Toppforsk@IE: Partial differential equations

Partial differential equations

Toppforsk@IE research group

Activities

Activities

Toppforsk@IE: PDE (drop-down3)

Introductory lectures on Wasserstein gradient flows

Introductory lectures on Wasserstein gradient flows

Date: December 14-15, 2026

Place: Auditorium EL1

Filippo Santambrogio, Institut Camille Jordan, Université Lyon 1, France

The motivation for this course dates back to the work of Jordan, Kinderlehrer and Otto who found a variational structure for the Fokker-Planck equation, i.e. the heat equation with a gradient drift in conservative form. They proved that this equation is the gradient flow of a relative entropy in the Wasserstein metric, induced by optimal transport, and analyzed a time-discrete scheme, now called JKO scheme, which approximates the solutions. This paved the way to the study of many other equations sharing the same structure. In the course we will provide the necessary background in terms of optimal transport and of the general theory of gradient flows, analyze the convergence of the JKO scheme, and study the properties of this discrete approximation.

Monday 10:15-12:00: Introduction to optimal transport, Wasserstein distances, and curves of measures.
Monday 13:15-15:00: Introduction to gradient flows in Euclidean and metric spaces, and to the case of the Wasserstein space, with a formal derivation of the corresponding PDE.
Tuesday 10:15-12:00: Rigorous proof of convergence for the easiest case (Fokker-Planck) and how to adapt to nonlinear equations.
Tuesday 13:15-15:00: Complementary material. Other PDEs with a gradient flow structure, applications and estimates.

Toppforsk@IE: PDE (drop-down2)

Introductory lectures on kinetic theory for conservation laws

Introductory lectures on kinetic theory for conservation laws

Date: May 19-20, 2026

Place: Auditorium F4

Nathaël Alibaud, Université Marie et Louis Pasteur, LmB, UMR CNRS 6623, France; and SUPMICROTECH, Franc

This course is devoted to the kinetic formulation of scalar conservation laws (SCLs), inspired by the analysis of kinetic models from statistical physics, such as the Boltzmann and Vlasov equations, and adapted to SCLs by Pierre-Louis Lions, Benoît Perthame, and Eitan Tadmor in 1991. The kinetic approach provides a robust framework for the analysis of SCLs, yielding well-posedness results in the purely integrable setting and allowing one to establish stability properties through weak compactness arguments alone. As a guiding example, particular attention will be devoted to the method of compensated compactness, originally introduced by Luc Tartar in 1979 to capture regularizing effects arising solely from the genuine nonlinearity of the flux. We will explain how to recover such results within the kinetic framework through Fourier-analytic techniques and averaging lemmas

Tuesday 10:15-12:00: Formulation via vanishing viscosity
Tuesday 13:15-15:00: Well-posedness
Wednesday 10:15-12:00: Well-posedness/Compensated compactness I
Wednesday 13:15-15:00: Compensated compactness II

Toppforsk@IE: PDE (drop-down)

An introductory lecture series on Stochastic PDEs and Interval Arithmetic in Computer-Assisted Proofs

An introductory lecture series on Stochastic PDEs and Interval Arithmetic in Computer-Assisted Proofs

Date: December 8-11, 2025

Place: Auditorium H1 (Monday-Wednesday), Auditorium EL2 (Thursday)

Julien Vovelle, UMPA, ENS de Lyon
We give an introduction to stochastic partial differential equations, discussing in particular regularization by noise, invariant measures, and the stochastic compactness method for first and second order conservation laws.

Monday 10:15: Quick introduction to the Wiener process and the stochastic integral
Monday 14:15: Itô formula, regularization by the noise in transport equations
Tuesday 10:15: The stochastic compactness method I
Tuesday 14:15: The stochastic compactness method II. Invariant measures

Javier Gomez-Serrano, Brown University
In this series of lectures I will discuss how computers can be used for discovery, intuition, computation and mathematical proofs, using a broad collection of problems.

Wednesday 10:15: Introduction
Wednesday 14:15: Computer-assisted proofs (I)
Thursday 10:15: Computer-assisted proofs (II)
Thursday 14:15: AI+Math

Background

Background

This Toppforsk@IE research group consists of members from the Department of Mathematical Sciences. The group has three research goals: systems with shocks and singularities, stochastic partial differential equations, and computer-assisted proofs.


Systems with shocks and singularities. We seek solution concepts that allow continuation of solutions beyond sudden changes in the qualitative behaviour. Examples are the development of shocks or wave breaking in water waves, or the transition between finitely and infinitely many particles or agents. The latter appears in so-called mean field games which investigates the transition between continuum models. This direction of investigation bridges the group's current expertise with some new areas where it can be applied. 


Stochastic partial differential equations. This type of differential equations has become a hot topic in recent years as its theoretical development is essential in the common ground between statistics and mathematics. It also plays a major role in disiplines with large data sets such as neuroscience and biologi. The group has started pursuing this direction, and intends to broaden its understanding of the topic.


Computer-assisted proofs. A recently developed new tool in mathematics which has, combinded with analytical theory, given surprising and major break throughs within areas where fine or precise estimates are needed, e. g. in fluid mechanics. The tool is not yet exploited in Norway, and the group believes it can be of great use to the community. 

 

Publications

Publications

  • TBA

Toppforsk@IE: PDE (picture)

Photo © Kai T. Dragland (2025)