Abstracts

Diego Cordoba : Instant blow-up for the generalized SQG equations

In this talk we present recent results on the existence of solutions of the generalized Surface Quasi-geostrophic equations (SQG) that initially are in C^k, C^k,γ or in super-critical Sobolev spaces, but lose that prescribe regularity for t > 0.

Michele Coti Zelati : Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations

We investigate the long-time properties of the two-dimensional inviscid Boussinesq equations near a stably stratified Couette flow. We prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo inviscid damping while the vorticity and density gradient grow. The result holds at least until a natural, nonlinear timescale. Notice that the density behaves very differently from a passive scalar, as can be seen from the inviscid damping and slower gradient growth. The proof relies on several ingredients: (A) a suitable symmetrization that makes the linear terms amenable to energy methods and takes into account the classical Miles-Howard spectral stability condition; (B) a variation of the Fourier time-dependent energy method introduced for the inviscid, homogeneous Couette flow problem developed on a toy model adapted to the Boussinesq equations, i.e. tracking the potential nonlinear echo chains in the symmetrized variables despite the vorticity growth.

Manuel del Pino : Dynamics of concentrated vorticities in 2d and 3d Euler flows

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices. We rigorously establish the law of motion of "leapfrogging vortex rings", initially conjectured by Helmholtz in 1858.

Raphaël Ducatez : Delocalization transition for critical Erdős–Rényi graphs

We analyse the spectrum of the (scaled) adjacency matrix A of the Erdős-Rényi graph G(N,d/N) in the critical regime d = b log N with b constant. We establish a one-to-two correspondence between vertices of degree at least 2d and nontrivial eigenvalues outside the asymptotic [−2,2]. This correspondence implies a transition at an explicit b*. For d > b^∗ log N the spectrum is just [−2,2] and the eigenvectors are completely delocalized. For d < b* log N we still have delocalization in [−2,2] but another phase appears. The spectrum outside [−2,2] is not empty and the associated eigenvectors concentrate around the large degree vertices. This is a joint work with Johannes Alt and Antti Knowles.

Francisco Gancedo : Global regularity for 2D Navier-Stokes free boundary with small viscosity contrast

We study the dynamics of two incompressible immiscible fluids in 2D modeled by the inhomogeneous Navier-Stokes equations. We show a new approach to prove that if initially the viscosity contrast is small then there is global-in-time regularity. The result allows to obtain preservation of the natural C^1+γ Hölder regularity of the interface for all 0 < γ < 1 with low Sobolev regularity of the initial velocity without any extra technicality. In particular, it uses new quantitative harmonic analysis bounds for C^γ norms of even singular integral operators on characteristic functions of C^1+γ domains.

Léo Girardin : Non-local pulling in reaction-diffusion equations

In recent years it has been established that, in reaction-diffusion models of monostable type, a favorable environment that flees away from a population density arising from a localized initial condition can in fact improve the spreading speed of this population density. In a simplified Fisher-KPP setting, when the speed of the favorable area is sufficiently small for the population to keep up, then the spreading speed is correctly predicted by the Fisher-KPP speed in this environment. But when the speed is too large for the population to keep up, the Fisher-KPP speed of the second, less favorable environment, is in general only a lower estimate for the spreading speed: the population can be “non-locally pulled” by its exponential tail in the favorable area, even though the distance between this area and the spreading front is linearly increasing in time. The exact spreading speed is given by an explicit formula and at least two methods of proof are known. In this talk I will review some results about non-local pulling in reaction-diffusion systems and equations and then I will present a work in progress in collaboration with Thomas Giletti and Hiroshi Matano.

François Huveneers : Slow diffusion in weakly non-linear disordered lattices

Disordered harmonic lattices behave as Anderson insulators: the transport of conserved charges is suppressed on all time scales. But it is expected that transport is restored once anharmonic interactions are introduced. However, very slow processes are involved for small anharmonic couplings, and the transport rates are hard to quantify. This is so far an open problem with unresolved conflicting conjectures. In this talk I will present new results, both numerical and analytical, indicating that an initially localized wave packet spreads slower than polynomially with time. From a joint work with Oskar Prosniak and Wojciech De Roeck.

Kihyun Kim : On recent developments on the long-term dynamics for the self-dual Chern-Simons-Schrödinger equation

The goal of this talk is to introduce the self-dual Chern-Simons-Schrödinger equation (CSS) which is a gauge-covariant 2D cubic nonlinear Schrödinger equation. The distinguished feature of CSS is the self-duality, and it turns out that CSS has a deep connection with critical geometric dispersive equations such as Schrödinger maps. Moreover, the dynamics of CSS under equivariant symmetry is surprisingly rigid, e.g. the non-existence of multi-solitons. This is based on joint works with Soonsik Kwon and Sung-Jin Oh.

Frédéric Klopp : On the spatial extent of localized eigenfunctions for random Schrödinger operators

On ℤ^d, consider φ, an l²-normalized function that decays exponentially at ∞ at a rate at least μ. One can define the onset length (of the exponential decay) of φ as the radius of the smallest ball, say, B, such that one has the following global bound |φ(x)| ≤ ∥φ∥_∞ e^{- μ dist(x,B)}.

The present talk will describe the onset lengths of the localized eigenfunctions of random Schrödinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than l and localization center in a ball of radius L is smaller than C L^d e^{-c l}, for l > 0 large (for some constants C,c > 0). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. The talk is based on joint work with Jeff Schenker.

Xavier Lamy : Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system

The anisotropic Ginzburg-Landau system

for u : ℝ^{^2} → ℝ^{^2} and δ ∈ (-1,1), models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that |u(x)| → 1 and u has a prescribed topological degree d ≤ -1 as |x| → ∞, for small values of the anisotropy parameter |δ| < δ₀(d). Unlike the isotropic case δ = 0, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup. This is joint work with M. Kowalczyk and P. Smyrnelis.

 Mickaël Latocca : Probabilistic well-posedness for the Schrödinger Equation posed for the Grushin Laplacian

 In this talk we study the local well-posedness of the equation

     i ∂_t u + Δ_G u = |u|² u

 where Δ_G = ∂_x² + x² ∂_y² is the Grushin Laplacian and u(t) : ℝ² → ℂ is the solution, to be constructed with initial data u(0) =u₀ ∈ H_G^s(ℝ^d) (the adapted Grushin-Sobolev spaces). From a deterministic perspective, the best local well-posedness theory is in C⁰([0, T),H_G^3/2+) and the proof only uses the Sobolev embedding. Our main goal is to provide a probabilistic construction of local solutions for initial data u₀ ∈ H_G^s where s < 3/2. This is achieved using linear and bilinear random estimates. In the first part of the talk I will introduce the random initial data which we will consider. Then I will explain why randomisation helps to lower the well-posedness threshold: this is a general argument in the study of dispersive equations with random initial data. Then I will explain how bilinear random estimates relate to our probabilistic well-posedness problem, which we will prove if time permits. This talk is based on a joint work with Louise Gassot [1].

References:
[1] L. Gassot and M. Latocca, Probabilistic Local Well-posedness for the Schrödinger equation posed for the Grushin Laplacian, Preprint, 2021.

Mathieu Lewin : New results on the Lieb-Thirring inequality

The Lieb-Thirring inequality provides a control on the negative eigenvalues of Schrödinger operators in terms of the size of the potential. It plays a central role in the analysis of large quantum systems. In this talk I will first introduce the inequality and then focus the discussion on its best constant. After reviewing what is believed, what is known and what is open, I will present new results on the value of the best constant as well as numerical simulations in 1D and 2D. Collaboration with Rupert L. Frank (Munich, Germany) and David Gontier (ENS & Paris-Dauphine, France).

Alberto Maspero : Full description of Benjamin-Feir instability of Stokes waves in deep water

Small-amplitude, traveling, space periodic solutions -- called Stokes waves -- of the 2 dimensional gravity water waves equations in deep water are linearly unstable with respect to long-wave perturbations, as predicted by Benjamin and Feir in 1967. We completely describe the behavior of the four eigenvalues close to zero of the linearized equations at the Stokes wave, as the Floquet exponent is turned on. We prove in particular the conjecture that a pair of non-purely imaginary eigenvalues depicts a closed figure eight, parameterized by the Floquet exponent, in full agreement with numerical simulations. This is a joint work with M. Berti and P. Ventura.

Maria Medina de la Torre : Blow-up analysis of a curvature prescription problem in the disk

We will establish necessary conditions on the blow-up points of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures, where a non local restriction will appear. Conversely, given a point satisfying these conditions, we will construct an explicit family of approximating solutions that explode at it. These results are contained in several works written in collaboration with L. Battaglia, A. Jevnikar, R. López-Soriano, A. Pistoia and D. Ruiz.

References:
[1] L. Battaglia, M. Medina, A. Pistoia, Large conformal metrics with prescribed Gaussian and geodesic curvatures. Calc. Var. 60, 39 (2021).
[2] L. Battaglia, M. Medina, A. Pistoia, A blow-up phenomenon for a non-local Liouville-type equation. To appear in Journal d’Analyse Mathematique.
[3] A. Jevnikar, R. López-Soriano, M. Medina and D. Ruiz, Blow-up analysis of conformal metrics of the disk with prescribed Gaussian and geodesic curvatures. To appear in Analysis and PDEs.

Vincent Millot : Torus and split solutions of the Landau-de Gennes model for nematic liquid crystals

In this talk, I will present the Q-tensor model of Landau-de Gennes for nematic liquid crystals in the so called Lyutsyukov regime dealing with maps with values in the 4-dimensional sphere.

We investigate the orbital stability of the Ginzburg-Landau vortex of degree one for the Gross-Pitaevskii equation in dimension two. Our main result is a nonlinear coercivity estimate on the renormalized energy, from which we can deduce the stability. This is a joint work with Philippe Gravejat and Didier Smets.

Robert Schippa : Strichartz estimates for Maxwell equations with rough coefficients

We consider Maxwell equations in media with rough material laws in two and three space dimensions. We prove local-in-time Strichartz estimates via phase space analysis. Depending on symmetry properties of the matrix-valued coefficients and in the absence of charges, we recover the Strichartz estimates for wave equations with rough coefficients.  The talk is based on joint work with Roland Schnaubelt (KIT).

Christian Seis : Asymptotics near extinction for nonlinear fast diffusion on a bounded domain

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by the spectral gap), or algebraically slow (which is only possible in the presence of non-integrable zero modes). In the first case, the nonlinear dynamics are well-approximated by exponentially decaying eigenmodes up to at least twice the gap. This improves on a result of Bonforte and Figalli, by providing a new and simpler approach which is able to accommodate the absence of a spectral gap, as occurs when the vanishing profile fails to be isolated (and may belong to a continuum of such profiles). Joint work with Beomjun Choi and Robert J. McCann.

Chenmin Sun : Quasi-invariant Gaussian measures for the nonlinear Schrödinger equations

In the statistical study of Hamiltonian PDEs out of the equilibrium (lack of invariant measures), it is a natural question to understand the transport properties for canonical Gaussian measures. In this talk, we prove that the Gaussian measures of high regularities are quasi-invariant along the 2D NLS flow. As a comparison, the invariance of Gibbs measures for 2D NLS has been shown by Bourgain (the cubic case) and by an impressive recent work of Deng-Nahmod-Yue (for high-order nonlinearities). Due to the low regularity issue, these results however do not imply transport properties for Gaussian free field along the real NLS flow (without renormalizatoin). This is based on a joint work with Yu Deng and Nikolay Tzvetkov.

We consider ground state solutions, suitably defined as energy minimizers, of a class of semilinear biharmonic (fourth-order) Schrödinger equations. By exploiting a connection to the adjoint Stein–Tomas inequality on the unit sphere and by using trial functions related to Knapp’s example, we prove a symmetry breaking result for ground state solutions, which is in striking contrast to the well-known results of radial symmetry for ground states of classical second-order nonlinear Schrödinger equations. We also discuss symmetry breaking for a minimization problem with constrained mass and for a related problem on the unit ball subject to Dirichlet boundary conditions.This is joint work with Enno Lenzmann (University of Basel).