A crash course on evolution (linear) PDEs

Universidad de La Habana, Cuba, June 2-6 2025




 
Course outline


Lecture 1 - The heat equation (3h)

• Topic 1. The heat equation and the Fourier transform

• Topic 2. The heat equation and the heat kernel

• Topic 3. The heat equation and the energy method (a priori estimates)

• Topic 4. Duhamel formula and perturbation argument


Lecture 2 - Transport equations (3h)

• Topic 5. The Gronwall lemma

• Topic 6. Transport equations and smooth data

• Topic 7. Ttransport equations and non-smooth data

• Topic 8. Duhamel formula and perturbation argument (bis)



Lecture 3 - Parabolic equations (3h)

• Topic 9. Introduction to the parabolic equations framework

• Topic 10. Existence of solutions - an implicit Euler scheme approach

• Topic 11. Existence of solutions - the variational approach of J.-L. Lions

• Topic 12. Generalities about evolution equations and semigroups


Lecture 4 - Uniqueness and qualitative properties (3h)

• Topic 13. Uniqueness - the Gronwall lemma approach

• Topic 14. Uniqueness - the duality approach

• Topic 15. Weak maximum principle and strong maximum principle

• Topic 16. Nash approach to ultracontractivity