Math Courses

Applied Partial Differential Equation. Part II: Nonlinear Schrödinger equations.

by Paolo Antonelli

→ Europe/Rome
Description

Lecturer:
Paolo Antonelli (GSSI)

Content:

  • Review of basic tools from harmonic analysis: real and complex interpolation.
  • Derivation of effective equations for nonlinear dispersive waves.
  • Invariances and conserved quantities: the Noether’s theorem.
  • Existence of local regular solutions: the energy method.
  • Local and global smoothing estimates associated to the linear propagator: dispersive estimates, Strichartz estimates, Kato smoothing estimates.
  • The local Cauchy problem for the nonlinear Schrödinger equation in H1 and L2.
  • Global existence and asymptotic behavior for repulsive nonlinearities; scattering theory.
  • Formation of singularities at finite times: blow-up results based on virial arguments.
  • Stability of solitary waves: concentration-compactness.
  • Instability of solitary waves in the mass-critical case, universality of the blow-up profile with minimal mass.