Notes of the CIMPA Research Summer School course
"Mathematical basis for evolution PDEs"
La Habana, June-July 2013
Lectures 1 & 2 (June 24th and 25th, 2013) :
Chapter 1 - Variational solution for parabolic equation
Existence of solutions for parabolic equations by the mean of the
variational approach and the existence Theorem of J.-L. Lions.
Lectures 3, 4 & 5 (June 26th and 27th, 2013) :
Chapter 2 - Transport equation : characteristics method and DiPerna-Lions renormalization theory
Existence of solutions by the mean of the characterics method.
Renormalization theory of DiPerna-Lions.
A remark on the transport theory and solutions built by duality.
Uniqueness of solutions thanks to Gronwall's argument and duality argument.
Duhamel formula and existence of solutions for equations with a source term.
Lecture 5, 6 & 7 (June 28th, July 1st and 2nd, 2013) :
Chapter 3 - More about the heat equation.
Long time asymptotic behaviour for the heat equation by nonlinear tools
Long time asymptotic behaviour for the heat equation by spectral analysis for semigroups tools
Initial program:
Lesson 1 - Parabolic equation.
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.
Lesson 2 - Transport equation.
Existence of solutions by the mean of the characterics method and
renormalization theory of DiPerna-Lions.
A remark on the transport theory and solutions built by duality.
Lesson 3 & 4 - Uniqueness and complements.
Uniqueness of solutions thanks to Gronwall's argument and duality argument.
Duhamel formula and existence of solutions for equations with (possibly nonlinear) source term.
Positivity, conservation laws and Lyapunov functionals.
A priori estimates (moment and Lebesgue norm) and compactness method.
Tutorial on the well-posedness problem for (simple) biological partial differential evolution
equations (if possible, in relation with the models studied in the other courses).
Lesson 5 - More about the heat equation.
Smoothing effect thanks to (robust) Nash's argument.
Rescaled (self-similar) variables and Fokker-Planck equation.
Poincaré inequality and long time asymptotic (with rate).
Lesson 6 - Entropy of Boltzmann.
Fisher information, log Sobolev inequality and long time
convergence to the equilibrium (with rate).
Lesson 7 - Spectral analysis for semigroups.
Resolvants and spectral analysis of generators.
Spectral mapping theorem.
Rate of convergence for solutions to the Fokker-Planck equation
in a weighted Lebesgue space thanks to the extension of the
spectral gap approach.