Math 567: Notes

• I. Linear Waves and Dispersion
  • Jan. 5: linear `wave' equations and dispersion relations
  • Jan. 10: solution by Fourier transform
  • Jan. 12: dispersion and group velocity
  • Jan. 17: measuring dispersion: decay estimates
  • Jan. 19: measuring dispersion: space-time estimates

• II. Nonlinear Dispersive PDE: Examples, Structure, Well-Posedness
  • Jan. 31: the water wave problem: shallow water, Boussinesq, and KdV
  • Feb. 2: origins of the nonlinear Schroedinger equation
  • Feb. 7: local existence theory (classical solutions)
  • Feb. 9: local existence theory (rougher solutions)
  • Feb. 14: Hamiltonian systems, symmetries, conservation laws: ODEs
  • Feb. 16: Hamiltonian systems, symmetries, conservation laws: PDEs
  • Feb. 28: Scaling and criticality
  • Mar. 2: Global existence
• III. Nonlinear Dispersive PDE: Qualitative Behaviour
  • Mar. 7: Nonlinear scattering theory: small-data scattering
  • Mar. 9: Nonlinear scattering theory: remarks on large-data scattering
  • Mar. 14: Solitary waves: existence
  • Mar. 16: Solitary waves: stability
  • Mar. 21: Finite-time blow-up

• IV. Introduction to Well-Posedness for Critical-Scaling PDE
  • Mar. 23: critical local theory (revisited)
  • Mar. 28: concentration compactness
  • Apr. 4: minimial blowup solution
  • Apr. 6: rigidity