NEW (updated December 22) :

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, chapter 5 (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, chapter 6 (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 chapter 2-2 (

De Giorgi proof of Holder regularity. Existence and uniqueness of solutions to parabolic equations in a Lp & M1 frameworks

See also the previous academic year 2022-2023