An introduction to evolution PDEs
Academic Master 2nd year
Paris-Dauphine, September-November 2014
In a first part, we will present several results about
the well-posedness issue for evolution PDE.
Chapter 1 - Parabolic equation, chapter 1
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.
Chapter 2 - Transport equation, chapter 2
Existence of solutions by the mean of the characterics 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.
In a second part, we will mainly consider the long term asymptotic issue.
Chapter 3 - More about the heat equation, chapter 3
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 4 - Entropy and applications, chapter 4
Dynamic system, equilibirum, 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.
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 5 - The parabolic-elliptic Keller-Segel equation
Existence, mass conservation and blow up
Uniqueness
Self-similarity and long time behavior