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)
The plan is the following :
Chapter 1 -
Variational solution for parabolic equation, chapter 1 (updated September
2024)
Existence of solutions for parabolic equations by the mean of J.-L.
Lions' variational approach.
Some exercises on chapter 1
(updated September 2024)
Chapter 2 - De
Giorgi-Nash-Moser theory and beyond for parabolic equations, chapter 2 (updated October
2024)
We establish the ultracontractivity property of parabolic equations
using different approaches developed by De Giorgi, Nash and Moser.
We also establish the Holder regularity and the Harnack inequality.
We next deduce
existence and uniqueness of the fundamental solution to
parabolic equations.
Some exercises on chapter 2
(updated October 2024)
Chapter 3 - Evolution equation,
semigroup and longtime behaviour,
chapter 3 (updated
October 2024)
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.
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,
with elements of correction exam2020+.
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