Math 251B (Topics in Partial Differential Equations).

Instructor: Monica Visan, MS 6167. Email address: visan@math.ucla.edu

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.
• Ill-posedness.
• 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. MR1646048
• 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. MR2233925
• 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 problems:

• Homework 1 is due in class on Friday, February 2nd.