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.
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.
Complements to Lecture 4 - The
McKean-Vlasov equation, lecture 4+
We establish the existence and uniqueness of solutions to the
McKean-Vlasov equation
Some exercises on lectures 3,4
& 4+
Chapter 2: The Landau equation
We establish the existence of weak solutions to the Landau equation
Chapter 3: More about the heat
equation (self-similarity)
Rescaled (self-similar)
variables and Fokker-Planck equation.
Poincaré inequality and
long time asymptotic (with rate) in $L^2$
Fisher information, log
Sobolev inequality and long time
convergence to the
equilibrium (with rate) in $L^1$.
Chapter 4: More about the longtime
asymptotic (for positive semigroups)
Brief introduction on
entropy technique.
Brief introduction to
positive and Stochastics semigroups as well as
quantitative asymptotic
through Doeblin-Harris technique.
Applications to a
parabolic equation will be considered.
Chapter 5: More
about the Landau equation
We
establish the uniqueness of solutions to the Landau equation
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