The aim of this course is to provide basic tools for the study of partial differential equations. The course is divided in two parts. The first part will focus on linear partial differential operators with emphasis on spectral theory. The second part will be devoted to the study of non-linear PDE’s.
- Introduction and recalls on Fourier transform, Sobolev spaces, bounded operators, compact operators, Krein Rutman Theorem.
- Linear PDE’s: – Unbounded operators: domains, closed and closable operators, adjoints, self-adjoint operators, Friedrich’s extension. – Spectrum of unbounded operators : resolvent, spectrum, essential spectrum, Weyl theorem’s, maxi-min principle, operators with compact resolvent. – Accretive operators : recalls on Hille-Yosida theorem, semigroups and application to PDE’s, maximum principle.
- Non-linear PDE’s: – The inviscid and viscous Burgers equation (blow-up, shocks, travelling waves). – The incompressible Euler equations (weak and strong solutions. The incompressible Navier-Stokes and the Keller-Segel equations. – The nonlinear Schrödinger equation (Strichartz estimates). – Nonlinear elliptic estimates. Various methods such as compactness and fixed point method will be used.