An introduction to evolution PDEs
PSL University, September-December 2023
NEW (updated December 22) :
the "written exam" will mainly focus on chapters 1, 2-1, 4, 2-2
and on the material of the tutorials (contrary to what has
been announced previously). The "oral exam" will take the form of a
discussion about the "written exam" and on all the material
presented during the course.
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 in the
Chapters 0
- On the
Gronwall Lemma, Gronwall (updated October
2020)
- ODE, ODE (updated January 2023)
In a first part, we will present several results about the well-posedness issue for evolution PDE.
Chapter 1 -
Variational solution for parabolic equation, chapter 1 (updated October
2023)
Existence of solutions for parabolic equations by the mean of the
variational approach and the existence Theorem of J.-L. Lions.
Some exercises
on chapter 1 and a related exam2020,
with elements of correction exam2020+.
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 4 - Transport equation:
characteristics method en DiPerna-Lions renormalization theory,
chapter 4 (updated November
2023)
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.
Chapter 5 - Evolution equation
and semigroup, chapter 5 (updated
December 2023)
Linear evolution equation and semigroup.
Semigroup and generator.
Duhamel
formula and mild solution. Coming back to the well-posedness
issue.
Chapter 6 - Semigroup and
longtime behaviour, chapter 6 (updated
December 15, 2023)
L1
convergence for the Fokker-Planck equation.
Doblin-Harris theorem of convergence.
Some exercices
on chapters 0 to 6 and some related exams.
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
See also the previous academic
year 2022-2023