Attractors and the Spectrum of a Zeroth-order Pseudo-differential - PowerPoint PPT Presentation

Attractors and the Spectrum of a Zeroth-order Pseudo-differential Operator Miniconference on Sharp Eigenvalue Estimates for Partial Differential Operators Javier Almonacid Department of Mathematics, SFU April 4, 2020 Joint work with Nilima Nigam (SFU) Maciej Zworski (UC Berkeley)

Motivation: Internal Waves Figure: Picture taken from the International Space Station (ISS) in February, 2013. It shows the north coast of the island of Trinidad in the southeastern Caribbean Sea and the huge internal waves that are visible in the top left. Image credit: NASA/JPL. ◮ Internal waves play an important role in ocean circulation. Generated by tides and winds, they propagate through the oceans and seas, redistributing momentum and energy before dissipating. ◮ Topography of the ocean floor may play a role in dissipation and generation of internal wave attractors. 2 / 19

Motivation: Internal Waves ◮ Usual simplification: non-rotating, linearly stratified fluid. ◮ Extensively studied in 2D: Maas & Lam ’95, Ogilvie ’05, Lam & Maas ’08, Rieutord, Georgeot & Valdettaro ’01 . ◮ And in 3 D: Manders & Mass ’03 , Drijfhout & Maas ’07 , Pillet et. al. ’18 . Figure: L: Experimental setting. R: A internal ray beam path leads to an attractor in a plane transverse to the along-slope, down-canal direction into which rays are initially launched. Source: Pillet et. al., JFM 2018 . 3 / 19

Motivation: Internal Waves Figure: L: (Filtered) Velocity amplitudes at different slices of the tank obtained by Particle Image Velocimetry (PIV). R: The attractor at x = 250 mm, with the superimposed ray tracing prediction. Source: Pillet et. al., JFM 2018. 4 / 19

Forced Internal Waves in a Periodic Tank Main ingredients and simplifications: ◮ Linearly stratified fl uid with density ρ , small perturbations η , ◮ Incompressible fl uid ( ∇ · u = 0) and no- fl ux BC, ◮ (Linearized) Conservation of mass: ∂ t η + u · ∇ ρ = 0 , ◮ (Linearized) Conservation of momentum: ρ ∂ t u + η g e 3 + ∇ Π = F e − i ω 0 t , 5 / 19

Forced Internal Waves in a Periodic Tank Main ingredients and simplifications: ◮ Linearly stratified fl uid with density ρ , small perturbations η , ◮ Incompressible fl uid ( ∇ · u = 0) and no- fl ux BC, ◮ (Linearized) Conservation of mass: ∂ t η + u · ∇ ρ = 0 , ◮ (Linearized) Conservation of momentum: ρ ∂ t u + η g e 3 + ∇ Π = F e − i ω 0 t , ◮ If the box were periodic and F = 0 , replace pressure as Π = ( − ∆) − 1 g ∂ z η. 5 / 19

Forced Internal Waves in a Periodic Tank Main ingredients and simplifications: ◮ Linearly stratified fl uid with density ρ , small perturbations η , ◮ Incompressible fl uid ( ∇ · u = 0) and no- fl ux BC, ◮ (Linearized) Conservation of mass: ∂ t η + u · ∇ ρ = 0 , ◮ (Linearized) Conservation of momentum: ρ ∂ t u + η g e 3 + ∇ Π = F e − i ω 0 t , ◮ If the box were periodic and F = 0 , replace pressure as Π = ( − ∆) − 1 g ∂ z η. ◮ Domain: 2-torus p ( z , ξ ) → ξ 3 ◮ Symbol: � | ξ | − r β ( y , z ) (to account for topography). 5 / 19

The Problems of Interest in T 2 × ( 0 , T ] , i ∂ t u − P ( x , D ) u = fe − i ω 0 t � t = 0 = 0 , u � ◮ u = u ( x , t ) , periodic BC ◮ P ( x , D ) self-adjoint pseudo-differential operator, ◮ The symbol p ( x , ξ ) is homogeneous of degree 0, of form ξ 2 p ( x , ξ ) = 1 + | ξ | 2 − periodic function . � We would like to figure out: 6 / 19

The Problems of Interest in T 2 × ( 0 , T ] , i ∂ t u − P ( x , D ) u = fe − i ω 0 t � t = 0 = 0 , u � ◮ u = u ( x , t ) , periodic BC ◮ P ( x , D ) self-adjoint pseudo-differential operator, ◮ The symbol p ( x , ξ ) is homogeneous of degree 0, of form ξ 2 p ( x , ξ ) = 1 + | ξ | 2 − periodic function . � We would like to figure out: ◮ What is the long-time behaviour of solutions? 6 / 19

The Problems of Interest in T 2 × ( 0 , T ] , i ∂ t u − P ( x , D ) u = fe − i ω 0 t � t = 0 = 0 , u � ◮ u = u ( x , t ) , periodic BC ◮ P ( x , D ) self-adjoint pseudo-differential operator, ◮ The symbol p ( x , ξ ) is homogeneous of degree 0, of form ξ 2 p ( x , ξ ) = 1 + | ξ | 2 − periodic function . � We would like to figure out: ◮ What is the long-time behaviour of solutions? ◮ What is the spectrum of P? 6 / 19

The Problems of Interest in T 2 × ( 0 , T ] , i ∂ t u − P ( x , D ) u = fe − i ω 0 t � t = 0 = 0 , u � ◮ u = u ( x , t ) , periodic BC ◮ P ( x , D ) self-adjoint pseudo-differential operator, ◮ The symbol p ( x , ξ ) is homogeneous of degree 0, of form ξ 2 p ( x , ξ ) = 1 + | ξ | 2 − periodic function . � We would like to figure out: ◮ What is the long-time behaviour of solutions? ◮ What is the spectrum of P? ◮ Does the spectrum of P affect the long-term evolution? 6 / 19

Discretization Techniques ◮ Fourth-order explicit method in time, ◮ Fourier collocation method in space, ◮ In particular, the operator Q ( x , D ) of symbol q ( x , ξ ) = ( 1 + | ξ | 2 ) − 1 / 2 ξ 2 is discretized as � � � � k 2 Q ( x , D ) w ≈ F − 1 � F w N diag 1 + | k | 2 where F is the discrete Fourier transform and w N corresponds to w evaluated at the N 2 grid points in real space. 7 / 19

The Evolution Problem Theorem (Colin de Verdière-Saint Raymond ’20, Dyatlov-Zworski ’20) Let P be a self-adjoint, pseudo-differential operator of order zero and ω 0 / ∈ Spec pp ( P ) . Then, under certain additional assumptions on P , for any f ∈ C ∞ ( T 2 ) , the solution u to the evolution problem satisfies u ( t ) = e − i ω 0 t u ∞ + b ( t ) + ǫ ( t ) , where ◮ u ∞ = lim ε → 0 ( P − ω 0 − i ε ) − 1 f ∈ H − 1 / 2 − := ∩ δ> 0 H − 1 / 2 − δ ( T 2 ) and is not in L 2 ( T 2 ) except if it vanishes, ◮ � b ( t ) � L 2 ( T 2 ) < ∞ , ◮ ǫ ( t ) → 0 as t → ∞ in H − 1 / 2 − . 8 / 19

The Evolution Problem Figure: Attractors and linear evolution of the squared L 2 -norm. The forcing is a centered Gaussian, r = 0 . 25 and β ( x ) = cos( x 1 ) + sin( x 2 ) . Video available at youtu.be/3b5 UQfRcyEk. 9 / 19

The Evolution Problem Figure: Attractors and linear evolution of the squared L 2 -norm. The forcing is a centered Gaussian, r = 0 . 25 and β ( x ) = cos( x 1 ) + sin( x 2 ) . Video available at youtu.be/3b5UQfRcyEk. Theorem (cont.) Moreover, for t > 0, � u ( t ) � 2 L 2 ( T 2 ) ∼ ct except if u ∞ vanishes. 9 / 19

The Evolution Problem ◮ How the Fourier coefficients of u N ( t ) decay? − → Radial Energy Density � � � ( 1 + | k | 2 ) s | � x ∈ R 2 : R ≤ | x | < R + 2 1 G s ( R ) = u N ( k ) | 2 , A R = . N 2 k ∈ A R ∩ Z 2 Fourier Space 6 5 A R 4 3 2 R 1 k 2 0 R+2 -1 -2 -3 -4 -5 -6 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 k 1 10 / 19

The Evolution Problem 10 10 0 Radial Energy Density G O(1) 10 0 T=200 T=400 T=600 T=800 T=1000 10 -10 T=1200 T=1400 T=1600 T=1800 T=2000 10 -20 10 1 10 2 R Figure: If a distribution v ∈ H s , then the radial energy density G s should decay faster than O ( R − 1 ) as R → ∞ . 11 / 19

The Evolution Problem 10 10 -0.7 O(R -1.4 )) Radial Energy Density G 10 0 T=200 T=400 T=600 T=800 T=1000 10 -10 T=1200 T=1400 T=1600 T=1800 T=2000 10 -20 10 1 10 2 R 10 10 -1.5 O(R -2.9 ) Radial Energy Density G 10 0 T=200 T=400 T=600 T=800 T=1000 10 -10 T=1200 T=1400 T=1600 T=1800 T=2000 10 -20 10 1 10 2 R Figure: Radial energy density as T → ∞ . Top: s = − 0 . 7. Bottom: s = − 1 . 5. 12 / 19

Spectrum of P Theorem (Colin de Verdière ’20) Let J := [min p , max p ] , where p is the symbol of P . Then, 1. σ ess ( P ) = J , 2. P has a fi nite number of eigenvalues in J and they have fi nite multiplicity. 13 / 19

Spectrum of P Theorem (Colin de Verdière ’20) Let J := [min p , max p ] , where p is the symbol of P . Then, 1. σ ess ( P ) = J , 2. P has a fi nite number of eigenvalues in J and they have fi nite multiplicity. Theorem (Galkowski-Zworski ’ 19) Under certain assumptions on P , there exists an open neighbourhood of 0 in C , U and a set R ( P ) ⊂ { Im ( z ) ≤ 0 } ∩ U such that for every compact set K ⊂ U , R ( P ) ∩ K is discrete, and spec L 2 ( P + i ν ∆) − → R ( P ) , ν → 0 + , uniformly on K . Moreover, R ( P ) ∩ R = spec pp , L 2 ( P ) ∩ U . 13 / 19

Spectrum of P = 0.001 0 -0.2 -0.4 -0.6 -0.8 -1.5 -1 -0.5 0 0.5 1 1.5 1 50 100 150 200 Zoomed view -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 -0.2 -0.22 -0.24 -0.26 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Figure: Superposition of the fi rst 200 eigenvalues (ordered by magnitude, then phase) for several values of ν decreasing until ν = 10 − 3 . Each color represents a position in the vector of eigenvalues. As ν decreases, eigenvalues move up to the real axis. Video available at youtu.be/qeNRxWSptu0. 14 / 19

Viscous Approximations ξ 2 � � p ( x , ξ ) = 1 + | ξ | 2 − 0 . 5 cos( x 1 ) + sin( x 2 ) � Figure: L: Solution to the evolution problem. R: First 4 eigenfunctions of the viscous approximation, ν = 10 − 3 . 15 / 19

Viscous Approximations ξ 2 p ( x , ξ ) = 1 + | ξ | 2 − 2 cos( x 1 ) � Figure: L: Solution to the evolution problem. R: First 4 eigenfunctions of the viscous approximation, ν = 10 − 4 . 16 / 19