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.