An introduction to evolution PDEs

Academic Master 2nd year

Paris-Dauphine, September-December 2022



 


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.

The following associated list 1 and list 2 of excercises will be dealt with during the tutorials. (updated November 2021)
 
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 (updated October 2020)

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 (updated October 2020)
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 A.1, A.2, A.4, A.6, A.7
See also the above list of exercises for tutorials.

Chapter  3 - Transport equation: characteristics method en DiPerna-Lions
renormalization theory, chapter 3 (updated October 2020)
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 and in the above list of exercises for tutorials.

Chapter 4 - Evolution equation and semigroup, chapter 4 (updated October 2020) and some complement to 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 (updated November 2021)
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 - More about longtime asymptotic, chapter 6 (updated January 16, 2023 !)
Brief introduction on entropy technique.
Brief introduction on Krein-Rutman theory.
Brief introduction to positive and Stochastics semigroups as well as
quantitative asymptotic through Doeblin-Harris technique.   



Last years exams :
Exam 2013-2014
Exam 2014-2015
Exam 2015-2016
Exam 2016-2017
Exam 2017-2018
Exam 2018-2019
Exam 2019-2020
Exam 2020-2021
Exam 2021-2022
Exam 2022-2023
Exam 2023-2024

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

PREFALC Master course: On the Keller-Segel equation
Santiago de Chile, March-April 2018

Chile, March-April 2018


2021 program:
Chapter 6 - More about longtime asymptotic, chapter 6
Brief introduction on entropy technique.
Brief introduction to positive and Stochastics semigroups as well as
quantitative asymptotic through Doeblin-Harris technique.   
Applications to a renewall equation will be considered. 

In a last part, we will probably investigate how the different tools we have introduced before
can be useful when considering a nonlinear evolution problem

Chapter 6 - A crash course about kinetic equations, chapter 7
Brief introduction to classical mathematical tools for kinetic equatopn (taking as an example the free
transport equation).
Brief introduction to the Landau equation (picked up from Exam2019-2020).
Brief introduction to hypocoercivity techniques.

2020 program:
Chapter 6 - Stochastiscs semigroup, chapter 6
Brief introduction to positive and Stochastics semigroups as well as
quantitative asymptotic through Doeblin-Harris technique.   
Applications to a renewall equation will be considered.
convergence to the equilibrium (with rate) in $L^1$.

Chapter 7- Navier-Stokes equation  chapter 7
A chapter about the Navier-Stockes equation and about kinetic equations replace
the usual chapter about the Keller-Segel equation.

and the associated list 1 and list 2 of excercises

 2017 program:
Chapter 7 - 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.

2019 program:
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