An introduction to evolution PDEs
PSL University, September-December 2024
Chapters 0 - Prerequisite
A prerequisite is also 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 presented here
- On the Gronwall Lemma, essentially
sections 1 & 2 (updated October 2020)
A tentative plan is the following :
Chapter 1 -
Variational solution for parabolic equation, chapter 1 (updated October
2023)
Existence of solutions for parabolic equations by the mean of J.-L.
Lions' variational approach.
Chapter 2 - De
Giorgi-Nash-Moser theory and beyond for parabolic equations
We establish the ultracontractivity property of parabolic equations
using different approaches developed by De Giorgi, Nash, Moser
and Boccardo-Gallouët. We deduce
existence and uniqueness of solutions to parabolic equations
in a Lp & M1 frameworks.
We next establish the Holder regularity and the Harnack inequality.
Chapter 3 - Evolution equation,
semigroup and longtime behaviour
Linear evolution equation and semigroup.
Semigroup and generator. Duhamel
formula and mild solution. Coming back to the well-posedness
issue.
Doblin-Harris theorem of
convergence and the Krein-Rutman theory.
See also the material
of previous academic years and the 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
A more extended version of the present lecture will be available here
soon