19-22 mai 2025 Neuville-sur-Oise (France)

Résumés

Scott Armstrong - Superdiffusivity for a diffusion in a critically-correlated incompressible random drift

We consider an advection-diffusion (or "passive scalar") equation with a divergence-free vector field, which is a stationary random field exhibiting "critical" correlations. Predictions from physicists in the 80s state that, almost surely, this equation should behave like a heat equation at large scales, but with a diffusivity that diverges as the square root of the log of the scale. In joint work with Ahmed Bou-Rabee (New York University) and Tuomo Kuusi (University of Helsinki, we give a rigorous proof of this prediction using an iterative quantitative homogenization procedure, which is a way of formalizing the renormalization group arguments proposed by the physicists. The idea is to consider a scale decomposition of the vector field, and "coarse-grain" the equation, scale-by-scale. The random swirls of the vector field at each scale enhance the effective diffusivity and, as we zoom out, we essentially obtain an ODE for the effective diffusivity as a function of the scale. Integrating this allows us to deduce that it diverges at the predicted rate. New coarse-graining techniques are needed to make this rigorous, permitting us to integrate out the smaller scales in the equation and prove the result.

Tobias Barker - Critical norm blow-up rates for the energy supercritical nonlinear heat equation

We study the behavior of the scaling critical Lebesgue norm for blow-up solutions to the nonlinear heat equation (the Fujita equation). For the energy supercritical nonlinearity, we give estimates of the blow-up rate for the critical norm. This is based on joint work with Jin Takahashi (Institute of Science Tokyo) and Hideyuki Miura (Institute of Science Tokyo).

Daniel Boutros - On analogues of Onsager’s conjecture for inviscid hydrodynamic PDEs

We consider Onsager’s conjecture in the case of a bounded domain, as boundary effects play a crucial role in hydrodynamic turbulence. We present a regularity result for the pressure in the Euler equations, which is essential for the proof of the conservation part of the conjecture (in the presence of boundaries). As an essential part of the proof, we introduce a new weaker notion of boundary condition which we show to be necessary by means of an explicit example. Moreover, we derive this new boundary condition rigorously from the weak formulation of the Euler equations.

In addition, we consider an analogue of Onsager’s conjecture for the hydrostatic Euler equations (also known as the inviscid primitive equations of oceanic and atmospheric dynamics). The anisotropic nature of these equations allows us to introduce new types of weak solutions and prove a range of independent sufficient criteria for energy conservation. Therefore there probably is a ‘family’ of Onsager conjectures for these equations. Furthermore, we employ the method of convex integration to show the nonuniqueness of weak solutions to the inviscid and viscous primitive equations (and also the Prandtl equations), and to construct examples of solutions that do not conserve energy in the inviscid case. These are joint works with Claude Bardos (University Paris Cité), Simon Markfelder (University of Würzburg) and Edriss S. Titi (Texas A&M University and University of Cambridge).

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 - Non-uniqueness of weak solutions to nonlinear heat equations

Jia and Sverák showed in 2015 that existence of forward self-similar solutions that are linearly unstable can be used to establish non-uniqueness of Leray-Hopf solutions to the unforced incompressible 3D Navier-Stokes equations. Although the aforementioned (non-)uniqueness question remains open to this day, a number of works have since utilized the ideas of Jia and Sverák to demonstrate non-uniqueness for fluid dynamics equations with non-zero forcing terms. In this talk, we consider the unforced focusing power nonlinearity heat equation, and rigorously implement the Jia - Sverák method, thereby showing non-uniqueness of local solutions for the full range of supercritical Lebesgue spaces. In particular, we rigorously verify the (analogue of the) spectral assumption made by Jia and Sverák for the Navier-Stokes equations. This is joint work with M. Hofmanová (University of Bielefeld), T. Lange (Scuola Normale Superiore di Pisa), and E. Luongo (University of Bielefeld).

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 - Vacuum free boundary problems in gas dynamics

We will discuss recent progress on the vacuum free boundary problems arising in the dynamics of isolated gases with or without gravity. We will give an overview of the well-posedness and stability theory, and present some new results on waiting time solutions.

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 Ct0(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 - Weak-strong uniqueness and singular limits for the Navier-Stokes-Poisson system

I will discuss some ongoing work involving using relative-entropy methods to investigate weak-strong uniqueness and the low Mach low Froude number limit for the gravitational compressible Navier-Stokes-Poisson system with degenerate, density-dependent viscosity in three space dimensions.

Bruno Premoselli - Least-energy solutions for the Brezis-Nirenberg problem in dimension 3 in the non-coercive case

We consider in this talk the celebrated Brezis-Nirenberg equation in the non-coercive case λ > λ1, where λ1 is the first eigenvalue of the Laplacian on a bounded open set of ℝn. We prove in dimension 3, and on a general bounded set, the existence of least-energy sign-changing solutions of very small energy when λ belongs to a left-neighbourhood (that we characterize) of any eigenvalue. We develop for this a new variational framework, inspired from eigenvalue-optimisation problems in conformal geometry: we consider the principal eigenvalue of - Δ - λ over an appropriate weighted L2 space and minimize it over the set of all admissible weight; we then show that the resulting infimum, if attained, provides least-energy solutions of the equation. Our framework applies in every dimension and when n ≥ 4 provides a new and accurate description of the energy function associated to the equation.

Mouad Ramil - Eyring-Kramers law for the underdamped Langevin process

The Eyring-Kramers (EK) law describes the asymptotic of the mean transition time between basins of a potential function when the temperature is low. The EK law was obtained first for the overdamped Langevin process and was recently extended to non-reversible elliptic diffusion processes. However, the scheme of proof relies on Potential theory tools which are ill-defined when considering non-elliptic diffusion process. In this presentation, we will talk about a recent work in collaboration with S. Lee and I. Seo (Seoul National University) where we extend the EK law for the underdamped Langevin process, which is a non-elliptic and non-reversible diffusion process, by implementing a novel approach which circumvents the previous technicalities.

Simona Rota Nodari - On a quasilinear Schrödinger equation: the small frequency limit

In this talk, I will present some recent results on a quasilinear Schrödinger equation with a power nonlinearity. After showing the uniqueness and the non-degeneracy of the positive radial solution uω for all ω > 0, I will describe its asymptotic behavior in the limit ω → 0. This gives some important information about the orbital stability of uω and the uniqueness of normalized ground states. This is joint work with François Genoud (EPFL).

Tobias Schmid - Blow up solutions for the critical 4D Zakharov system

The Zakharov model is a nonlinear coupling of a Schrödinger equation to a wave interaction term and approximates the electric field/ion density of a weakly magnetized plasma undergoing rapid oscillations. In 1994 Glangetas and Merle proved the existence of blow up for the 2D Zakharov equation through self-similar collapse of conformal type. In this talk we consider the 4D Zakharov equation and present a different result on finite-time concentration blow up near the solitary ground state solution. We first outline the construction of ``well behaved'' approximate solutions near the solitary bulk with a fast decaying error and which are inspired by matched asymptotic expansions. Finally we explain how to obtain exact solutions using distorted Fourier techniques for the linearized flow and modulation theory. The talk is based on joint work with Joachim Krieger (EPFL).

Leonardo Tolomeo - A statistical version of the soliton resolution conjecture

In this talk, I will discuss the construction of the Gibbs measure for the mass-subcritical Schrödinger equation. 

 

After reviewing the construction of the measure in finite volume, I will show that in the infinite volume regime, in the correct regime, the measure concentrates around a single soliton over a Gaussian background. This provides a version of the soliton resolution conjecture at equilibrium. 

 

This talk is based on joint works with T. Oh (University of Edinburgh), M. Okamoto (University of Osaka), H. Weber (University  of Münster) and J. Forlano (Monash University).

Juan José López Velázquez - Kinetic equations in open systems

In this talk I will describe several examples of kinetic equations for which the boundary conditions imply the exchange of some quantity (mass, momentum or energy) with the surroundings. A consequence of this is that the resulting solutions do not converge to an equilibrium solution, although in some cases the solutions can converge to steady states, but they can yield also periodic oscillations or in some cases solutions that can be described as self-similar solutions in some regions of the space of parameters. Specific examples of equations that will be described include the classical Boltzmann equation, the Smoluchowski coagulation equation and he Becker-Döring equation.

Yao Yao - Stability and growth for 2D Euler equation

In this talk, I will discuss some recent results on the 2D incompressible Euler equation, where we show the orbital stability of some traveling wave solutions, and also use the stability to establish vorticity gradient growth. Namely, the results include the nonlinear stability of vortex quadrupoles with odd-odd symmetry (joint with Kyudong Choi (Ulsan National Institute of Science and Technology) and In-Jee Jeong (Seoul National University)), nonlinear stability of multiple Lamb dipoles (joint with Ken Abe (Osaka Metropolitan University) and In-Jee Jeong), and using stability to obtain growth of vorticity gradient (joint with In-Jee Jeong and Tao Zhou).

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