Arbeitsgruppe Analysis und Partielle Differentialgleichungen


S4B1 - Graduate Seminar on Analysis (Summer term 2015)



Dozenten:


Termin:

    Monday, 16 (c.t.) - 18 , SemRaum 0.006
    Preliminary meeting: Monday, 09.02., 16:15 pm, SemRaum 0.006

Topic: Blow-up for dispersive equations

    In this seminar we will study blow-up solutions to various dispersive equations. It is a striking feature of critical and supercritical equations (the terms will be explained in the seminar) that there are solutions which blow-up in finite time. Blow-up of an analytic solution is related to a break down of the mathematical model. We will be interested in sufficient criteria on initial data for blow-up, as well as in a precise description of the blow-up.

Literature:

Prerequests:

    Basic knowledge in Fourier transform, Partial differential equations, Harmonic analysis

Talks
  • Semilinear wave equations: Scaling, criticality, focusing/defocusing, blow-up for negative energy data, John's example, ...
  • Sobolev inequality, local well-posedness in the subconformal case, global wellposedness for defocusing equations, persistence of regularity
  • dilation identity, monotonicity formulas, similarity coordinates, Antonini-Merle Lyapunov functional
  • Merle-Zaag proof for universality of blow-up speed
  • The blow up for the critical nonlinear Schroedinger equation (heuristic, statement of rigoros results)
  • Selfsimilar solutions to the supercritical nonlinear Schroedinger equation