) = Rewritten 2 u 2 + G(x, u, u) dx, u G = Q t u = i grad u H NLS In many cases the Schrödinger equation admits a gauge symmetry under phase translation, in which case G = G(x, u 2 )

17 example 4: Shallow water equations Shallow water equations Model equations for water waves t η = x ((h + η) x ξ ) (4) t ξ = gη 2 xξ 2 This is a Hamiltonian PDE with Hamiltonian H SW = 2 (h + η(x)) x ξ 2 + gη 2 dx The mathematically rigorous study of the shallow water limit of the equations of water waves was initiated by T. Kano & T. Nishida (979). The work of D. Lannes (2000) extended their results from the analytic category to to Sobolev spaces, and to more subtle limiting situations such as the Green - Naghdi system

18 example 5: Boussinesq systems Boussinesq system on spatial domain D R The Hamiltonian is H Boussinesq (p, q) = T 2 p2 + g 2 q2 ± 2 ( xp) 2 + G(x, q) dx The symplectic form is given by J Boussinesq = ( 0 ) x x 0 ṗ = x grad q H Boussinesq (q, p) = x (gq + q G) (5) q = x grad p H Boussinesq (q, p) = x (p 6 2 x p + p G) The sign is badly ill-posed (McKean, the bad Boussinesq), while the + sign is well posed (the good Boussinesq). Completely integrable nonlinear cases include G = p 3 (Zakharov) and G = 2 qp2 (Kaup, Sachs)

19 Example 6: the generalized KdV Korteweg de Vries equation t r = 6 3 x r x ( r G(x, r)), x T (6) The Hamiltonian is H KdV (r) = T 2 ( xr) 2 + G(x, r) dx Rewritten t r = J grad r H KdV, where J = x Completely integrable cases are G = r 3 and G = r 4 Rigorous studies of water waves in the KdV scaling regime and limit are given in Craig (985), T. Kano & T. Nishida (986) and others

20 Water waves Hamiltonian in general coordinates Following a question of T. Nishida in 206 Fluid domain Ω(t) R 2 with free surface γ(t, s). Evolution determined by free surface conditions T txy N txy = 0, p(t, γ(t, s)) = 0 Energy = kinetic + potential H = K + P, K = 2 Potential energy P = Ω V dydx, Ω ϕ(x, y) 2 dydx V = (0, gy2 2 ) Dirichlet Neumann operator, with ϕ(γ(s)) = ξ(s) G(γ)ξ(s) := N ϕ(γ(s)), K = 2 ξ(s)g(γ)ξ(s) ds γ γ

21 Legendre transform Lagrangian L = K P The kinematic boundary condition states that N γ = N ϕ(γ) = G(γ)ξ Decmpose the vector field γ(s) along the curve γ = (γ (s), γ 2 (s)) in terms of its Frenet frame (T(s), N(s)) n(t, s) = N γ(t, s), τ(t, s) = T γ(t, s) then the Lagrangian is L = 2 n(t, s)g (γ)n(t, s) ds γ The Legendre transform γ δ γ L = G (γ)n(t, s) = ξ(s), H = 2 ξ(s)g(γ)ξ(s) ds γ + γ γ γ V N ds γ g 2 γ2 2(s) sγ (s) ds γ s γ

22 Hamilton s canonical equations variations δγ and δξ of the Hamiltonian N γ = n = δ ξ K = G(γ)ξ the kinematic boundary conditions Decomposing boundary variations δγ = (N δγ) N + (T δγ) T variations of the potential energy δ γ P δγ = δ N δγ P δγ = gγ 2 (s) N δγ ds γ Finally δ γ K has normal and tangential components, with the result that ( t ξ = gγ 2 2 s γ 2 ( sξ) 2 (G(γ)ξ) 2 ) s γ sξτ The tangential component τ = T γ depends upon the manner in which the surface is parametrized, which in most cases imposes a holonomic constraint γ

23 Thank you