PSL University, September-December 2023

NEW (updated December 22) :  the "written exam" will mainly focus on chapters 1, 2-1, 4, 2-2 and on the material of the tutorials (contrary to what has been announced previously). The "oral exam" will take the form of a discussion about the "written exam" and on all the material presented during the course.

A prerequisite for the analysis of evolution PDE (in order to establish pointwise estimates for both
existence theory and long time asymptotic analysis) is the so-called Gronwall lemma presented in the

Chapters 0
- On the Gronwall Lemma,  Gronwall (updated October 2020)
- ODE, ODE (updated January 2023)

In a first part, we will present several results about the well-posedness issue for evolution PDE.

Chapter  1 - Variational solution for parabolic equation, chapter 1 (updated October 2023)
Existence of solutions for parabolic equations by the mean of the
variational approach and the existence Theorem of J.-L. Lions.
Some exercises on chapter 1 and a related exam2020, with elements of correction exam2020+.

Chapter  2 - De Giorgi-Nash-Moser theory and beyond for parabolic equations - first part, chapter 2-1 (updated November 2023)
We establish the ultracontractivity property of parabolic equations using different approaches developed by De Giorgi, Nash, Moser
and Boccardo-Gallouët.
Some exercises on chapter 1 & 2.

Chapter  3 -  The Fokker-Planck equation, the Poincaré inequality and longtime behaviour, chapter 3 (updated November 2023)
We deduce the Fokker-Planck equation as the evolution equation for the solutions of the heat equation in self-similar variables.
We establish the exponential convergence to the Gaussian equilibrium of the solutions to the Fokker-Planck equation with the help of
the Poincaré inequality.
Some exercises on chapter 3 and a related exam2022.

Chapter 4 -  Transport equation: characteristics method en DiPerna-Lions renormalization theory, chapter 4 (updated November 2023)
Existence of solutions by the mean of the characteristics method and  renormalization theory of DiPerna-Lions.
Uniqueness of solutions thanks to Gronwall argument and duality argument.

Chapter 5 -  Evolution equation and semigroup, (updated December 2023)
Linear evolution equation and semigroup. Semigroup and generator.

Duhamel formula and mild solution. Coming back to the well-posedness issue.

Chapter 6 -  Semigroup and longtime behaviour, (updated December 15, 2023)
L1 convergence for the Fokker-Planck equation.
Doblin-Harris theorem of convergence.

Some exercices on chapters 0 to 6
and some related exams.

Chapter 2 -  De Giorgi-Nash-Moser theory and beyond for parabolic equations - second part updated December 22, 2023)
De Giorgi proof of Holder regularity. Existence and uniqueness of solutions to parabolic equations in a Lp & M1 frameworks