The purpose of the course is to introduce the students to the topic of (mainly nonlinear) partial differential equations showing on few paradigmatic examples how they can be treated thinking of them as
ordinary differential equations in an infinite dimensional setting, using tools of functional analysis and of harmonic analysis. The course is very advanced, and should be taken only by students with a strong background in functional analysis.
Functional analysis, specifically Sobolev spaces and, broadly speaking, the topics of
the 1st year courses ADVANCED ANALYSIS parts A e B.
The course considers advanced topics in partial differential equations.
After an introductory part focused on harmonic analysis, the course
introduces various equations, such as Navier Stokes and nonlinear
Schroedinger, and looks at the initial value problem.
Preliminaries. Riesz 's Interpolation Theorem and some applications. Maximal function . Marcinkiewicz Interpolation Theorem. Theorem by Hardy Sobolev Littlewood. Sobolev's Embedding for homogenous spaces in R^n). Inequality of Gagliardo Nirenberg. Bochner Integral. Stokes Equation . Weak solutions, uniqueness, energy identity. Incompressible Navier Stokes Equations. Weak solutions. Leray 's Theorem on global existece of weak solutions in dimensions 2 and 3. Well posedness in Sobolev spaces. Leray 's Thorem of the uniqueness, well posedness in 2D.
In the 2nd part of the course we will consider semilinear Schroedinger Equations. Keel-Tao's proof of Strichartz estimates. Local well posedness in H^1. Examples of global well posedness. Tsutsumi's of global well posedness in L^2.
The course consists of lectures during which the Instructor discusses all the details of the topics covered, answers student's questions and tries to get them involved. The students will receive before the lectures the lecture notes of the Instructor.
The exam consists of a student seminar of about 30 minutes on a topic arranged with the Instructor, during which the student will show whether or not is able to apply the main ideas presented during the lectures by the Instructor in specific and analogous contexts. The Instructor might ask some questions on the topics covered during the course in class.
The lecture notes and other information will be available through Moodle
Along with some instructor's notes, we will use the following bibliography
1) Bahouri, Chemin, Danchin: Fourier analysis and nonlinear partial differential equations. Springer
2) Cazenave, Haraux: An introduction to semilinear evolution equations. Oxford Univ.Press.
3) Chemin, Desjardins, Gallagher, Grenier: Mathematical Geophisics. Oxford Univ.Press.
5) Robinson, Rodrigo, Sadowski: The three dimensional Navier Stokes Equations,
Cambridge Univ. Press.
6) Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press.
7) Stein: analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press.