Past Partial Differential Equations Seminar

The talk is focused on the clamped plate problem, initially formulated by Lord Rayleigh in 1877, and solved by M. Ashbaugh & R. Benguria (Duke Math. J., 1995) and N. Nadirashvili (Arch. Ration. Mech. Anal., 1995) in 2 and 3 dimensional euclidean spaces. We consider the same problem on both negatively and positively curved spaces, and provide various answers depending on the curvature, dimension and the width/size of the clamped plate.

Quasiconvexity is the fundamental existence condition for variational problems, yet it is poorly understood. Two outstanding problems remain: 

• 1) does rank-one convexity, a simple necessary condition, imply quasiconvexity in two dimensions? 

• 2) can one prove existence theorems for quasiconvex energies in the context of nonlinear Elasticity? 

In this talk we show that both problems have a positive answer in a special class of isotropic energies. Our proof combines complex analysis with the theory of gradient Young measures. On the way to the main result, we establish quasiconvexity inequalities for the Burkholder function which yield, in particular, many sharp higher integrability results. 
The talk is based on joint work with Kari Astala, Daniel Faraco, Aleksis Koski and Jan Kristensen.

In his pioneering work in 1860, Riemann proposed the Riemann problem and solved it for isentropic gas in terms of shocks and rarefaction waves. It eventually became the foundation of the theory of one-dimension conservation laws developed in the 20th century. We prove the non-nonlinear structural stability of the Riemann problem for multi-dimensional isentropic Euler equations in the regime of rarefaction waves. This is a joint work with Tian-Wen Luo.

Spacetimes formed from the cartesian product of Minkowski space and a flat torus play an important role as toy models for theories of supergravity and string theory. In this talk I will discuss an upcoming work with Huneau and Stingo showing the nonlinear stability of such a Kaluza-Klein spacetime. The result is also connected to a claim of Witten.

In mathematical modelling, data and solutions are often represented as measurable functions, and their quality is being captured by their membership to a certain function space. One of the core questions arising in applications of this approach is the comparison of the quality of the data and that of the solution. A particular attention is being paid to optimality of the results obtained. A delicate choice of scales of suitable function spaces is required in order to balance the expressivity (the ability to capture fine mathematical properties of the model) and the accessibility (the level of its technical difficulty) for a practical use. We will give an overview of the research area which grew out of these questions and survey recent results obtained in this direction as well as challenging open questions. We will describe a development of a powerful method based on the so-called reduction principles and demonstrate its use on specific problems including the continuity of Sobolev embeddings or boundedness of pivotal integral operators such as the Hardy - Littlewood maximal operator and the Laplace transform.

***CANCELLED*** In this talk, I will discuss recent results related to the mathematical justification of PDEs which model multi-phase flows at the macroscopic level from mesoscopic descriptions with jump conditions at interfaces. We will also present interesting and difficult open problems.

We use entropy methods to show that the heat equation with Dirichlet boundary conditions on the complement of a compact set in R^d shows a self-similar behaviour much like the usual heat equation on R^d, once we account for the loss of mass due to the boundary. Giving good lower bounds for the fundamental solution on these sets is surprisingly a relatively recent result, and we find some improvements using some advances in logarithmic Sobolev inequalities. In particular, we are able to give optimal asymptotic bounds for large times for the fundamental solution with an explicit approach rate in dimensions larger than 2, and some new bounds in dimension 2.

This is a work in collaboration with Alejandro Gárriz and Fernando Quirós.

This talk will be devoted to our recent developments in the analysis of emerging models for complex flows. I will start from presenting a general PDE system describing two-fluid flows, for which we prove existence of global in time weak solutions for arbitrary large initial data. I will explain where the famous approach of Lions developed for the compressible Navier-Stokes equations fails and how to use a more direct, weighted Kolmogorov criterion to prove compactness of approximating sequences of solutions. Through a formal limit, I will link the two-fluid model to the constrained two-phase models. Applications of such models include modelling of granular flows, crowd motion, or shallow water flow through a channel. The last part of my talk will focus on the rigorous derivation of these models from the compressible Navier-Stokes equations via the vanishing singular pressure or viscosity limit.

I will present a model for an optimal portfolio allocation and consumption problem for a portfolio composed of a risk-free bond and two illiquid assets. Two forms of illiquidity are presented, both illiquidities based on Lévy processes. The goal of the investor is to maximise a certain utility function, and the optimal utility is found as a solution of a nonlinear PIDE of the Hamilton-Jacobi-Bellman kind.

We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10-dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

In this talk I will discuss well-posedness of hyperbolic Cauchy problems with multiplicities and the role played by the lower order terms (Levi conditions). I will present results obtained in collaboration with Christian Jäh (Göttingen) and Michael Ruzhansky (QMUL/Ghent) on higher order equations and non-diagonalisable systems.

So-called Schauder estimates are a standard tool in the analysis of linear elliptic and parabolic PDE. They have been originally obtained by Hopf (1929, interior case), and by Schauder and Caccioppoli (1934, global estimates). The nonlinear case is a more recent achievement from the ’80s (Giaquinta & Giusti, Ivert, Lieberman, Manfredi). All these classical results hold in the uniformly elliptic framework. I will present the solution to the longstanding problem, open since the ‘70s, of proving estimates of such kind in the nonuniformly elliptic setting. I will also cover the case of nondifferentiable functionals and provide a complete regularity theory for a new double phase model. From joint work with Giuseppe Mingione (University of Parma).

The synthetic theory of Ricci curvature lower bounds introduced more than 15 years ago by Lott-Sturm-Villani has been largely succesful in describing the geometry of metric measure spaces. However, this theory fails to include sub-Riemannian manifolds (an important class of metric spaces, the simplest example being the so-called Heisenberg group). Motivated by Villani's ``great unification'' program, in this talk we propose an extension of Lott-Sturm-Villani's theory, which includes sub-Riemannian geometry. This is a joint work with Barilari (Padua) and Mondino (Oxford). The talk is intended for a general audience, no previous knowledge of optimal transport or sub-Riemannian geometry is required.

*** Cancelled *** We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the current distribution. Such transport metrics arise naturally in mean field limits of recent Ensemble Kalman methods for inverse problems. We show how the transport problem splits into two separate minimisation problems: one for the evolution of mean and covariance of the interpolating curve, and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities, as well as the dynamics of the associated gradient flows. This is joint work of Martin Burger, Matthias Erbar, Daniel Matthes and André Schlichting.

The aim of this talk is to present some novel inequalities for the k-plane transform acting on the modulus square of solutions of the linear time-dependent Schrodinger equation. Our motivation for studying these tomographic expressions comes for virial identities in the context of Schrodinger equations, where tomographic Strichartz estimates of the type we will discuss here appear naturally.

I will introduce a new parabolic system for the flow of nematic liquid crystals, enjoying a free boundary condition. After recent works related to the construction of blow-up solutions for several critical parabolic problems (such as the Fujita equation, the heat flow of harmonic maps, liquid crystals without free boundary, etc...), I will  construct a physically relevant weak solution blowing-up in finite time. We make use of  the so-called inner/outer parabolic gluing. Along the way, I will present a set of optimal estimates for the Stokes operator with Navier slip boundary conditions. I will state several open problems related to the partial regularity of the system under consideration. This is joint work with F.-H. Lin (NYU), Y. Zhou (JHU) and J. Wei (UBC). 

In the plane, we know that area of a set is monotone with respect to the inclusion but perimeter fails, in general. If we consider only bounded and convex sets, then also the perimeter is monotone. This property allows us to estimate the minimum number of convex components of a nonconvex set.

When studying integral functionals of the calculus of variations, convexity with respect to minors of the Jacobian matrix is a nice tool for proving existence and regularity of minimizers.

Sometimes it happens that the infimum of a functional on a set is less then the infimum taken on a dense subset: we usually refer to it as Lavrentiev phenomenon. In order to avoid it, convexity helps a lot.

I will review the series of recent results with Merle (IHES), Rodnianski (Princeton) and Szeftel (Paris Sorbonne) concerning the description of implosion mechanisms for viscous three dimensional compressible fluids. I will explain how the problem is connected to the description of blow up mechanisms for classical super critical defocusing models. 

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).