Abstracts

Scott Armstrong - To be announced

Tobias Barker - To be announced

Daniel Boutros - To be announced

Raphaël Côte - Perturbation at blow up time of self similar solutions for the modified Korteweg-de Vries equation

The modified Korteweg-de Vries equation (mKdV) is an asymptotic model for fluid dynamics, and its self-similar solutions are connected to the formation of spirals and corners in a vortex patch. In this talk, I will present some recent results in collaboration with Simão Correia (University of Lisbon) and Luis Vega (Basque Center for Applied Mathematics) regarding the description, stability and perturbation of the blowup dynamic of self-similar solutions of (mKdV).

Paul Dario - Hydrodynamic limit for a class of degenerate convex grad phi interface models

In this talk, we will consider a classical model of random interfaces known as the grad phi model and introduced by Brascamp-Lieb-Lebowitz in 1975. This model has been extensively studied by the mathematical community, and one of its important aspects is that it can be studied using tools of elliptic regularity and stochastic homogenization. In this talk, we will introduce the model, some of its main properties, and discuss a generalization of an important result, known as the hydrodynamic limit and originally established by Funaki and Spohn, making use of techniques developed in the field of quantitative stochastic homogenization.

Lucas Ertzbischoff - On the hydrostatic limit of the Euler-Boussinesq equations

I will talk about the hydrostatic approximation of the 2d Euler-Boussinesq system, describing the evolution of an inviscid stratified fluid where the vertical length scale is much smaller than the horizontal one. Even though of importance in oceanography, the justification of the hydrostatic limit in this context has remained an open problem.

I will discuss some recent results showing that some instability mechanisms may prevent this limit to hold. I will also try to provide some key features of the associated equations and highlight several related challenges.

This is a joint work with Roberta Bianchini (IAC-CNR Rome) and Michele Coti Zelati (Imperial College London).

Irfan Glogic - To be announced

Cécile Huneau- High frequency limits in General Relativity

In this talk, I will present a joint work with Jonathan Luk (Stanford University), in which we prove a conjecture made by the physicist Burnett in 1989. If we consider a sequence of metrics solutions to Einstein vacuum equations, converging uniformly, but for which the derivatives only converge weakly, the metric obtained at the limit does not satisfy Einstein vacuum equations. Instead, Burnett conjectured that the equation satisfied at the limit has a very specific structure : it corresponds to Einstein equations coupled to a massless Vlasov field. The proof we give rely on a specific choice of coordinates, and we obtain at the end a characterisation of the Vlasov field in term of the microlocal defect measure of the sequence of metrics we consider.

Juhi Jang - To be announced

Hyunju Kwon - Non-conservation of generalized helicity in 3D Euler flows

Recently, there has been significant research into the non-conservation of total kinetic energy in Euler flows, which has led to Onsager’s theorem and its intermittent version. In this talk, I will discuss an analogous question for another conserved quantity: helicity. I will present the first example of a weak solution to the 3D Euler equations in C_t⁰(H^(1/2)- ∩ L^∞-) for which the helicity, defined in a generalized sense, is not conserved in time. The talk will be based on recent collaboration with Matthew Novack (Purdue University) and Vikram Giri (ETH Zurich).

Jonas Luhrmann - Asymptotic stability of the sine-Gordon kink outside symmetry

We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the ϕ⁴ model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes.

We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known ϕ⁴ model.

This is joint work with Gong Chen (Georgia Tech).

Angeliki Menegaki - Stability of Rayleigh-Jeans equilibria in the kinetic FPUT equation

In this talk we consider the four-wave spatially homogeneous kinetic equation arising in weak wave turbulence theory from the microscopic Fermi-Pasta-Ulam-Tsingou (FPUT) oscillator chains.  This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the global existence and stability of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibrium solutions. This is a joint work with Pierre Germain (Imperial College London) and Joonhyun La (Korea Institute for Advanced Study).

Shrish Parmeshwar - To be announced

Bruno Premoselli - To be announced

Mouad Ramil - To be announced

Simona Rota Nodari - To be announced

Tobias Schmid - To be announced

Leonardo Tolomeo - To be announced

Juan Luis Velazquez - To be announced

Yao Yao - To be announced