Math 251B (Topics in Partial Differential Equations).
Lectures: MWF 10:00-10:50am in MS 6201.
Instructor: Monica Visan, MS 6167. Email address: firstname.lastname@example.org
Office Hours: by appointment.
Topics: We investigate local and global well-posedness for the
semilinear Schrodinger equation. The lectures will cover
- Strichartz, bilinear Strichartz, and local smoothing estimates.
- Subcritical well-posedness and unconditional uniqueness.
- Critical well-posedness, unconditional uniqueness, blowup alternative.
- Stability theory.
- Conservation laws and subcritical global well-posedness.
- Monotonicity formulae and scattering.
- Linear profile decomposition.
- Minimal blowup solutions.
- Energy-critical global well-posedness and scattering.
- T. Cazenave, Semilinear Schrodinger equations. Courant Lecture
Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR2002047
- M. Christ, J. Colliander, and T. Tao, Ill-posedness for
nonlinear Schrodinger and wave equations. Preprint arXiv:math/0311048.
- M. Keel and T. Tao, Endpoint Strichartz estimates. Amer. J.
Math. 120 (1998), no. 5, 955-980.
- R. Killip and M. Visan, Nonlinear Schrodinger equations at critical regularity.
Lecture notes prepared for the Clay Mathematics Institute Summer School, Zurich, Switzerland, 2008.
- H. Koch, D. Tataru, and M. Visan, Dispersive Equations and Nonlinear Waves. Oberwolfach Seminars, 45.
Birkhauser/Springer Basel AG, Basel, 2014.
- T. Tao, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006.
- T. Tao, M. Visan, and X. Zhang, The nonlinear Schrodinger equation
with combined power-type nonlinearities. Comm. Partial Differential
Equations 32 (2007), no. 7-9, 1281-1343. MR2354495
Homework: There will be a small number of homework problems, whose completion is required to pass the class.
- Homework 1 is due in class on Friday, February