An introduction to evolution PDEs

Academic Master 2nd year

Paris-Dauphine, September-November 2019


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.

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 which several
variants are presented in a
Chapter 0 - On the Gronwall Lemma, chapter 0

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 - Markov semigroup, chapter 6
Brief introduction to positive and Markov semigroups as well as
quantitative asymptotic through Doeblin-Harris technique.   
Applications to a general Fokker-Planck equation and to the
scattering 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, chapter 5
Existence, mass conservation and blow up
Self-similarity and long time behavior

Last years examens :
Exam 2013-2014
Exam 2014-2015
Exam 2015-2016
Exam 2016-2017
Exam 2017-2018
Exam 2018-2019

Internship projects (2015-2016):
Project 1 about Fractional diffusion
Project 2 about kinetic Fokker-Planck equation
Project 3 about stability of interacting biological population

2017 program: - Entropy and applications, chapter 7
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.