Analysis and Hamiltonian PDEs: remembering Thomas Kappeler

05-09 June 2023

**Organisiert von:** G. Genovese, R. Montalto, B. Schlein

• Speaker: Patrick Gerard

Title: A low regularity phase space for the Benjamin-Ono equation with periodic boundary conditions

Abstract: We construct a Hilbert space of real valued distributions on the torus, containing all the Sobolev spaces H^s for s > -1/2,
to which the flow map of the Benjamin--Ono equation can be extended continuously, but not weakly continuously.
This is based on an improvement of the construction of the Birkhoff map constructed with Thomas Kappeler and Peter Topalov,
and is a joint work with Peter Topalov.

• Speaker: Dario Bambusi

Title: A Nekhoroshev theorem for some (smoothing) perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

Abstract: We consider the Benjamin Ono equation with periodic boundary conditions on a segment. We add a small Hamiltonian perturbation and consider the case where the corresponding Hamiltonian vector field is analytic as a map form energy space to itself. Let $\epsilon$ be the size of the perturbation. We prove that for initial data close in energy norm to an $N$-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $\cO(\epsilon^{\frac{1}{2(N+1)}})$ close to their initial value for times exponentially long with $\epsilon^{-\frac{1}{2(N+1)}}$.
The result is made possible by the use of Gerard-Kapeller's formulae for the Hamiltonian of the BO equation in Birkhoff variables.
Joint work with Patrick Gerard.

• Speaker: Jean-Claude Saut

Title: Boussinesq and integrability

Abstract: We will survey various questions (most of them open) concerning the (integrable) Boussinesq equation and system.

• Speaker: Michael Struwe

Title: Plateau flow, or the heat flow for half-harmonic maps

Abstract: Using the interpretation of the half-Laplacian on the circle as the
Dirichlet-to-Neumann operator for the Laplace equation on the ball, we
devise a classical approach to the heat flow for half-harmonic maps from
the circle to a closed target manifold, recently studied by Wettstein,
and for arbitrary finite-energy data we obtain a global existence result
fully analogous to the classical existence result for the harmonic map
heat flow of surfaces, and in similar generality.
When the target is a smoothly embedded, oriented closed curve, the
half-harmonic map heat flow may be viewed as an alternative gradient
flow for the Plateau problem of disc-type minimal surfaces.

• Speaker: Michiel van den Berg

Title: Localisation for the torsion function and first Dirichlet eigenfunction

Abstract: We discuss localisation phenomena for the torsion function and first Dirichlet eigen- function on sequences of non-empty, open sets in Euclidean space with finite Lebesgue measure. We show that localisation for the torsion function in L1 implies localisation for the first Dirichlet eigenfunction in L2 Joint work with Dorin Bucur and Thomas Kappeler.

• Speaker: Camillo De Lellis

Title: Anomalous dissipation and flows of rough vector fields

Abstract: Consider smooth solutions to the 3d Navier-Stokes for divergence-free vector fields $u^\varepsilon$ with vanishing viscosity $\varepsilon$.
The total kinetic energy dissipation is $-\varepsilon \int |Du^\varepsilon|^2\, dx$ and
it is a tenet of the theory of fully developed turbulence that in a variety of situations its size should typically be independent of $\varepsilon$.
Producing rigorous mathematical examples of this prediction is hard.
In a recent joint work with Elia Bru\'e we study what happens if we introduce a forcing term $f^\varepsilon$. The problem is compatively easier and we show that there is a regularity threshold for $\{f^\varepsilon\}$ below which anomalous dissipation can be shown to happen in some examples and above which it would only be possible through a blow-up scenario. Surprisingly one side of the problem is linked with some fundamental questions about solving ODEs with rough coefficients.

• Speaker: Sergei Kuksin

Title: Averaging and mixing for random perturbations of linear and integrable equations

Abstract: I will discuss stochastic \epsilon-perturbations of
deterministic integrable Hamiltonian systems in R^{2n}, linear and
non-linear. I will show that, firstly, on time intervals of order
1/\epsilon the actions of solutions for perturbed equations are close to
those of solutions for specially constructed effective stochastic
equations. Secondly, if the effective equation is mixing, then the
approximation of the actions of solutions for perturbed equations,
provided by it, is uniform in time. The mixing assumption admits an easy
sufficient condition. The results generalise to random perturbations of
linear and integrable Hamiltonian PDEs.

• Speaker: Peter Topalov

Title: Perfect fluid flows on $\mathbb{R}^d$ with growth/decay conditions at infinity

Abstract: We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on
$\mathbb{R}^d$ with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^\beta$ with $\beta<1/2$.
In particular, we show that the solution generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data.

• Speaker: Massimiliano Berti

Title: Quasi-periodic vortex patches of 2d Euler

Abstract: We prove the existence of time quasi-periodic vortex patch solutions of the 2d-Euler equations in ℝ2, close to uniformly rotating Kirchhoff elliptical vortices, with aspect ratios belonging to a set of asymptotically full Lebesgue measure. A major difficulty of the KAM proof is the presence of a zero normal mode frequency, which is due to the conservation of the angular momentum. The key novelty to overcome this degeneracy is to perform a perturbative symplectic reduction of the angular momentum, introducing it as a symplectic variable in the spirit of the Darboux-Carathéodory theorem of symplectic rectification, valid in finite dimension. This approach is particularly delicate in a infinite dimensional phase space: our symplectic change of variables is a nonlinear modification of the transport flow generated by the angular momentum itself. This is the first time such an idea is implemented in KAM for PDEs. Joint work with Z. Hassaina and N. Masmoudi.

• Speaker: Nikolay Tzvetkov

Title: On invariant measures for the Benjamin-Ono equation

Abstract: We will discuss invariant measures under the
Benjamin-Ono equation flow and the relevance of the Thomas work for this
problem.

• Speaker: Eugene Wayne

Title: Breathers and the Implicit Function Theorem.

Abstract: Breather solutions are spatially localized, temporally periodic solutions of partial differential equations and lattice systems. They are extremely rare for partial differential equations with constant coefficients but are known to exist in some PDEs with inhomogeneous coefficients. In this talk I’ll describe a new approach to the construction of breathers for special classes of PDEs with spatially periodic coefficients, based on the implicit function theorem.

• Speaker: Maciej Zworski

Title: Internal waves in 2D aquaria and homeomorphisms of the circle

Abstract: The connections between the formation of internal waves in
fluids, spectral theory, and homeomorphisms of the circle were
investigated by oceanographers in the 90s and resulted in novel
experimental observations (Leo Maas et al, 1997). The specific
homeomorphism is given by a ``chess billiard" and has been considered
by many authors (Fritz John 1941, Vladimir Arnold 1957, Jim Ralston
1973... ). The relation between the nonlinear dynamics of this
homeomorphism and linearized internal waves provides a striking
example of classical/quantum correspondence (in a
classical and surprising setting of fluids!). I will illustrate the
results with numerical and experimental examples and explain how
classical concepts such as rotation numbers of homeomorphisms
(introduced by Henri Poincare) are related to solutions of the
Poincare evolution problem (so named by Elie Cartan). The talk is
based on joint work with Semyon Dyatlov and Jian Wang. I will also
mention recent progress by Zhenhao Li on the case of irrational
rotation numbers.

• Speaker: Herbert Koch

Title: The KdV hierarchy at H^{-1} regularity

Abstract: I report on joint work with Friedrich Klaus and Baoping Liu on
wellposedness for the whole KdV hierarchy at this rough regularity
level. Basic ingredients are the Miura map, which has very rich and
surprising properties, and the seminal ideas introduced by Killip and
Visan.

• Speaker: Nicola Visciglia

Title: Modified energies for 1-d periodic NLS and applications

Abstract: We show how to construct suitable energies for pure power NLS posed on the circle.
As a consequence we shall deduce on one hand polynomial upper bounds on the growth of Sobolev norms, on the other hand we obtain the quasi-invariance of the transported Gaussian measures along the deterministic flow.
The tools that we use are elementary, in particular at the level of dispersion we only rely on the rather classical L4 estimate proved by Zygmund. This is a joint work with D. Berti, F. Planchon, N. Tzvetkov.

• Speaker: Enno Lenzmann

Title: Half-Wave Maps with Hyperbolic Target.

Abstract: The half-wave maps equation is a geometric Hamiltonian PDE, which exhibits both energy-criticality and complete integrability (with a Lax pair structure). After reviewing the current state of affairs for half-wave maps with target S^2 (the unit two-sphere), we will consider the case of the hyperbolic plane H^2 as target space. In striking contrast to critical wave-maps and Schrödinger maps with target H^2, the half-wave maps equation with hyperbolic target is found to have traveling solitary wave solutions with finite energy. As a main result, we derive a complete classification of these solutions. This talk is based on joint work with P. Gérard, J. Hilken and A. Schikorra.

• Speaker: Catherine Sulem

Title: A Hamiltonian Dysthe equation for deep-water gravity waves
with constant vorticity.

Abstract: This is a study of the water wave problem in a two-dimensional domain of
infinite depth in the presence of nonzero constant vorticity. The goal
is to describe the effects of uniform shear flow on the modulation of
weakly nonlinear quasi-monochromatic surface gravity waves. Starting
from the Hamiltonian formulation of this problem (Wahlén 2007,
Constantin-Ivanov-Prodanov 2008), and using techniques from Hamiltonian
transformation theory, we derive a Hamiltonian Dysthe (high-order
nonlinear Schrödinger) equation for the time evolution of the wave
envelope. Consistent with previous studies, we observe that the uniform
shear flow tends to enhance or weaken the modulational instability of
Stokes waves depending on its direction and strength. Our method also
provides a non-perturbative procedure to reconstruct the surface
elevation from the wave envelope, based on the Birkhoff normal form
transformation that eliminates all non-resonant triads. This model is
tested against direct numerical simulations of the full Euler equations
and against a related Dysthe equation recently derived by Curtis,
Carter and Kalisch (2018). This is a joint work with P. Guyenne and A.
Kairzhan.

• Speaker: Peter A. Perry

Title: Direct and Inverse Scattering for the Intermediate Long Wave Equation

Abstract: This is joint work with Joel Klipfel and Allen Wu. The intermediate long wave equation (ILW) is a completely integrable equation for dispersive waves in a stratified medium of finite depth $\delta$. It has the Korteweg-de Vries equation (KdV) and the Benjamin-Ono equation (BO) as asymptotic limits in $\delta$. The inverse scattering map for KdV is defined by a local Riemann-Hilbert problem while the inverse scattering map for ILW and BO equations are defined by nonlocal Riemann-Hilbert problems. We give a complete analysis of the direct scattering map and formulate the inverse scattering map as a nonlocal Riemann-Hilbert problem in $\mathbb{C}\setminus [0,\infty)$. We also comment on an alternate Lax pair and its spectral properties.

• Speaker: Michela Procesi

Title: Reducibility of the linear wave equation with unbounded perturbation

Abstract: We prove the perpetual boundedness of the Sobolev norm of all the solutions
of a quasi-periodically forced linear Klein-Gordon equation on the circle
$$
u_{tt}-u_{xx}+m u+Q(\omega t)u =0\,,
$$
where $Q$ is an unbounded pseudo-differential operator of order 2, parity preserving and reversible,
provided that the forcing frequency belongs to a Borel set of asymptotically full measure.
This result is obtained by reducing the Klein-Gordon equation to constant coefficients, applying first
a pseudo-differential normal form reduction and then a KAM diagonalization scheme.
A main point is that the equation is equivalent to a first order pseudo-differential system
which, at the highest order, is the sum of two backward/forward transport equations, with non-constant coefficients,
respectively on the subspaces of functions supported on positive/negative Fourier modes.
The key idea is to straighten such operator through a novel quantitative Egorov analysis.

• Speaker: Remi Carles

Title: Pathological set with loss of regularity for nonlinear Schrödinger equations

Abstract: We consider the mass-supercritical defocusing nonlinear Schrödinger
equation. We prove loss of regularity in arbitrarily short times for
regularized initial data belonging to a dense set of any fixed Sobolev
space for which the nonlinearity is supercritical. The proof resumes the
general strategy introduced for the nonlinear wave equation. We overcome
the lack of finite speed of propagation in Schrödinger equations by
using properties of solutions to the compressible Euler equation, which
appear in (monokinetic) WKB analysis. The error analysis is performed
thanks to a renormalized modulated energy functional. Finally, we prove
spatially localized estimates for the exact solution. The talk is based
on a joint work with Louise Gassot.

• Speaker: Chiara Saffirio

Title: Uniqueness criteria for the Vlasov-Poisson system and applications to semiclassical problems.

Abstract: The Vlasov-Poisson system is a non-linear kinetic equation describing the mean-field time-evolution of particles forming a plasma.
In this talk I will present uniqueness criteria for Vlasov-Poisson type equations, emerging as corollaries of stability estimates in strong topologies (associated with Lebesgue norms) or in weak topologies (induced by Wasserstein distances), and show how they serve as a guideline to study the classical limit from the Hartree equation to the Vlasov equation in different settings.

• Speaker: Thierry Paul

Title: On Quantum Topologies

Abstract: We will present recent generalizations of the 2-Wasserstein metric suitable for quantum mechanics, with applications to low regularity semiclassical approximation, uniform in the Planck constant time-dependent perturbation theory and quantum optimal transport.

• Speaker: Renato Luca'

Title: Phase blow up for the cubic NLS and connections with the binormal flow.

Abstract: In this talk, we will discuss a suitable family of borderline regularity solutions of the 1d cubic Schrodinger equation exhibiting a phase blow-up. We will also discuss the connection between these solutions and the evolution of curves under the binormal curvature equation. This is a joint work with V. Banica, N. Tzvetkov and L. Vega.

• Speaker: Alberto Maspero

Title: Hamiltonian Birkhoff normal form for gravity-capillary water waves with constant vorticity: almost global existence

Abstract: We prove an almost global in time existence result of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity. The result holds for any value of gravity, vorticity and depth and any surface tension belonging to a full measure set. The proof demands a Hamiltonian paradifferential Birkhoff normal form reduction for quasi-linear PDEs in presence of resonant wave interactions: the normal form may be not integrable but it preserves the Sobolev norms thanks to its Hamiltonian nature. This is a joint work with M. Berti and F. Murgante.

• Speaker: Benoit Grébert

Title: Typical dynamics of some resonant Hamiltonian PDEs.

Abstract: I will discuss some results about long time stability of small typical solutions of some resonant Hamiltonian PDEs (NLS, Kdv and BO). In particular I will focus on a recent result where we (with J. Bernier and Z. Wang) prove exponential stability of small typical solutions of Schrödinger-Poisson equation.
For these resonant Hamiltonian PDEs the linear frequencies are fully resonant and we have to use the nonlinearity to break the resonances.