An introduction to evolution PDEs

Additional material


 


Possible prerequisite is
- On the Gronwall Lemma, chapter 0 (updated October 2020)
- ODE, ODE (updated January 2023)
- Review of differential calculus for ODEs and PDEs 1st course, 2nd&3rd courses (updated September 2022)
- applied functional analysis as one can find in the book of Brézis, of Kavian (first chapter) and of Lieb & Loss
A quite stable program is about well-posedness for parabolic and transport equations
as well as longtime behaviour (mainly for parabolic equations). Additional topics are
the study of particular nonlinear models, some regularity effect in general parabolic
equation and for kinetic equations.

See also the previous academic year 2022-2023
See also the previous academic year 2023-2024


First Part - on the well-posedness issue for evolution PDE.

Chapter - 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 - 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.


Second part - on 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.   


2023 program:
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 2 -  De Giorgi-Nash-Moser theory and beyond for parabolic equations - second part chapter 2-2 (updated December 22, 2023)
De Giorgi proof of Holder regularity. Existence and uniqueness of solutions to parabolic equations in a Lp & M1 frameworks


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.

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


Excercises:
- list 1 and list 2 of excercises
- Some exercises on chapter 1 and a related exam2020, with elements of correction exam2020+.
- Some exercises on chapter 1 & 2.
- Some exercises on chapter 3 and a related exam2022.
- Some exercices on chapters 0 to 6 and some related exams.


 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