Prerequisite is the basis of applied functional analysis as one can find (for instance) in the two following classical references:

H. Brézis, (French) [Functional analysis, Theory and applications], Masson, Paris, 1983:

Chap 1, Chap 2, Chap 3, Chap 4, Chap 5, Chap 6, Chap 8, Chap 9

Lieb & Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997:

Chap 1, Chap 2, Chap 5, Chap 6, Chap 7 (from 7.1 to 7.10), Chap 8 (from 8.1 to 8.12), Chap 9

The color notation means that one must know absolutely, be familiarized with, have read at least once that matter.

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

Existence of solutions for parabolic equations by the mean of the

variational approach and the existence Theorem of J.-L. Lions.

A remark on the uniqueness of solutions and the semigroup theory.

A list of important exercises are: Exercises B.1, B.2, B.4, B.8, B.9, B.10.

Chapter 2 - Transport equation: characteristics method en DiPerna-Lions

renormalization theory, chapter 2

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.

Duhamel formula and existence of solutions for equations with (possibly nonlinear) source term.

A list of most important exercises are: Exercises 2.1, 2.3, 2.5, 2.6.

See also the exercises in the Appendix sections.

Chapter 3 - Evolution equation and semigroup, chapter 3

Linear evolution equation and semigroup. Semigroup and generator.

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

Semigroup Hille-Yosida-Lumer-Phillips' existence theory. Complements and discussion.

A list of important exercises are: Exercises 1.4, 4.4, 6.9.

In a second part, we will mainly consider the long term asymptotic issue.

Chapter 4 - More about the heat equation, chapter 4

Smoothing effect thanks to Nash argument.

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 5 - Entropy and applications, chapter 5

Dynamic system, equilibrium, entropy (dissipation

of entropy & Lyapunov-La Salle) methods.

Dissipative operator with compact resolvent, self-adjoint operator

and Krein-Rutman theorem for positive semigroup.

Relative entropy for linear and positive PDE

Applications to a general Fokker-Planck equation, to the

scattering equation and to the growth-fragmentation equation.

In a last part, we will investigate how the different tools we have

introduced before can be useful when considering a nonlinear

evolution problem

Chapter 6 - The parabolic-elliptic Keller-Segel equation

Existence, mass conservation and blow up

Uniqueness

Self-similarity and long time behavior

Notes can be found here and there

Last years examens :

Examen 2013-2014

Examen 2014-2015

Examen 2015-2016

Internship projects (2015-2016):

Project 1 about Fractional diffusion

Project 2 about kinetic Fokker-Planck equation

Project 3 about stability of interacting biological population