An introduction to evolution PDEs
Academic Master 2nd year
Paris-Dauphine, September-November 2020
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.
During the two first weeks, we will focus on part of the material of Chapter 1 and Chapter 2.
We will also deal with the following associated exercises.
A chapter about the Navier-Stockes equation and about kinetic equations will probably replace
the usual chapter about the Keller-Segel equation.
**** program of academic year 2019-2020 ****
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 1 - On the Gronwall Lemma, chapter 1
In a first part, we will present several results about
the well-posedness issue for evolution PDE.
Chapter 2- Variational solution for parabolic equation, chapter 2
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 3 - Transport equation: characteristics method en DiPerna-Lions
renormalization theory, chapter 3
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 4 - Evolution equation and semigroup, chapter 4
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 5 - More about the heat equation, chapter 5
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 6 - 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 7 - The parabolic-elliptic Keller-Segel equation, chapter 7
Existence, mass conservation and blow up
Uniqueness
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
Exam 2019-2020
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.