**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.