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

by Paolo Antonelli

Sunday 1 Dec 2019, 09:00 → Tuesday 31 Dec 2019, 11:00 Europe/Rome

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.