Galina Perelman: 2 soliton collision in NLS

$$
i \psi_t = – \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R
$$
where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$

This family of equations has solitary wave solutions
$$
e^{i \theta(x,t) \phi (x – b(t), E)}
$$
where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, …

If I set $\epsilon^2 = E$ and write $\phi(y, \epsilon^2) = \epsilon \hat{\phi}(\epsilon, \epsilon).$ We have then that $\hat{\phi}(z, \epsilon) = \phi_0 (z) + O(\epsilon^2)$ where $\phi_0$ is the standard soliton for cubic NLS. A calculation shows that
$$\| \phi(\cdot, \epsilon^2) \|_{H^1} = O(\epsilon^{1/2}).$$ Let’s collect the parameters $\sigma = (\beta, E, b, v) \in R^4.$

The question I’d like to address:

Question: As $t \rightarrow -\infty$, suppose that $\psi(t) = w(\cdot, \sigma_0 (t)) + w(\cdot, \sigma_1 (t)) + o_{H^1} (1)$. Because of the galilean invariance we can arrange so that $\sigma_0$ does not move and we assume that $v_1 > 0$. So, we can arrange this data to have completely decoupled solitons as $t \rightarrow – \infty$. The question is then to understand the soliton collision and also what happens afterwards.

Perturbative regime:
$$\epsilon^2 = E_1 \ll 1, E_0 \thicksim 1, v_1 \thicksim 1.$$

Collision Scenario:

  1. $w(\cdot, \sigma_0 (t))$ is ‘preserved’.
  2. $w(\cdot, \sigma_1 (t))$ splits into two outgoing waves of the cubic NLS. The splitting is controlled by the linearized operator associated to the large soliton $w_{\sigma_0}$.

Collision: $|t| \lesssim \epsilon^{-1-\delta}, ~ \delta > 0$.
pre-interaction: $t leq – \epsilon^{-1-\delta}$
post-interaction: $t leq – \epsilon^{-1-\delta}$

She draws a picutre:

Long wide soliton to the left of a big soliton at the origin before the collision. After the collision the small soliton splits into two waves, one moving left and one moving right. The big soliton at the origin is drawn not centered at the origin.

$s = s(\frac{v_1}{2}), r = r (\frac{v_1}{2})$ where $s(k), r(k)$ are the translation and reflection coefficients of the linearized operator corresponding to $w(\cdot, \sigma_0 (t))$. Here we have $|s|^2(k) + |r|^2 (k) =1$. The only trace of nonlinearity appears in the phase.

This phenomena has been observed before by Holmer-Mazuola-Zworski and earlier by physicists.
H-M-Z conisdered the cubic NLS with an external delta potential. For small incoming solitons, they have observed the small soliton splitting caused by the Dirac function potential.

Hypotheses:

(H0): $F \in C^\infty, F(\xi) = – 2 \xi + O(\xi^2), \xi \rightarrow 0.$
$F(\xi ) \geq – C\xi^q, C>0, q<2, \xi \geq 1$. (GWP in $H^1$)
$\exists !$ ground state.

Linearization around $w(x, \sigma(t)) = e^{i\theta} \phi(x – b(t), E)$. We substitute $\psi = w + f$ and expand to obtain the following equation for $f$:

$$
i {\bf{f}}_t = L(E) {\bf{f}}.
$$

Here ${\bf{f}}$ is a (column) vector $(f, \overline{f})$.
$$
L(E)= (-\partial_y^2 + E) \sigma_3 + V(E).
$$
Here $\sigma_3$ is the Pauli matrix and $V$ is a certain matrix involving $V_1 = F(\phi^2) + F’ (\phi^2) \phi^2$ and $V_2 = F’ (\phi^2) \phi^2$.

She draws a spectral plane. Essential spectrum along real line in region $|x| > E$ and some eigenvalues drawn as x’s inside the gap and one above and below the real line on the imaginary axis. 0 is an eigenvalue. We have two explicit eigenfunctions $\xi_0$ and $\xi_1$.

$M(E)$ is the generalzied null space of $L(E)$. We have the following equivalence:

$$\sigma(L(E)) \subset R, {\mbox{dim}} M(E) = 4 \iff \frac{d}{dE} \| \phi(E) \|_2^2 > 0.$$

These conditions imply the orbital stability of $\Phi$.

$Lf = \lambda f, ~\lambda \geq E, \lambda = E + k^2, ~ k \in R$. If $k^2 + I \notin \sigma_p (L(E))$ then $\exists ~! ~ f(x,k) = s(k) e^{i k x} (1, 0)^t + O(e^{-\gamma x})$ as $ x\rightarrow + \infty, ~ \gamma > 0$ and $f(x,k) = e^{ikx} (1,0)^t + r(k) e^{-ikx}(1,0)^t + O(e^{\gamma x}), x \rightarrow – \infty$.

$w(x,\sigma, t), ~ j=0,1$ normalized as before.

(H1):
$$\frac{d}{dE} \| \phi(E) \|2^2 |{E=E_0} > 0$$

(H2): $\epsilon^2 = E_1$ sufficiently small

(H3): $M(E + \frac{v_1^2}{4}) \notin \sigma_p (L(E_0))$ (Nobody knows how to prove no embedded eignevalues.)

Proposition: $\exists ~! ~ \psi \in C(R, H^1)$ such that ….

Theorem: For $\epsilon^{-1-\delta} \leq t \leq \delta \epsilon^{-2} | \ln \epsilon |$

$$ \psi (t) = w (\cdot, \sigma(t)) + \psi_+ (t) + \psi_{-} (t) + h(t)$$

  1. $\sigma(t) = (\beta(t), E_0, b(t), v_0), ~V_0 = \epsilon \kappa$ where $\kappa$ is an explicit constant and
    $$
    |\beta(t) – \beta_0 (t)|, |b(t) – v_0 t| \leq C \epsilon^2 t.
    $$
  2. $\Psi_{\pm} (x,t) = ….ack too fast to type…
    $$\Psi_{\pm}$$

is expressed as an explicit phase times a function $S^{\pm}$ which solves cubic NLS emerging from data built using thre reflection, transmission coefficients and $\phi_0 (y)$.
3. error estimates in terms of $\epsilon.$


Edriss Titi: Loss of smoothness in 3d Euler Equations

(joint work with Claude Bardos)

Overview:

  1. Background
    • Euler
    • Classical
    • Nonuniqueness: De Lellis – Sh…
  2. Shear flow
    • DiPerna Majda example: weak limit of Euler solutions whose limit is not a solution
    • Illposedness of Euler in C^{0,\alpha}
  3. Vortex sheets induced by 3d shear flows
    • Examples
    • Differences between 2d and 3d Kelvin-Helmholtz problems
    • Comments on numerics

Euler equations

Euler equations on the 3-torus. $\omega$ is the vorticity. Recast using Biot-Savart.

Vorticity stretching term distinguishes 2d and 3d.

Classical Wellposedness:

  • global existence and uniquenes for initial data $\omega_0 \in L^\infty$.
    This result is due to Yudovich (1963). Some extension….
  • For data in $C^{1,\alpha}$, Euler equations are short time well-posed and the solution conserves energy. [Lictenstein (1925)]
  • The same result holds the context of Sobolev spaces $H^s, ~ s > \frac{5}{2}$. (Basically same result in more modern spaces)

Question: Does there exist a regular solution (say in $C^{1,\alpha}$) of the 3d Euler equation that becomes singular in a finite time (blows up problem)? This is in osome sense as difficult as the millenium problem. There are different opinions….

“I spoke with Necas about this…near end of his life…on Wendesday’s he thinks it blows up and on Thursdays he thinks no…so he has bad dreams about it…”

DeLellis-Szekelyhidi: There exists a set of initial data $u_0 \in L^2 (\Omega)$ (not explicitly constructed, Baire argument) for which the Cauchy problem has, for the same inital data, an infinite family of weak solutiosn of the 3d Euler equations: a residual set in the space $C(R; L^2_{weak} (\Omega))$.

These are also in $L^\infty$ so they have finite energy. (Built on Shnirelman and others….). This is a breakthrough…but it is not so physical. Maybe a selection mechanism….for NS we don’t have such a result. Leray solutions are not known to be unique. Any result like this for NS would be extremely important….connect it with turbulence. The lack of uniqueness, according to Leray, relates to turbulence.

Shear flows:

$$u(x,t) = (u_1 (x_2), 0, u_3 (x_1 – t u_1 (x_2))).$$

For $u_1, u_3 \in C^1$, the above shear flow is a classical solution of the Euler equations with pressure $p=0$. Yudovich used these to show the existence of solutions with exponentially growing high regularity norms.

This example due to DiPerna-Majda (1987).

Theorem (DiPerna-Lions): Norm explosion in $W^{1,p}$ for Euler, for any $p \geq 1$.

Idea of the proof:
$ \partial_{x_2} u_3 (x_1 – t u_1 (x_2))=…$

Theorem: The shear flow is a weak solution of the Euler equations in the sense of distribtuions in $R^3$, provided $u_1, u_3 \in L^2_{loc} (R^3)$. On the periodic box, we can do same thing and in this case we have finite energy.

Why do I stress the finite energy? This relates to the Onsager conjecture.

Theorem [Ill-posedness of the Euler equations in $C^{0,\alpha}$]:

The shear flow with $C^{1,\alpha}$ components $u_1, u_3$. However, for $u_1, u_3 \in C^{0,\alpha}$ then the above shear flow is always in $C^{0, \alpha^2}$ which is a much larger space. We instantly lose the $C^{0, \alpha}$. There exists a shear flow which starts in $C^{0, \alpha}$ which, at any positive time, is not in $C^{0, \beta}$ for any $\beta > \alpha^2$.

This family of solutions is compactly supported in space and time.

Other spaces and optimal spaces:

There are many layers of spaces between these H”older spaes. He writes a tower of inclusions between $C^{1,\alpha } \subset C^{0, \alpha}$. In fact, there is well-posedness
[Pak and Park] vs. failure of wp in $B^1_{\infty, infty}$ (Zygmund class) and failure in certain Triebel-Lizorkin spaces.

Weak limit of oscillating initial data:

DiPerna-Majda example…

Shear flow with vorticity interface. Vortex sheet flows are irrotational off an interface. To build such solutions he takes $u_1, u_3$ as (parametrized) Heaviside functions.

…wow…this talk is coming pretty fast, slides are changing…I stop typing and start to just try to keep up.

Numerical investigation of blowup for the 3d Euler

John Gibbon gave a talk a few years ago on the history of these investigations. Tom Hou and Bob Kerr are competing and disagreeing in this direction….is there a singularity…maybe not?

Question: Does the soluton of the following PDE blow up?
$$
\partial_t u – \nu \Delta u = |\nabla u |^4?
$$

What would you try numerically to determine if it blows up or not? You can even collapse it to the corresponding 1d problem?

Postlude Discussion:
Yudovich explored the DiPerna-Lions shear flow examples to see that norms measuring high regularity can grow exponentially in time. Chemin has studied the vortex patch and shown some measures of regularity of the boundary of the patch grow doubly exponentially fast. It was not explicitly clear to me yet how to relate Chemin’s rough patch boundary example to the growth of norms measuring regularity of the solution. Also, Chemin’s examples emerge from non-smooth initial data. I remain interested in the question: Does there exist nice data for 2D Euler which evolves with high regularity norms growing doubly exponentially?


Benoit Grébert: Hamiltonian Interpolation for Approximation of PDEs.

(joint work [Grébert-Faou] with Erwan Faou)

Aim:

Take a PDE with solution u. Consider a numerical approximation $u^n$ built with a symplectic integrator which approximates $u(nh)$. We build a hamiltonian $H_h$ such that
$$u^n = \Phi_{Hh}^{nh}(u_0) + very ~small.$$

I am concerned with the long time behavior of the numerical trajectory.

My concern right now is not in estimating the quality of the approximation. Instead, I want to understand the numerical flow.

Outline:

  1. Finite dimensional Context (ODE)
  2. PDE Context
  3. Ideas of the proof (time permitting)

Finite Dimensional Context

We go back to Moser’s theorem. A discrete symplectic map close to the identity can be approximated by a Hamiltonian flow. Consider an analytic symplectic map
$$
R^{2n} \ni (p,q) \longmapsto \Psi(p,q) \in R^{2n}
$$
with $\Psi = Id + O(\epsilon)$. Then $\exists~ H_\epsilon$ such that
$$\Psi = \Phi_{H\epsilon}^\epsilon + O(e^{-\frac{1}{c\epsilon}}).$$
([Moser 1968], [Benettin-Giorgilli 1994])

Numerical Context: Suppose I have a Hamiltonian ODE system
$$ (\dot{p}, \dot{q}) = X_H (p,q)
$$
and an associated numerical discrete-time-step symplectic integrator
$$
(p_n, q_n)= \Psi_h^n (p_0, q_0).
$$
We then have that $\Psi_{h} = \Phi_{Hh} + O(e^{-1/ch}).$ We obtain that $H_h (p_n, q_n) = H_h (p_0, q_0) + n e^{-1/ch}$. So, we are observing that the modified energy is essentially conserved for exponentially long times.

Backward Error Analysis

PDE Context

$$ H = H_0 + P$$
Here we imagine $H_0$ is the linear part and P is the nonlinear part. As an example, consider the cubic NLS on $T^d$. We can treat other equations as well. Let’s recall the Hamiltonian formalism in the Fourier variables:

Expand $u$ to get
$$
u = \sum \xi_j e^{ijx}, ~ {\overline{u}}= \sum \eta_j e^{-ijx}.
$$
We can then write, for each $j \in Z^d$,
$$
{\dot{\xi}} = -i \frac{\partial H}{\partial \eta}
$$
$$
{\dot{\eta}} = i \frac{\partial H}{\partial \xi}.
$$
For the cubic NLS case, we obtain
$$
H = \sum |j|^2 \xi_j \eta_j + \sum^* \xi_{k_1} \xi_{k_2} \eta_{l_1} \eta_{l_2}
$$
where $\sum^*$ is the sum over all the parameters subject to the constraint $k_1 + k_2 = l_1 + l_2$.

The problem we face here is that the linear part is unbounded, and we have infinitely many dimensions as first obstructions in passing from the ODE to the PDE context.

Splitting Method:
$$\Phi_{P+H0 } \thicksim \Phi_p^h \circ \Phi_{H0}^h =^? \Phi_{Hh}^h.$$

First naive idea: Use the Baker-Campbell-Haussdorf formula. We can then expand as a Lie series…
to write
$$\Phi_p^h \circ \Phi_{H0}^h = e^{h\mathcal{L}p}e^{h\mathcal{L}H0} = e^{h\mathcal{Hh}}
$$
with $H_h = H_0 + P + \frac{h}{2}{ P, H_0 } + \dots.
$$

To proceed, we will need conditions $small = h^N C(N,, \| num sol \|_H^N)$ NOT FAIR! So we need to work harder.

First Idea:
Replace $hH_0$ by $A_0$ by cutting off to low frequencies. We can splt and impose the CFL condition.
Midpoint + split. He considers different cutoffs.

We then consider $\Phi_p^h \circ \Phi_{A_0}^1$.

Second Idea: Use the Wiener Algebra. Space of functions with Fourier coefficients in $l^1$.


Theorem (Grébert-Faou): For the approximation scheme $\Phi_p^h \circ \Phi_{A_0}^1$ there exists a (polynomial) modified energy $H_h$ such that

$$
\| \Phi_p^h \circ \Phi_{A_0}^1 (\xi, \eta) – \Phi_{Hh}^h(\xi, \eta) \|_{l^1} \leq h^{N+1} (cN)^N
$$

uniformly for $\|(\xi, \eta)\|_{l^1} \leq M.$

So, assuming that the numerical trajectory is bounded in $l^1$ (as opposed to the stronger claim that it is bounded in $H^k$ for $k$ large) then
$$
H_h (u^n) = H_h (u_0) + Cn h^{N+1}.
$$


Of course, I have to explain: what is $N$? This is related to a regularization condition. We know that $N = \frac{r-2}{r_0 – 2}$ where $r_0$ is the degree of $P$ (so 4 for cubic NLS). The parameter $r$ is determined by the condition:
$ \forall ~ j = 1, \dots, r$ and for any $j$-tuple of integers $(k_1, \dots, k_j) \in Z^d$, we have
$$|\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj}| \leq 2 \pi.$$

CFL: $|\lambda_k | \leq C$.

He describes some examples where $N = 3, 4$ and $N=7$.

For cubic NLS, we end up obtaining
$$Hh = \frac{1}{h} A_0 + Z_1 + h Z_2 + \dots
$$
where
$$
Z_1 = \sum^* \frac{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})}}{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})} – 1}.
$$
You can now see how the zero divisor issue emerges and is resolved.


,


Sergei Kuksin (École Polytechnique): Nonlinear Schrödinger Equation

We consider Hamiltonian PDE. This is of course very interesting. In physics, there is a class of pdes which is also of interest:

Hamiltonian PDE = small damping + small forcing

Why is it so important?

  1. This class contains a very important equation: Navier-Stokes.
    $$
    \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0.
    $$
    We are interested in cases $d = 2,3.$ For $d=3$, this problem seems impossible. So, let’s collapse to the 2d case.
  2. Nonlinear Schrödinger equation with some damping and forcing
    $$
    \dot{u} + i \Delta u – i |u|^2 u = \epsilon \Delta u + force.
    $$
    Similarly, we might want to study the PKdV equation
    $$
    \dot{u} + u_{xxx} + u u_x = \epsilon u_{xx} + force.
    $$

We are interested in the small viscosity $\epsilon \ll 1$ and $t \rightarrow \infty$ extremes. At least we want to study $t \gtrsim \epsilon^{-1}$.

Two papers on my web page:

We introduce the slow time $\tau = \epsilon t$.

Perturbations of linear Hamiltonian PDEs

$$
\frac{\partial u}{\partial \tau} + i \epsilon^{-1} (- \Delta u + V(x)u ) = \Delta u – \gamma_R |u|^{2p}u – i \gamma_I |u|^{2q} u + (random force).
$$
Both of the parameters $\gamma > 0$ and satisfy $\gamma_R^2 + \gamma_I^2 =1$. The parameters $p,q$ are natural numbers, possibly 0. WE weill look at the case $d=1$ on $x \in [0,\pi]$ with Dirichlet boundary conditions.

Some more information about the random force,
$$
(random force) = \frac{d}{d\tau} \sum_{j=1}^\infty b_j \beta_j (\tau) e_j (x)
$$
Here the $\beta_j$ are complex valued standard, independent random variables.

We will work in the Sobolev space $H^2$.

Theorem 1: If $u_0 \in H^1$ then $\exists ~! ~ u^\epsilon (\tau, x)$ such that
$$
E ( \|u\|1^2 + \int0^\tau \| u(s)\|_2^2 ds) < \infty.
$$

Let $u_0^\omega \in H^1$ be a random.

  • Let $\mathcal{P}(u_0^\omega) = \mu$ denote the measure in H^1
  • Calculate $u^\omega (\tau)$.

Definition: A measure $\mu$ is called a stationary measure if $\forall ~ \tau$ we have $\mathcal{P} (\mu_\tau ) = \mu.$

Bogolyubov-Krylov: A stationary measure almost always exists.

Theorem 2 (Hairer, Odasso, AS): If $b_j \neq 0 ~ \forall ~j$ then $\exists ~ !$ stationary measure $\mu^\epsilon.$ For any solution $u(\tau)$, we have
$$
\mbox{dist} (\mathcal{P}(u(\tau)), \mu^\epsilon) \rightarrow 0 ~\mbox{as}~ \tau \rightarrow 0.
$$

The measure $\mu_\epsilon$ depends upon the force but not on the data.

Fourier Tranform

For the operator $A = – \Delta + V(x)$ consider the eignefunctions $\phi_1, \phi_2, \dots$ with associated eigenvalues $\lambda_1, \lambda_2, \dots$. Assume that

  1. $\lambda_1 >0$
  2. $\lambda \cdot s \neq 0 ~ \forall s \in {\mathbb{Z}}^\infty, ~ 0 <|s| < \infty.$

For any $u \in H^1$, we can expand $u$ w.r.t. the basis and denote the associated coefficients by $v_1, v_2, \dots$. The Fourier transform is the map $u \longmapsto v$ and the inverse goes the other way.

We can pass from $v_j$ to polar coordinates $I_j, \phi_j$. He recasts the dynamics w.r.t the polar coordinate variables and started speaking about averaging lemmas.

Effective Equations

These objects are somehow analogs of the kinetic equations in the theory of weak turbulence….some notation….I want to understand this better….an average of the nonlinear potential energy term. This is a semilinear heat equation with a nonlocal heat equation. The term proportional to $\gamma_I$ does not influence the effective equation. This equation really takes complete control when $\epsilon$ is very small.

The advance obtained here uses randomness in the forcing. “I expect that the effective equation is relevant even without the randomness but I don’t know how to prove it.”


J. Colliander: Numerical Simulations of Radial Supercritical Defocusing Waves

My slides


F. Bouchet (ENS-Lyon): Invariant measures

Collaborators:

  • A. Venaille
  • E. Simonnet
  • H. Morita
  • M. Corvellec

Physical phenomena. I am interested in self-organization in turbulent flows. Examples: stripes and spots on Jupiter. Ocean currents. Height differences in ocean surface. Stable jets.

I will mainly speak about the 2d Navier-Stokes equation with random forcing. This is not such a good model for these phenomena. There are others that are quite similar that might be better to describe the phenomena listed above like the quasigeostrophic and shallow water layer models.

Equilibrium will be related to 2D Euler. For 2D, we have the vorticity-stream formulation. Steady solutions to the Euler equation satisfying $\omega = f(\psi)$ or, equivalently, ${\bf{u}} \cdot \nabla \omega = 0,$ play a crucial role in describing the dynamics. Degeneracy: what is the selection mechanism leading to $f$? The main advance is that $f$ can be predicted using equilibrium statistical mechanics ideas.

Outline:

  1. Invariant measures of the 2D Euler equation
    • Equilibrium stat mech
    • applications of equilibrium stat mec
    • invariant measures of the 2d euler equation
  2. Irreversible relaxation of the 2D Euler equations
    • irreversibility in fluid mechanics
    • …..slide switched….ack
  3. 2D stochastic Navier-Stokes equation: non-equilibrium phase transitions

Statisitical mechanics for 2d and geopphysical flows.

Statistical equilibrium. very old idea. famous contributions

  • Onsager 1949
  • Joyce-Montgomery 1970
  • Caglioti Marhioro plvirenti lions 1990
  • Robert-Sommeria 1991
  • Miller 1991
  • Eyink-Spohn 1994

Robert-Sommeria-Miller (RSM) theory:

The most probable vorticity field. We want to measure the number of microscopic fields $\omega$ which correspond to a probabiility $\rho$. The number of such configuarations is quantified by the Boltzmann-Gibbs Entropy. This is the mixing entropy. Microcanonical RSM variational problem. Critical points are startionary flows of the QG model.

Microcanonical measures for Hamiltonian systems:

  • Hamilton’s equations
  • Liouville Theorem
  • Define the microcanonical measures which are the natural invariant measures taking into account the constraints in the dynamics.

Detailed Liouvilles thEorem for 2D Euler:

Lee 1952, Kraichnan JFM 1975, Robert 2000

We want to take into account the casimirs and the constraints. He describes a limiting process based on galerkin approximations. Mean field behavior? Large deviations? Sanov theorem?

……lots of discussion…..ideas vs. proofs…..nontribvial…what’s going on? Audience is confusing me…speaker seems clear.

Young measures….entropy…

The claim is that the theory he and his collaborators hav developed explains the emergence and stabiltiy of coherent structures like the great spot on Jupiter. Similar statements about ocean structures.

Are microcanonical measures invariant measures for the 2D Euler dynamics? Is the setof invariant Young measures for the 2D Euler dynamics larger than the set of microcanonical measures?

Two conjectures:

  • Weak perturbations of the 2D Euler equations close to steady states converge to invariant Young measures.
  • The 2D Euler equations converge to invariant Young measures.

Wave breaking is an irreversible mechanism in fluids that does not require viscosity.


Sebastian Reich: Data Assimilation

Data Assimilation

Nature Physical Laws

Measurements Model

        Data

        Optimal prediction

He drew arrows between these frameworks of understanding and highlights the assembly of processing at the data assimilation level.

Sequential Data Assimilation in a nutshell.

Model + Observations $\longmapsto$ Prediction

Ingredients of Data Assimilation:

  1. Mathematical and numerical model. solutions and their undertainties caused by approximation errors as well as state and parameter undertainties.
  2. Data/observations with measurements as well as approximation (forward operators) errors –> Inverse problems
  3. Numerical approximations to the data assimilation problem within a statistical (Bayesian) framework, assessment of the induced predictions and their uncertainties.

Mathematical problem statement

Consider an evolution problem for which the initial state is treated as a random variable with some given probability density function. For simplicity assume finite-d phase space. The uncertainty in the initial conditions will generally lead to unpredictability over long time intervals. Weather prediction is a nice example.

To counterbalance this increase in uncertainty, we collect observations at discrete times subject to some random measurement errors. We wish to find a trajectory that makes optimal use of the available information in terms of initial data, observations and model dynamics. The task of data assimilation is to combine the model, the measurements and then we want to make the optimal prediction.

Theoretical solution

i) Model dynamics

Lift the dynamics to the level of the Liouville equation on the probability distribution function.

ii) Data assimilation

Assimilate data using Bayes’ theorem
$$\pi (x|y) \thicksim \pi(y|x) X \rho_{pr} (x).
$$

Here $\pi(x|y)$ is the know conditional PDF (likelihood) for observing $y$ given a state $x$. Given an actual measurement, we can correct and proceed.

Under Bayes’ theorem, we always reuce uncertainty.

Ensemble Prediction…

ack….slides are changing fast.

Particle filter. We give better weight to points that are closer to the observed data. If we repeat this a few times, there will be very few particles contributing to the final answer.

Assimilation as a continuous deformation of probability: McKean-Vlasov

We can think of Bayes theorem as an optimal transportation problem.

Crisan-Xiong 2010 did something similar in the context of continuous time filter problem.

Otto 2001 for an application in gradient flow dynamics.

We started with an ODE, spoon fed the measurement data to update the dynamics, and encounter a more complicated dynamical description of the system. We encounter a McKean-Vlasov system, a modified Liouville equation, which is closed by an elliptic PDE.

Numerical filter implementations will now rely on appropriate approximations to the lliptic PDE. We use the ensemble of solutions to define an appropriate statistical model and then solve via numerics or by quadrature.

Obvious choices for the numerical version of $\rho$ include a Gaussian PDF parameterized by the ensemble mean and covariance matri (ensemble Kalmna filter) or Gaussian mxture modes.


N. Faou: 2d Submarines

2D Euler equation on 2-torus….I was a bit tired and did not type notes during this talk.


 

,

These are notes from a meeting entitled Advanced Numerical Studies in Nonlinear PDEs in Edinburgh, Scotland.

Walter Craig (McMaster): Water Wave Interactions

I’m an analyst but I’m going to talk about numerics and experiments as well as analysis. We will discuss the problem of water waves and then I’ll talk about two specific settings in which the theory has led to good and quite elegant numerics and the numerics have started to answer some questions.

(joint work with P. Guyenne and C. Sulem)

Outline

  • Free surface water waves
  • Hamiltonian PDEs
  • Periodic Traveling wave patterns
  • Solitary wave Interactions
  • The KdV scaling limit

Free surface water waves

Euler’s equations of hydrodynamics, incompressible and irrotational flow. This is therefore given as a potential flow. The irrotational assumption is really an oceanographers assumption. Of course, there is vorticity but we follow the models of oceanographers.

The fluid domain is $-h < y < \eta (x,t)$. So, the domain is changing. Free surface boundary conditions hold on $y = \eta (x,t)$.

Zakharov’s Hamiltonian

  • The energy functional

$$ H = K+P $$
$$
K = \int_x \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi|^2 dy dx.
$$
$$
P = \int_x \frac{g}{2} \eta^2 dx.
$$

This could also include surface tension effects.

  • Zakharov’s choice of variables,
    $$
    z = ( \eta(x), \xi(x) = \phi(x, \eta(x))),
    $$
    for which we consider $\phi = \phi[\eta, \xi] (x,y)$.
  • Express the energy in terms of $\xi$ and $\eta$. This involves the Dirichlet-Neumann operator $G(\eta)$.

Dirichlet-Neumann operator

  • Laplace’s equation on the fluid domain: $\Delta \phi = 0$ subject to bottom Neumann boundary condition. Free surface boundary data $\phi (x, \eta(x)) = \xi(x)$, for which the D-N operator is given by
    $$
    \xi(x) \longmapsto \phi(x,y) \longmapsto N \cdot \nabla \phi (1+ |\nabla_x \eta|^2)^{1/2} := G(\eta) \xi(x).
    $$
  • In these coordinates, we can rewrite the boundary conditions in a new (and nicer) form. This reexpresses the water wave problem as a Hamiltonian system in Darboux coordinates.

Hamiltonian PDEs

  • KdV is a Hamiltonian PDE with a different symplectic structure.
  • Other Hamiltonian PDEs
    • shallow water equations
    • Boussinesq
    • KP
    • NLS
    • Dysthe equation

Many of these problems arise in scaling limits of the water wave problem.

Lemma (Properties of D-N operator):

  1. $G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
  2. $G(\eta)^* = G(\eta)$ Hermitian Symmetric
  3. $G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$. There is an operator valued power series expansion of $G(\eta)$ (using a theorem of Christ-Journé 1987).
  4. Some explicit calculations of the Taylor expansion (I couldn’t keep up….)
  5. Conservation Laws
    • Mass: $M = \int \eta dx$ (He shows the calculation using properties of $G$.)
    • Momentum: Similar calculation
    • Energy: Easy since the commutator of $H$ with itself vanishes.
  6. Taylor expansion of the Hamiltonian
  7. Linearized equations; comparison with the harmonic oscillator.

Periodic Traveling wave patterns

  • Can I find traveling wave solutions?
    $$ \eta(x,t) = \eta ( x-tc); \xi(x,t) = \xi(x – tc) $$
  • Spatially periodic, $\Gamma \subset {\mathbb{R}^{d-1}}$.
    $$
    \eta(x + \gamma, \cdot) = \eta(x, \cdot), \xi(x+\gamma, \cdot) = \xi(x, \cdot), ~ \forall \gamma \in \Gamma.
    $$

On such domains, we can use the Fourier tranform.

Rk: Notice this is a mathematician imposing a period rather than the physics making that selection. More can be said in this direction, but let’s proceed this way.

Rk: These (time independent) traveling wave patterns can be imagined to emerge in transient interactions in seas. The nonlinear actions create large amplitudes and this might be related to the phenomena of freak waves.

Equations for traveling waves.

Periodic traveling wave patterns are critical points of the Hamltonian on the variety $I = const$, with Lagrange multiplier $c \in {\mathbb{R}^{d-1}}.$

This leads to a bifurcation problem.

brief history (dimension $d=2$):

  • Levi-Civita 1925; existence of traveling waves
  • Struik 1926; traveling waves case
  • Zeidler 1971
  • Beale 1979
  • Jones-Toland 1985

brief history (dimension $d=3$):

  • Reeder Shinbrot 1981
  • Sun 1986
  • Craig-Nicholls 2000
  • Iooss-Plotnikov-Toland 2000 (small divisor problem)

He shows a picture from the wave tank at Penn State. He then shows some numerics which are trying to model those observations and they look beautiful.

Kuksin Question: Stability of these patterns?

Craig Answer: This is a very good question. I don’t know results like that. This is related to Benjamin-Feir. McLean showed instability for $d = 3$. Some further discussion….We need the Bloch theory of stability for these wave patterns. This appears to be difficult analytically so might need some numerical studies at first. There are instability zones….

Solitary wave Interactions

Solitary waves in 2-dimensions (Friedreichs-Hyers 1954, Amick-Fraenkel-toland 1980s)

  • Head-on collisions of solitons.

The numerics reveal some inelasticity in the collision. We’d like to understand those. If we make the amplitude of the solitons bigger, the dispersive ripples are more visible.

The KdV scaling limit

Titi’s Question: Can we reduce to the surface equations including rotation?

Craig’s Answer: Yes and No. You can make a rotation depending purely on y and impose that. Then it is reducible. But this is rather artificial. There is stuff that happens in the middle which is not a surface effect. Therefore, this problem requires a more complete analysis of the Euler equation and will not collapse to a system on the surface.


Sergey Nazarenko (Warwick): Assumptions, Techniques, Cahllenges in Wave Turbulence

This is not so much about new result. Instead, this is an attempt by a physicist trying to explain wave turbulence ideas being explored by physicists to mathematicians. My view is that there is a lot of interesting work to be done. Lots of open problems….

What is wave turublence?

He shows a picture of a relatively calm seashore from Nice. He emphasizes there is a wide range fo scales in these problems. WT is a statistical system of nonlinear waves.

Examples:

  • Water waves
  • Waves in rotating and stratified fluids (internal and inertial waves, Rossby waves)
  • Plasma waves
  • Waves in Bose-Einstein condensates
  • Kelvin waves on quantized vortex filnments
  • MHD turbulence in interstellar turbulence and solar wind
  • Nonlinear optics
  • Solids: phonons, spin waves. Kinetics of phonons in weakly anharmonic crystals is a first example of study in tis direction (1920s). I didn’t catch the name….

He shows a picutre of a wave take of Lukaschuk.

Waves in fusion plasmas. Shows a picture of a Tokamak. Drift wave turbulence causes anomalous heat and particle loss – major problem for fusion. The devices have grown larger and larger basically to carry out the confinement for a longer period of time.

MHD turbulence in astrophysics. He shows some data from the Ulysses/Swoops (los alamos) solar wind studies.

Bose Einstein Condensates Nazarenko-Onorato 2006:

  • Inverse cascade – condensation
  • Condensate strongly affects WT

Quantum Turbulence (see Lvov et. al. 2007) (Superfluid turbulence)

  • Kelvin waves on quantized vortex filaments
  • Interaction with hydro eddies (vortex bundles) is important
  • Kelvin Wave Turbulence

Optical Turbulence

  • Bortolozzo et. al 2008
  • This project studies nonlinear corrections (coming from the optical physics) which are included beyond the 1d NLS model.

Kuksin Question: Which corrections? Can you write them down?

Nazareknko: Something like a DNLS correction…not so clear.

Ingredients in the approach

He writes $NLS_3^\pm (T^d)$ and comments that this is a physically reasonable model but we are really interested in the study in infinite space with finite energy density.

He reexpresses the NLS equation in Fourier language.

Set of wave modes: amplitudes and phases.

N-mode joint probability density function. Some notation….probability…sectors in the wave modes setting.

Random Phase (RP) and Random Phase Amplitude (RPA) systems

RP:

All phases are independent random variables such that uniformly distributed on $S^1$.

RPA:

  1. All amplitudes and all phases are independent random variables.
  2. All phases are uniformly distributed on $S^1$.

Note: RPA does not mean Gaussian. Nevertheless, we have obtained successful closures without assuming the Gaussian statistics.

Frog Jumps!

  • expanding in small nonlinearity
  • Assuming RP at $t=0$.
  • Taking limit of a large box followed by the limit of small nonlinearity.

(The order of these steps is important.)

Evolution of joint PDF? We can derive the evolution equation under these assumptions. The derivation is rather systematic, in fact it is perhaps rigorous.

Mathematical Challenges:

  • WT is formally derived for $t=0$.
  • Does it work at the long time of nonlinear evolution?
  • Does RPA survive over this time?
  • Adding forcing and dissipation: will WT describe the steady state?

Hmmmm….This RPA condition at $t=0$ reminds me a bit of the assumption of product wave function in the QMB theory. The dynamics in the Hartree derivation might drive the multiparticle wave function away from the product case. Here we have a dynamic that might drive us away from the RPA condition.

Evolution of 1-mode PDF.

Kinetic equation (Hasselmann 1962).

Kolmogorov-Zakharov state.

  • Explained a steady state spectrum corresponding to energy cascade.
  • Exact solution of the asymptotic closure.

Numerics and Analysis of KE.

  • What is the role of KZ solutions with respect to the thermodynamic Rayleigh-Jeans state?
  • Similar issues for the classical Boltzmann equation.

Zakharov was awarded the 2003 Dirac Medal for “putting the theory of wave turbulence on a firm mathematical ground”! What is it that we want to do?


Gregor Tanner (Nottingham): A wave chaos approach towards describing the vibro-acoustic response of engineering structures

(Joint work with D. Chappel, Stefano Gianai, Hanya Ben Hamdin, Dmitrii Maksimov)

This talk is more directed toward engineering applications. inuTech is an industrial collaborator.

Overview:

  • Introduction – the need for numerical short wavelenght methods in vibroacoustics
  • From wave equations to the Liouville equation
  • Solving the Liouville equation – a boundary integral approach (Dynamical Energy Analysis – DEA)
  • Tackling the Midfrequency problem – hybrid methods
  • Numerical results

Aim: predicting wave intensity distributions for the vibro-acoustical response of mechanical structures. Think of a car. Companies like Bombardier and Airbus use these methods. It is a difficult problem. You want these structures to be quite and with no noise in the interior.

Where is the problem?

Techniques:

  • Low frequencies – wavelength around the size of the object
  • Finite Element method
  • Boundary elemtn method
  • plane wave methods

High Frequencies:

  • Ray tracing
  • Statistical energy analysis

Midfrequency problems:

  • Structures with large variations in the local wavelength. (Large variations in the stiffness of components, ie body frame and side panels.)
  • Hybrid methods. Try to connect exact numerical methods with the statistical methods.

Short wavelength approximations – from wave chaos to statistical methods

  • Wave chaos -short wavelength asymptotics
    • Keller
    • Gutzwiller
    • Berry
    • Bogolmolny
    • Smilansky
  • Nonlinear dynamics – thermodynamic formalism
    • Ruelle
    • Arnold
    • Sinai
    • Eckmann
    • Cvitanovic – chaosbook.org
  • Wave transport – statistical methods in vibro-acoustics
    • Lyon – SEA (1967 paper)
    • Langley – WIA
    • Heron
    • Weaver – diffusion equation
    • Le Bot – radiative transformation

Linear wave equation. WKB ansatz. Hamiltoninan equations for the amplitude and phase. Characteristics of JH; nonlinear ODE; Liouville equation (linear).

Linear wave –> WKB –> HJ equation –> Liouville Equation

Think of polygonal billiards, not necessarily convex.

We want to understand the influence of a source (transfmitting at frequency $\omega$) at one location on the wave amplitude at another point. He writes this as a green’s function $G(r, r_0, \omega)$.

Small wavelength limit, so low frequency waves.

Write things as sums over all paths.

Perron-Frobenius operator…

Tandem Satellite images of coastline of Madagascar

I started wondering about connections between these ideas and quantum ergodicity…

Typical Wave Function in a stadium Billiard

Bouncing Ball Modes

Postlude:

I had a nice conversation with Gregor after the break. I learned from him about microlasers. The idea is to build a circular region out of a lasing material. We energize the material somehow with hopes to excite the whispering gallery mode. The laser light propagates near the boundary but can be arranged to exit the medium by raising the curvature at a specific location. These appear to be rather hard to control to create a unidirectional beam. Since the losses take place all along the boundary, there is very little power in the output beam. Some web searching revealed an advance made by the Capasso group at Harvard.
Elliptic Notched Microlaser Cavity Drawing
Elliptic notched microlaser cavity SEM photograph.
Microlaser Cavity (Artistic Rendition)
Schematic Image
Artistic Rendering


David Dritschel (St. Andrews): CLAM, The Combined Lagrangian Advection Method

(Many many collaborators)

Courbet's "The Wave"

I’ll be speaking a bit about a numerical method. I’ll focus mostly on the results we’ve obtained to understand the large scale atmospheres, like Jupiter and perhaps also the ocean.

The numerical method (CLAM) emerges from a Lagrangian method from the 50s for studying fluid dynamics. Zabusky then built from these developments to develop new methods in plasmas. We’ve been extending these ideas to treat certain geophysical fluid flows.

The atmosphere and the oceans are extremely complex, turbulent flows. Accurate computer simulation is immensely difficult to achieve. However, much of this difficulty is inherent in the computational methods employed:

  • None take direct advantage of the natural inherent Lagrangian advection of dynamical, chemical and biological tracers. (Exploit Lagrangian Descriptions.)
  • None seek to separate slow vortical (eddying) and fast wave-like motions and use appropriate, optimal numerical methods for each. (Slow Rossby waves interacting with fast waves inertial-gravity waves.)

We can build the mathematical theory of the separation into the numerical methods and this will lead to better predictions.

Contour Advection (CASL) Dritschel & Ambaum 1997

geostropic and hydrostatic balances are basic features for describing atmospheric wave dynamics.

This talk reminds me somehow of Bourgain’s high/low method for proving low regularity GWP.

The idea is to use the advection of the vorticity to resolve some (especially relevant) sub-grid scales.


Dugald Duncan: IDE equation

Overview:

  • full IDE equation and how it looks like, where it arises
  • Linear part of the IDEbehaviour and approximation
  • the full problem – behanviour and approximation
  • examples

$$
u_t = \sigma \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy + f(u) dx ~ \forall x \in \Omega, t>0.
$$
Typically, $f(u) = u – u^3$. This should be contrasted with the Allen-Cahn equation

$$
u_t = \sigma \Delta u + f(u) dx ~ \forall x \in \Omega, t>0.
$$

There are no spatial derivatives. Therefore, there are now boundary conditions. Instead, this is some kind of integral dynamical equation. It is similar to the Cahn-Allen equation.

This equation is also related to sandpiles, neurons, phase transitions.

Other variations recently: Rossi, Perez-Llanos, Andreu, Mazon, Toledo et. al. They study a nonlocal version of the $p$-laplacian.

Linear IDE:

  • Ignore the nonlinear reaction term for now and take $\sigma \geq 0$ and $\Omega \subset {\mathbb{R}}$:
    $$ u_t = Lu.$$
  • L is a linear operator – partly a convolution:
    $$
    Lu = \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy = J * u
    $$
    …ack slide changed….

Discontinuities don’t move. The solution collapses to the average value. Ther eis acomparison principle.

Snapshots of linear behavior.

He does a Fourier analysis of the behavior of plane waves. Instead of having an $\omega^2$, we have $$\hat{J} (\omega) – \hat{J} (0) \thicksim \frac{\omega^2}{2} \frac{d^2}{d \omega^2} {\widehat{J}} (\omega).$$

Peter Bates and Paul Fife did some of the earliest analysis on this equation.

,