Forthcoming events in this series


Mon, 19 Feb 2024

16:30 - 17:30
L5

Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

Rupert Frank
(LMU Munich)
Abstract

The sharp constant in the Sobolev inequality and the set of optimizers are known. It is also known that functions whose Sobolev quotient is almost minimial are close to minimizers. We are interested in a quantitative version of the last statement and present a bound that not only measures this closeness in the optimal topology and with the optimal exponent, but also has explicit constants. These constants have the optimal behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proof relies on several ingredients:

• a discrete flow based on competing symmetries;

• a continuous rearrangement flow;

• refined estimates in the neighborhood of the optimal Aubin-Talenti functions.

The talk is based on joint work with Dolbeault, Esteban, Figalli and Loss. 


 
Mon, 12 Feb 2024

16:30 - 17:30
L5

OxPDE-WCMB seminar - From individual-based models to continuum descriptions: Modelling and analysis of interactions between different populations.

Mariya Ptashnyk
Abstract

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles.

Mon, 05 Feb 2024

16:30 - 17:30
L5

Characterising rectifiable metric spaces using tangent spaces

David Bate
(Warwick)
Abstract

This talk will present a new characterisation of rectifiable subsets of a complete metric space in terms of local approximation, with respect to the Gromov-Hausdorff distance, by finite dimensional Banach spaces. Time permitting, we will discuss recent joint work with Hyde and Schul that provides quantitative analogues of this statement.
 

Mon, 29 Jan 2024

16:30 - 17:30
L5

Asymptotic stability of traveling waves for one-dimensional nonlinear Schrodinger equations

Charles Collot
(CY Cergy Paris Université )
Abstract

We consider one-dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, and of Krieger and Schlag, and on precise renormalizations, computations, and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.

Mon, 22 Jan 2024

16:30 - 17:30
L5

Cross-diffusion systems for segregating populations with incomplete diffusion

Ansgar Jungel
(TU Wien)
Abstract

Busenberg and Travis suggested in 1983 a population system that exhibits complete segregation of the species. This system can be rigorously derived from interacting particle systems in a mean-field-type limit. It consists of parabolic cross-diffusion equations with an indefinite diffusion matrix. It is known that this system can be formulated in terms of so-called entropy variables such that the transformed equations possess a positive semidefinite diffusion matrix. We consider in this talk the case of incomplete diffusion, which means that the diffusion matrix has zero eigenvalues, and the problem is not parabolic in the sense of Petrovskii. 

We show that the cross-diffusion equations can be written as a normal form of symmetric hyperbolic-parabolic type beyond the Kawashima-Shizuta theory. Using results for symmetric hyperbolic systems, we prove the existence of a unique local classical solution. As solutions may become discontinuous in finite time, only global solutions with very low regularity can be expected. We prove the existence of global dissipative measure-valued solutions satisfying a weak-strong uniqueness property. The proof is based on entropy methods and a finite-volume approximation with a mesh-dependent artificial diffusion. 

Mon, 15 Jan 2024

16:30 - 17:30
L5

Functions of bounded variation and nonlocal functionals

Panu Lathi
(Academy of Mathematics and Systems Science of the Chinese Academy of Sciences)
Abstract

In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems. 

Mon, 27 Nov 2023

16:30 - 17:30
L3

Schoen's conjecture for limits of isoperimetric surfaces

Thomas Körber
(University of Vienna)
Abstract

R. Schoen has conjectured that an asymptotically flat Riemannian n-manifold (M,g) with non-negative scalar curvature is isometric to Euclidean space if it admits a non-compact area-minimizing hypersurface. This has been confirmed by O. Chodosh and M. Eichmair in the case where n=3. In this talk, I will present recent work with M. Eichmair where we confirm this conjecture in the case where 3<n<8 and the area-minimizing hypersurface arises as the limit of large isoperimetric hypersurfaces. By contrast, we show that a large part of spatial Schwarzschild of dimension 3<n<8 is foliated by non-compact area-minimizing hypersurfaces.

Tue, 21 Nov 2023

17:00 - 18:00
L1

THE 16th BROOKE BENJAMIN LECTURE: Advances in Advancing Interfaces: The Mathematics of Manufacturing of Industrial Foams, Fluidic Devices, and Automobile Painting

James Sethian
((UC Berkeley))
Abstract

Complex dynamics underlying industrial manufacturing depend in part on multiphase multiphysics, in which fluids and materials interact across orders of magnitude variations in time and space. In this talk, we will discuss the development and application of a host of numerical methods for these problems, including Level Set Methods, Voronoi Implicit Interface Methods, implicit adaptive representations, and multiphase discontinuous Galerkin Methods.  Applications for industrial problems will include modeling how foams evolve, how electro-fluid jetting devices work, and the physics and dynamics of rotary bell spray painting across the automotive industry.

Mon, 20 Nov 2023
16:30
L3

Recent developments on evolution PDEs on graphs

Antonio Esposito
(Mathematical Institute (University of Oxford))
Abstract

The seminar concerns the study of evolution equations on graphs, motivated by applications in data science and opinion dynamics. We will discuss graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance, using Benamou--Brenier formulation. The underlying geometry of the problem leads to a Finslerian gradient flow structure, rather than Riemannian, since the resulting distance on graphs is actually a quasi-metric. We will address the existence of suitably defined solutions, as well as their asymptotic behaviour when the number of vertices converges to infinity and the graph structure localises. The two limits lead to different dynamics. From a slightly different perspective, by means of a classical fixed-point argument, we can show the existence and uniqueness of solutions to a larger class of nonlocal continuity equations on graphs. In this context, we consider general interpolation functions of the mass on the edges, which give rise to a variety of different dynamics. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen. The latter study can be extended to equations on co-evolving graphs. The talk is based on works in collaboration with G. Heinze (Augsburg), L. Mikolas (Oxford), F. S. Patacchini (IFP Energies Nouvelles), A. Schlichting (University of Münster), and D. Slepcev (Carnegie Mellon University). 

Mon, 20 Nov 2023
15:45
L5

OXPDE-WCMB seminar: From individual-based models to continuum descriptions: Modelling and analysis of interactions between different populations.

Mariya Ptashnyk
(Heriot-Watt University, Edinburgh)
Abstract

First we will show that the continuum counterpart of the discrete individual-based mechanical model that describes the dynamics of two contiguous cell populations is given by a free-boundary problem for the cell densities.  Then, in addition to interactions, we will consider the microscopic movement of cells and derive a fractional cross-diffusion system as the many-particle limit of a multi-species system of moderately interacting particles. 

Fri, 17 Nov 2023

14:00 - 15:00
L2

Self-similar solutions to two-dimensional Riemann problems involving transonic shocks

Mikhail Feldman
(University of Wisconsin)
Abstract

In this talk, we discuss two-dimensional Riemann problems in the framework of potential flow
equation and isentropic Euler system. We first review recent results on the existence, regularity and properties of
global self-similar solutions involving transonic shocks for several 2D Riemann problems in the
framework of potential flow equation. Examples include regular shock reflection, Prandtl reflection, and four-shocks
Riemann problem. The approach is to reduce the problem to a free boundary problem for a nonlinear elliptic equation
in self-similar coordinates. A well-known open problem is to extend these results to a compressible Euler system,
i.e. to understand the effects of vorticity. We show that for the isentropic Euler system, solutions have
low regularity, specifically velocity and density do not belong to the Sobolev space $H^1$ in self-similar coordinates.  
We further discuss the well-posedness of the transport equation for vorticity in the resulting low regularity setting.

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Mon, 13 Nov 2023

16:30 - 17:30
L3

MRA Filters

Hrvoje Šikić
(University of Zagreb)
Abstract

I will present some results from the newly developed theory of wavelets; based on the joint work with the following authors:

P.M. Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson.One-DimensionalDyadic Wavelets.Mem. Amer. Math. Soc. 280 (2022), no 1378, ix+152 pp.

About two and a half decades ago and based on the influential book by Fernandez and Weiss, an approach was developed to study wavelets from the point of view of their connections with Fourier analysis. The idea was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. Among the most thorough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk, I will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and the newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results

Mon, 06 Nov 2023

16:30 - 17:30
L3

On Hookean models of dilute polymeric fluids.

Tomasz Dębiec
(University of Warsaw)
Abstract

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent.

The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. In a simplified case where the drag term is corotational, we prove global existence of weak solutions and discuss some of their properties: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model: a corotational Oldroyd-B model with stress-diffusion.

In the general noncorotational case, we consider “generalised dissipative solutions” — a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation, which accounts for the lack of compactness in the polymeric extra-stress tensor. Joint work with Endre Suli (Oxford).

Mon, 30 Oct 2023

16:30 - 17:30
L3

Elasto-plasticity driven by dislocation movement

Filip Rindler
(University of Warwick)
Abstract

This talk presents some recent progress for models coupling large-strain, geometrically nonlinear elasto-plasticity with the movement of dislocations. In particular, a new geometric language is introduced that yields a natural mathematical framework for dislocation evolutions. In this approach, the fundamental notion is that of 2-dimensional "slip trajectories" in space-time (realized as integral 2-currents), and the dislocations at a given time are recovered via slicing. This modelling approach allows one to prove the existence of solutions to an evolutionary system describing a crystal undergoing large-strain elasto-plastic deformations, where the plastic part of the deformation is driven directly by the movement of dislocations. This is joint work with T. Hudson (Warwick).

Mon, 23 Oct 2023

16:30 - 17:30
L3

Graph Limit for Interacting Particle Systems on Weighted Random Graphs

Nastassia Pouradier Duteil
(Sorbonne Université)
Abstract

We study the large-population limit of interacting particle systems posed on weighted random graphs. In that aim, we introduce a general framework for the construction of weighted random graphs, generalizing the concept of graphons. We prove that as the number of particles tends to infinity, the finite-dimensional particle system converges in probability to the solution of a deterministic graph-limit equation, in which the graphon prescribing the interaction is given by the first moment of the weighted random graph law. We also study interacting particle systems posed on switching weighted random graphs, which are obtained by resetting the weighted random graph at regular time intervals. We show that these systems converge to the same graph-limit equation, in which the interaction is prescribed by a constant-in-time graphon.

Mon, 16 Oct 2023

16:30 - 17:30
L3

Plateau's problem via the theory of phase transitions

Stephen Lynch
(Imperial College London )
Abstract

Plateau's problem asks whether every boundary curve in 3-space is spanned by an area minimizing surface. Various interpretations of this problem have been solved using eg. geometric measure theory. Froehlich and Struwe proposed a PDE approach, in which the desired surface is produced using smooth sections of a twisted line bundle over the complement of the boundary curve. The idea is to consider sections of this bundle which minimize an analogue of the Allen--Cahn functional (a classical model for phase transition phenomena) and show that these concentrate energy on a solution of Plateau's problem. After some background on the link between phase transition models and minimal surfaces, I will describe new work with Marco Guaraco in which we produce smooth solutions of Plateau's problem using this approach. 

Mon, 09 Oct 2023

16:30 - 17:30
L5

Exponential mixing by random velocity fields

Rishabh Gvalani
(Max Planck Institute in Leipzig)
Abstract

We establish exponentially-fast mixing for passive scalars driven by two well-known examples of random divergence-free vector fields. The first one is the alternating shear flow model proposed by Pierrehumbert, in which case we set up a dynamics-based framework to construct such space-time smooth universal exponential mixers. The second example is the statistically stationary, homogeneous, isotropic Kraichnan model of fluid turbulence. In this case, the proof follows a new explicit identity for the evolution of negative Sobolev norms of the scalar. This is based on joint works with Alex Blumenthal (Georgia Tech) and Michele Coti Zelati (ICL), and Michele Coti Zelati and Theodore Drivas (Stony Brook), respectively.

Wed, 13 Sep 2023

14:00 - 15:00
C6

Nonlinear SPDE approximation of the Dean-Kawasaki equation

Professor Ana Djurdjevac
(Free University Berlin)
Abstract

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density, the so-called Dean-Kawasaki equation.

 

Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that approximate the Dean-Kawasaki equation, in the special case of non-interacting particles. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation.  This is the joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).

Tue, 04 Jul 2023

17:00 - 18:00
N3:12

Fractional Sobolev lsometric lmmersions of Planar Domains

Siran Li
(NYU Shanghai)
Abstract

We discuss $C^1$-regularity and developability of isometric immersions of flat domains into $\mathbb{R}^3$ enjoying a local fractional Sobolev $W^{1+s;2/s}$-regularity for $2/3 \leq s < 1$, generalising the known results on Sobolev (by Pakzad) and H\"{o}lder (by De Lellis--Pakzad) regimes. Ingredients of the proof include analysis of the weak Codazzi equations of isometric immersions, the study of $W^{1+s;2/s}$-gradient deformations with symmetric derivative and vanishing distributional Jacobian determinant, and the theory of compensated compactness. Joint work with M. Reza Pakzad and Armin Schikorra.

Wed, 28 Jun 2023

16:00 - 17:00
L6

Schauder estimates at nearly linear growth

Giuseppe Rosario Mingione
(University of Parma)
Abstract

Schauder estimates are a classical tool in linear and nonlinear elliptic and parabolic PDEs. They describe how regularity of coefficients reflects in regularity of solutions. They basically have a perturbative nature. This means that they can be obtained by perturbing the estimates available for problems without coefficients. This paradigm works as long as one deals with uniformly elliptic equations. The nonuniformly elliptic case is a different story and Schauder's theory turns out to be not perturbative any longer, as shown by counterexamples. In my talk, I will present a method allowing to bypass the perburbative schemes and leading to Schauder estimates in the nonuniformly elliptic regime. For this I will concentrate on the case of nonuniformly elliptic functionals with nearly linear growth, also covering a borderline case of so-called double phase energies. From recent, joint work with Cristiana De Filippis (Parma). 

Mon, 12 Jun 2023

16:30 - 17:30
L4

Breaking glass optimally and Minkowski's problem for polytopes

Jian-Guo Liu
(Duke University)
Abstract
Motivated by a study of least-action incompressible flows, we study all the ways that a given convex body in Euclidean space can break into countably many pieces that move away from each other rigidly at constant velocity, following geodesic motions in the sense of optimal transport theory. These we classify in terms of a countable version of Minkowski's geometric problem of determining convex polytopes by their face areas and normals. Illustrations involve various intriguing examples both fractal and paradoxical, including Apollonian packings and other types of full packings by smooth balls.
Mon, 05 Jun 2023
16:30
L4

KPP traveling waves in the half-space

Cole Graham
(Brown University)
Abstract

Reaction–diffusion equations are widely used to model spatial propagation, and constant-speed "traveling waves" play a central role in their dynamics. These waves are well understood in "essentially 1D" domains like cylinders, but much less is known about waves with noncompact transverse structure. In this direction, we will consider traveling waves of the KPP reaction–diffusion equation in the Dirichlet half-space. We will see that minimal-speed waves are unique (unlike faster waves) and exhibit curious asymptotics. The arguments rest on potential theory, the maximum principle, and a powerful connection with the probabilistic system known as branching Brownian motion.

This is joint work with Julien Berestycki, Yujin H. Kim, and Bastien Mallein.

Mon, 29 May 2023

16:30 - 17:30
L4

In Search of Euler Equilibria Via the MR Equations

Susan Friedlander
(University of Southern California)
Abstract

The subject of “geometric” fluid dynamics flourished following the seminal work of VI.
Arnold in the 1960s. A famous paper was published in 1970 by David Ebin and Jerrold
Marsden, who used the manifold structure of certain groups of diffeomorphisms to obtain
sharp existence and uniqueness results for the classical equations of fluid dynamics. Of
particular importance are the fixed points of the underlying dynamical system and the
“accessibility” of these Euler equilibria. In 1985 Keith Moffatt introduced a mechanism
for reaching these equilibria not through the Euler vortex dynamics itself but via a
topology-preserving diffusion process called “Magnetic Relaxation”. In this talk, we will
discuss some recent results for Moffatt’s MR equations which are mathematically
challenging not only because they are active vector equations but also because they have
a cubic nonlinearity.


This is joint work with Rajendra Beckie, Adam Larios, and Vlad Vicol.

 

Mon, 22 May 2023

17:30 - 18:30
L6

Scaling Optimal Transport for High dimensional Learning

Gabriel Peyre
(École Normale Supérieure )
Further Information

Please note a different room and that there are two pde seminars on Monday of W5 (May 22).

Abstract

Optimal transport (OT) has recently gained a lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport".

Mon, 22 May 2023
16:30
L6

Optimal mass transport and sharp Sobolev inequalities

Zoltan Balogh
(Universitat Bern)
Further Information

Please note a different room and that there are two pde seminars on Monday of W5 (May 22).

Abstract

Optimal mass transport is a versatile tool that can be used to prove various geometric and functional inequalities. In this talk we focus on the class of Sobolev inequalities.

In the first part of the talk I present the main idea of this method, based on the work of Cordero-Erausquin, Nazaret and Villani (2004).

The second part of the talk is devoted to the joint work with Ch. Gutierrez and A. Kristály about Sobolev inequalities with weights.