Non-Linear Dispersive Partial Differential Equations
Aims
- To know basic and advanced tools of Fourier analysis, and their application to the study of nonlinear partial differential equations;
- To be able to investigate local and global well-posedness for the Cauchy problem of semi- and quasi-linear Wave and Schrödinger equations;
- To understand the physical relevance of nonlinear partial differential equations.
Program
- Basic tools of Fourier analysis (Inequalities, Sobolev spaces and their embeddings);
- Littlewood-Paley theory, Besov spaces;
- Notions of solution of a (nonlinear) partial differential equation, boundary and initial value problems;
- Fixed point arguments, conservation laws, and energy method;
- TT* argument and Strichartz estimates;
- Applications to nonlinear dispersive equations: Wave and Schrödinger equations.
Suggested Bibliography
Books
- H. Bahouri, J.-Y. Chemin, R. Danchin. Fourier Analysis and Nonlinear Partial Differential Equations. Springer Verlag, 2011.
- L. C. Evans. Partial Differential Equations: Second Edition. AMS Graduate Studies in Mathematics, 2010.
- T. Cazenave. Semilinear Schrödinger Equations. Courant Lecture Notes, AMS (2003).
Freely available online lecture notes
- C. E. Kenig. The Cauchy Problem for the Quasilinear Schrödinger Equation (Following Kenig-Ponce-Vega).
- T. Tao. Nonlinear dispersive equations: local and global analysis.
- J.-Y. Chemin. Notes du Cours "Introduction aux Équations aux Dérivées Partielles d'Évolution".
Organization
- Exercise Classes and Homework
One exercise class / complementary lecture per week. The homework assignments will be given biweekly. Half of the total homework points are required to be admitted to the final exam.
- Final Exam
The final examination will be an oral verification of the knowledge acquired during the course.