Confirmed Speakers
Title of Talk: The nonlinear Brascamp-Lieb inequality and applications
Abstract: The Brascamp-Lieb inequality is a broad generalisation of many well-known multilinear inequalities in analysis, including the multilinear Hölder, Loomis-Whitney and sharp Young convolution inequalities. There is by now a rich theory surrounding this classical inequality, along with applications in convex geometry, harmonic analysis, partial differential equations, number theory and beyond. In this talk we present a certain nonlinear variant of the Brascamp-Lieb inequality, placing particular emphasis on some of its applications. Most of this is joint work with Stefan Buschenhenke, Neal Bez, Michael Cowling and Taryn Flock.
Title of Talk: Stability in Gagliardo-Nirenberg-Sobolev inequalities: nonlinear flows, regularity and the entropy method
Abstract: We discuss stability results in Gagliardo-Nirenberg-Sobolev inequalities, a joint project with J. Dolbeault, B. Nazaret and N. Simonov.
We have developed a new quantitative and constructive "flow method", based on entropy methods and sharp regularity estimates for solutions to the fast diffusion equation (FDE). This allows to study refined versions of the Gagliardo-Nirenberg-Sobolev inequalities that are nothing but explicit stability estimates. Using quantitative regularity estimates for the FDE, we go beyond the variational results and provide fully constructive estimates, to the price of a small restriction of the functional space which is inherent to the method.
Title of Talk: Microlocal analysis of singular measures
Abstract: In this talk; I will investigate the structure of singular measures from a microlocal perspective. I will introduce a notion of $L^1$-regularity wave front set for scalar and vector distributions. Our main result is a proper microlocal characterisation of the support of the singular part of tempered Radon measures and of their polar functions at these points. We deduce a sharp $L^1$ elliptic regularity result which appears to be new even for scalar measures and which enlightens the interest of the techniques from geometric measure theory to the study of harmonic analysis questions. We also deduce several consequences including extensions of the results in \cite{DePRi} giving constraints on the polar function at singular points for measures constrained by a PDE. Finally, we also illustrate the interest of this microlocal approach with a result of propagation of singularities for constrained measures.
This is a joint work with V. Banica (Sorbonne Université)
Title of Talk: Cavitation and Concentration in Solutions of the Euler/Euler-Poisson Equations and Related Nonlinear PDEs
Abstract: In this talk, we will discuss the intrinsic phenomena of cavitation/decavitation and concentration/deconcentration in global entropy solutions of the compressible Euler/Euler-Poisson equations and related nonlinear PDEs, which are fundamental to the understanding of the well-posedness and solution behaviour of nonlinear PDEs. We will start to discuss the formation process of cavitation and concentration in the entropy solutions of the isentropic Euler equations with respect to the initial data and the vanishing pressure limit. Then we will analyse a longstanding fundamental problem in fluid dynamics: Does the concentration occur generically so that the density develops into a Dirac measure at the origin generically in spherically symmetric entropy solutions of the multidimensional compressible Euler/Euler-Poisson equations? We will report our recent results and approaches developed for solving this longstanding open problem for the Euler/Euler-Poisson equations and related nonlinear PDEs and discuss its close connections with entropy methods and the theory of divergence-measure fields. Further perspectives and open problems in this direction will also be addressed.
Title of Talk: Boundary behaviour of nonlocal minimal surfaces
Abstract: Surfaces which minimize a nonlocal perimeter functional exhibit quite different behaviors from the ones minimizing the classical perimeter. We will investigate some structural properties of nonlocal minimal surfaces, and in particular we will discuss the "stickiness phenomenon", namely the strong tendency of adhering at the boundary of the reference domain.
Title of Talk: Approximation of mappings with derivatives of low rank
Abstract:
Title of Talk: Schrödinger-Proca constructions in the closed setting
Abstract: We discuss recent developments on Schrödinger-Poisson-Proca and Bopp-Podolsky-Schrödinger-Proca systems of equations in the closed setting. We discuss the building of the equations in two blocks of equations, their meaning and a recent result obtained by the speaker on the strong convergence of BPSP to SPP as the Bopp-Podolosky parameter goes to zero.
Title of Talk: Morrey's problem in classes of homogeneous integrands
Abstract: The question of whether rank-one convexity implies quasiconvexity is often called Morrey's problem. Sverak has shown that the answer is no in general and so a number of modifications have been proposed over the years. In this talk I will discuss some of these where positive answers have been found recently. The talk is based on joint work with Kari Astala (Helsinki), Daniel Faraco (Madrid), Andre Guerra (IAS) and Aleksis Koski (Helsinki).
Title of Talk: The low-density limit of the Schrodinger equation: random vs periodic potentials
Abstract: A major challenge in the kinetic theory of gases is to establish the convergence of the dynamics to a macroscopic transport process described by the appropriate kinetic equation. In the case of the quantum Lorentz gas with random potentials, important studies of Spohn, Erdos-Yau, Eng-Erdos and others have identified the linear Boltzmann equation as the correct limit. In this lecture I will discuss periodic potentials in the Boltzmann-Grad scaling, and report on a new limiting random process which is incompatible with the linear Boltzmann equation. Based on joint work with Jory Griffin (Oklahoma), J. Stat. Phys. 184 (2021).
Title of Talk: Stochastic Nash evolution
Abstract: The isometric embedding problem for Riemannian manifolds was solved by Nash in the 1950s. In the past ten years, Nash’s work has been adapted into $h$-principles for several (non-geometric) PDE following the work of De Lellis and Szekelyhidi on the Euler equations and turbulence. I will present a set of ideas that use this connection to re-investigate Nash’s work from a probabilistic standpoint. Our main contribution is a class of stochastic algorithms for isometric embedding and several other hard constraint systems.
Title of Talk: Hopf, Caccioppoli and Schauder, reloaded.
Abstract: So called Schauder estimates are in fact a contribution, at various stages, of Hopf, Caccioppoli and Schauder, between the end of the 20s and the beginning of the 30s.
Later on, they were extended, with various degrees of precision, to nonlinear uniformly elliptic equations. I will present the solution to the longstanding open problem of proving estimates of such kind in the nonuniformly elliptic case.I will also discuss the related problem of proving gradient continuity of minima of non-differentiable functionals, again in the nonuniformly elliptic case. From joint work with Cristiana De Filippis.
Title of Talk: Weighted Korn inequalities and linear stability of collisional charged particles
Abstract: We prove functional inequalities on vector fields on the Euclidean space when it is equipped with a bounded measure that satisfies a Poincaré inequality, and study associated self-adjoint operators. The weighted Korn inequality compares the differential matrix, once projected orthogonally to certain finite-dimensional spaces, with its symmetric part and, in an improved form of the inequality, an additional term. We also consider Poincaré-Korn inequalities for estimating a projection of the vector field by the symmetric part of the differential matrix and zeroth-order versions of these inequalities obtained using the Witten-Laplace operator. The constants depend on geometric properties of the potential, and the estimates are quantitative and constructive. These inequalities are motivated by kinetic theory and related with the Korn inequality (1906) in mechanics, on a bounded domain. We will explain how they are used to prove the linear collisional stability of charged particles subject to a confining field. This is based on two joint works with Carrapatoso, Dolbeault, Hérau, Mischler and Schmeiser.
Title of Talk: Gibbs measures and canonical stochastic quantization
Abstract: In this talk, I will discuss the (non-)construction of the focusing Gibbs measures and the associated dynamical problems. This study was initiated by Lebowitz, Rose, and Speer (1988) and continued by Bourgain (1994), Brydges-Slade (1996), and Carlen-Fröhlich-Lebowitz (2016). In the one-dimensional setting, we consider the mass-critical case, where a critical mass threshold is given by the mass of the ground state on the real line. In this case, I will show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988).
In the three dimensional-setting, I will first discuss the construction of the $\Phi^3_3$-measure with a cubic interaction potential. This problem turns out to be critical, exhibiting a phase transition: normalizability in the weakly nonlinear regime and non-normalizability in the strongly nonlinear regime.
If time permits, I will discuss the dynamical problem for the canonical stochastic quantization of the $\Phi^3_3$-measure, namely, the three-dimensional stochastic damped nonlinear wave equation with a quadratic nonlinearity forced by an additive space-time white noise (= the hyperbolic $\Phi^3_3$-model).
The first part of the talk is based on a joint work with Philippe Sosoe (Cornell) and Leonardo Tolomeo (Bonn), while the second part is based on a joint work with Mamoru Okamoto (Osaka) and Leonardo Tolomeo (Bonn).
Title of Talk: Markovian Solutions for Scalar Conservation Laws
Abstract: Groeneboom in 1989 discovered an explicit formula for the law of the entropy solution to Burgers' equation when the initial condition is a white noise. The method of his proof relied extensively on probabilistic methods and in particular on the sophisticated excursion theory for diffusions. Recently, by verifying a conjecture of Menon and Srinivasan, Kaspar and Rezakhanlou managed to prove a closure theorem for Markovian solutions to scalar conservation laws which bridged the probabilistic problem to kinetic theory. In this talk, I present a new and significantly shorter proof of Groeneboom's results (Joint work with Mehdi Ouaki). This approach builds on these recent developments, and a central limit theorem for certain Markovian jump processes. I also discuss how a kinetic theory can be developed when we add an external force to the Burgers' equation.
Title: On the wave turbulence theory for a stochastic KdV type equation
Abstract: This talk is a summary of a recent work completed with Binh Tran. Starting from the stochastic Zakharov-Kuznetsov (ZK) equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is d>1, the smallness of the nonlinearity is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schr ̈odinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only 3-wave interactions but also all m-wave interactions are allowed to happen. To the best of our knowledge, the work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.
Title of Talk: A constructive approach to nonlinear wave equations
Abstract: In this talk, a new attempt to construct solutions to nonlinear wave equations will be explained. The existence of self-similar solutions to semilinear wave equations with power nonlinearity has been already established by Pecher (2000), Kato-Ozawa (2003), etc. based on the standard fixed point theorem. We will rediscuss it by a constructive approach using the theory of hypergeometric differential equations. The same approach to damped wave equations will be also discussed.
Title of Talk: Nonlocal capillarity theory
Abstract: We present a nonlocal version of the classical Gauss free energy functional used in capillarity theory which considers surface tension energies of nonlocal type. The case of interaction kernels that are possibly anisotropic and not necessarily invariant under scaling can be also dealt with, and in this general setting we determine a nonlocal Young's law for the contact angle and discuss the unique solvability of the corresponding equation in terms of the interaction kernels and of the relative adhesion coefficient.
Title of Talk: The Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces
Abstract: The lecture will discuss recent joint work with Costante Bellettini at UCL. The work develops a general existence theory for prescribed-mean-curvature hypersurfaces in any given compact Riemannian manifold of dimension $\geq 3$. The method is PDE theoretic. It brings to bear on the question elementary, classical variational and gradient flow principles in semi-linear PDE via the use of elliptic and parabolic Allen--Cahn equations and a garden-variety PDE min-max lemma. These principles serve as a substantially simpler alternative to the varifold min-max methods pioneered by Almgren and Pitts several decades ago for the special case of mean-curvature prescribing function $g=0$ i.e. to prove existence of a minimal hypersurface; a decisive advantage of the PDE method (vis-a-vis the generality of the conclusions it can reach) is seen in the case $g$ not identically zero. For regularity conclusions the method relies on a new ``black-box-tool'' varifold regularity theory of independent interest. This theory identifies a set of non-variational conditions (i.e. conditions that do not require the varifold to be a critical point of a functional) under which a codimension 1 integral varifold lying close to a hyperplane of some positive integer multilpicity is regular.