An introduction to evolution PDEs
PSL University, September-December 2025
Prerequisite material can be found here.
A quite stable program
is about well-posedness for parabolic and transport equations
as well as longtime
behaviour (mainly for parabolic equations). Additional topics are
the study of particular
nonlinear models, some regularity effect in general parabolic
equation and for kinetic
equations.
An provisional plan is the following :
Chapter 1: A crash
course on evolution PDEs.
Lecture 1 - The heat
equation, lecture 1 (updated September
2025)
We tackle the heat equation with the help of the Fourier, heat
kernel and energy methods.
We next consider general parabolic equations which is handled with
the help of semigroup/perturbation arguments.
Some exercises on lecture 1
(updated September 2025)
Lecture 2
- Transport equations, lecture 2 (updated September
2025)
We solve transport equations by using the characteristics
method.
We next apply the semigroup/perturbation arguments for solving a
kinetic equation.
Some exercises on lecture 2
(updated September 2025)
Lecture 3
- Parabolic
equations, lecture
3
Existence of solutions for parabolic equations by the mean of J.-L.
Lions' variational approach.
Some exercises on lecture 3
Lecture 4 - Uniqueness and qualitative
properties, lecture 4
We carry on our analysis of the solutions of transport and
parabolic equations, establishing uniqueness,
weak maximum principle, strong maximum principle and
ultracontractivity property.
Some exercises on lecture 4
See also the material
of previous academic years and the last years exams:
Exam 2013-2014
Exam 2014-2015
Exam 2015-2016
Exam 2016-2017
Exam 2017-2018
Exam 2018-2019
Exam 2019-2020,
with elements of correction exam2020+.
Exam 2020-2021
Exam 2021-2022
Exam 2022-2023
Exam 2023-2024
Exam 2024-2025
A more extended version of the present lecture will be available here
soon