Vortex filament dynamics Walter Craig The Fields Institute - PowerPoint PPT Presentation

Vortex filament dynamics Walter Craig The Fields Institute McMaster University Legacy of Vladimir Arnold November 25, 2014 The Fields Institute

Work in collaboration with Carlos Garcia – Azpeitia The Fields Institute, McMaster University and UNAM, Mexico Chi-Ru Yang The Fields Institute, McMaster University and National Tsing Hua University, Taiwan Acknowledgements: NSERC, Canada Research Chairs Program, The Fields Institute

Outline Vortex filaments Natural questions in Hamiltonian dynamics Hamiltonian PDEs Invariant tori as critical points

Vortex filaments in R 3 ◮ one vortex filament: linear stationary, with uniform vortex strength γ = 1 b ( s ) = ( 0 , 0 , s ) It generates a flow field in R 3 described by u = ( ∂ x 2 ψ, − ∂ x 1 ψ, 0 ) where the velocity field is given by a stream function ψ = 1 2 log ( x 2 1 + x 2 2 ) = 1 2 log ( | z | 2 ) and z = x 1 + ix 2 are complex horizontal coordinates.

Vortex filament pairs Two exactly parallel linear vortex filaments evolve as described by point vortices in R 2 ◮ Opposite vorticity γ 1 = 1 = − γ 2 , initial configuration b 1 ( s ) = ( 1 b 2 ( s ) = ( − 1 2 a + i 0 , s ) , 2 a + i 0 , s ) then ballistic linear evolution 2 a + i t 2 a + i t b 1 ( s , t ) = ( 1 b 2 ( s , t ) = ( − 1 a , s ) , a , s ) ◮ Same vorticity γ 1 = 1 = γ 2 with the above initial configuration have circular orbits with angular frequency ω = a − 2 2 ae it / a 2 , s ) , 2 ae i ( t / a 2 + π ) , s ) b 1 ( s , t ) = ( 1 b 2 ( s , t ) = ( 1

◮ Question: Consider two near-vertical vortex filaments, slightly perturbed from exactly vertical. Do there persist similar orbital motions, whose time evolution is periodic or quasi-periodic. Configuration to be 2 π periodic in the vertical x 3 variables. ◮ In ‘center of vorticity’ coordinates, the horizontal separation of the two vortex filaments is w ( s , t ) = x 1 ( s , t ) + ix 2 ( s , t ) In a frame rotating with angular frequency ω w i ∂ t w + ∂ 2 s w − ω w + | w | 2 = 0 (1) the Klein, Majda & Damodaran model of near parallel vortex filaments. ◮ NB: For configurations which are greatly deformed from vertical, this is not an accurate approximation

Hamiltonian PDE ◮ This is a PDE in the form of a Hamiltonian system Set w = a + v ( s , t ) with a ∈ R and v ( s , t ) a perturbation term, a + v i ∂ t v + ∂ 2 s v − ω ( a + v ) + | a + v | 2 = 0 (2) by the choice ω = a − 2 then v = 0 is stationary ◮ The Hamiltonian is � 2 π 2 | ∂ s v | 2 + 1 2 a 2 | a + v | 2 − 1 2 log | a + v | 2 ds 1 H = (3) 0 Writing v ( s , t ) = X ( s , t ) + iY ( s , t ) the dynamics are given by Hamilton’s canonical equations ∂ t X = g rad Y H ∂ t Y = − g rad X H ◮ Small � v � H 1 solutions exist globally in time (C. Kenig, G. Ponce & L. Vega (2003), V. Banica & E. Miot (2012))

Linearized equations ◮ The tangent plane approximation is given by the linearization The linearized equations at equilibrium ( X , Y ) = 0 are derived from the quadratic Hamiltonian � 2 π ( ∂ s X ) 2 + ( ∂ s Y ) 2 + 2 H ( 2 ) = 1 � a 2 X 2 � ds (4) 2 0 ◮ Linearized equations ∂ t X = g rad Y H ( 2 ) = − ∂ 2 s Y s X − 2 ∂ t Y = − g rad X H ( 2 ) = ∂ 2 a 2 X

Linear flow ◮ Writing in a Fourier basis and using the Plancherel identity √ k ∈ Z ˆ 2 π ) � X k e iks X ( s ) = ( 1 / √ k ∈ Z ˆ Y k e iks 2 π ) � Y ( s ) = ( 1 / ( k 2 + 2 H ( 2 ) = � X k | 2 + k 2 | ˆ Y k | 2 � � a 2 ) | ˆ 1 2 k ∈ Z An infinte series of uncoupled harmonic oscillators, with k 2 + ( 2 / a 2 ) . � frequencies ω k ( a ) = ± k ◮ The solution operator, or the linear flow � X ( s , t ) � � X ( s , 0 ) � = Φ( t ) Y ( s , t ) Y ( s , 0 ) k 2 sin ( ω k t ) /ω k � � ˆ � � cos ( ω k t ) X k � e iks = ˆ − ω k sin ( ω k t ) / k 2 cos ( ω k t ) Y k k ∈ Z Angles evolve with linear motion θ k �→ θ k + t ω k

Elementary facts ◮ All solutions are Periodic, or Quasi-Periodic, or in general Almost Periodic functions of time ◮ More specifically, for initial data ( X 0 , Y 0 ) the active wavenumbers are K := { k : (ˆ k , ˆ X 0 Y 0 k ) � = 0 } The dimension of the frequency basis is � � m := dim Q span Q { ω k : k ∈ K } ◮ Orbit space consists of tori orbit ( X 0 , Y 0 ) = { Φ( t )( X 0 , Y 0 ) : t ∈ R } = T m Periodic (P): m = 1 Quasi-Periodic (QP): 1 < m < + ∞ Almost Periodic (AP): m = + ∞ NB: For generic a then ω k ( a ) satisfy 1 ≤ m ≤ + ∞

Elementary facts ◮ Energy is conserved �� k 2 + 2 � ˆ �� ˆ � X k 0 X k � H ( 2 ) ( X , Y ) = 1 a 2 ˆ ˆ 2 k 2 Y k 0 Y k k ∈ Z � k 2 + 2 � ˆ � ˆ � � � X k 0 X k � H ( 2 ) (Φ( t )( X , Y )) = 1 Φ( t ) T a 2 = Φ( t ) ˆ ˆ 2 k 2 Y k 0 Y k k ∈ Z ◮ Indeed each action variable is conserved k 2 + ( 2 / a 2 ) � | k | | X k | 2 + | Y k | 2 I k = k 2 + ( 2 / a 2 ) 2 | k | � 2 � k 2 + 2 � � � d 0 Φ k ( t ) T a 2 Φ k ( t ) = 0 k 2 dt 0 Hence all Sobolev energy norms are preserved H ( 2 ) = � � � ( X , Y ) � 2 | k | 2 r I k ω k I k , r := k ∈ Z k

Natural general questions 1. Whether any solutions of the nonlinear problem are Periodic, Quasi Periodic or Almost Periodic This refers to the KAM theory for PDEs 2. Whether the action variables I k ( z ) are approximately conserved (averaging theory), giving upper bounds on growth of action variables, or on higher Sobolev norms This is in the realm of averaging theory for PDEs, including Birkhoff normal forms and Nekhoroshev stability 3. Whether there exist some solutions which exhibit a growing lower bound on the growth of the action variables These would be cascade orbits, related to the question of Arnold diffusion

Results Theorem (C Garcia, WC & CR Yang (2012)) There exist Cantor families of periodic (i.e. m = 1 ) solutions of the vortex filament equations (2) near the uniformly rotating solution v = 0 Theorem (C Garcia, WC & CR Yang (in progress)) Given wavenumbers k 1 , . . . k m there is a set a ∈ A of full measure and an ε 0 = ε 0 ( a , k 1 , . . . k m ) such that for a Cantor set of amplitudes ( b 1 , . . . b m ) ∈ B ε 0 ⊆ C m there exist QP solutions of (2) with m-many Q independent frequencies Ω j ( b ) , of the form m b j e ik j s e i Ω j ( b ) t + O ( ε 2 ) � v ( s , t ) = j = 1 Actually, these two theorems hold for any central configuration of vortices. The case of more complex configurations of near-vertical vortices is part of our future research program.

Hamiltonian PDEs ◮ Flow in phase space , where z ∈ H a Hilbert space with inner product � X , Y � H , z ( x , 0 ) = z 0 ( x ) , ∂ t z = J g rad z H ( z ) , (5) ◮ Symplectic form J T = − J . ω ( X , Y ) = � X , J − 1 Y � H , ◮ The flow z ( x , t ) = ϕ t ( z 0 ( x )) , defined for z ∈ H 0 ⊆ H ◮ Theorem The flow of (5) preserves the Hamiltonian function: H ( ϕ t ( z )) = H ( z ) , z ∈ H 0 Proof: d dt H ( ϕ t ( z )) = � g rad z H , ˙ z � = � g rad z H , J g rad z H ( z ) � = 0 .

Invariant tori ◮ Mapping a torus S ( θ ) : T m θ �→ H to be flow invariant S ( θ + t Ω) = ϕ t ( S ( θ )) Angles evolve linearly, with frequency vector Ω ∈ R m ◮ This implies that both ∂ t S = Ω · ∂ θ S , and ∂ t S = J grad z H ( S ) hence Ω · ∂ θ S = J grad z H ( S ) (6) ◮ Problem of KAM tori: Solve (6) for ( S ( θ ) , Ω) . This is generally a small divisor problem. Rewrite (6) in self-adjoint form J − 1 Ω · ∂ θ S − grad z H ( S ) = 0 . (7)

Invariant tori - linear theory - small divisors ◮ The tangent space approximation for the mapping S Linearize at S , set δ S = Z and use the self adjoint form Ω · J − 1 ∂ θ Z − ∂ 2 z H ( S ) Z = F (8) k 2 + ( 2 / a 2 ) . � Frequencies of the linearized flow are ω k = ± k ◮ The eigenvalues of (8), the linearized operator for a solution with m temporal quasi-periods Ω = (Ω 1 , . . . Ω m ) ∈ R m � jk := k 2 + 1 (Ω · j ) 2 + 1 λ ± a 2 ± a 4 Eigenvalues λ ± jk are the small divisors. Analysis: resolvant expansion methods developed by Fr¨ ohlich & Spencer, WC & Wayne, Bourgain, Berti & Bolle, . . .

Space of torus mappings Consider the space of mappings S ∈ X := { S ( θ ) : T m �→ H} ◮ Define average action functionals � 1 T m � S , J − 1 ∂ θ j S � d θ I j ( S ) = 2 J − 1 ∂ θ j S δ S I j = The moment map for mappings ◮ The average Hamiltonian � H ( S ) = T m H ( S ( θ )) d θ δ S H = grad z H ( S )

A variational formulation Consider the subvariety of X defined by fixed actions M a = { S ∈ X : I 1 ( S ) = a 1 , . . . I m ( S ) = a m } ⊆ X Variational principle: critical points of H ( S ) on M a correspond to solutions of equation (7), with Lagrange multiplier Ω . NB: All of H ( S ) , I j ( S ) and M a are invariant under the action of the torus T m ; that is τ α : S ( θ ) �→ S ( θ + α ) , α ∈ T m .

Two questions ◮ Two questions. 1. Do critical points exist on M a ? Note that the following operators are degenerate on the space of mappings X : Ω · J − 1 ∂ θ S , Ω · J − 1 ∂ θ S − δ 2 S H ( 0 ) 2. How to understand questions of multiplicity of solutions? ◮ Answers – proposal in some cases: 1. Use infinite dimensional KAM theory or the Nash – Moser method, with parameters The latter relies on solutions of the linearized equations, via resolvant expansions (Fr¨ ohlich – Spencer estimates) 2. Morse – Bott theory of critical T m orbits.