**Course description**: Continuation of Math 251A. The local and global wellposedness theory of subcritical and critical semilinear Schrodinger, wave, and Korteweg de Vries equations; this theory is then used to develop further properties of such equations such as scattering, stability of solitons, and nature of blowup.

**Announcements:**

· (May 18) I will be traveling from May 31 onwards. Hence, the final class will be on Monday May 29.

·
(May 18) Clarification: the HW is due **Monday** May 22. There will be no further homework
assignments.

· (Apr 19) Due to a typo in one of the homework questions (and because I am behind in lectures), I am rolling back the due date for HW 1 to Friday Apr 21. Also, the second homework question turns out to be more difficult than anticipated (it requires either the Hardy-Littlewood maximal inequality, or the Fefferman-Stein vector-valued generalization of that inequality), so I am replacing it with a simplified version, see below.

·
(Apr 17) I will be in

· (May 15) I will be traveling until Apr 6, so the classes will begin on Fri Apr 7.

·
**Instructor**: Terence Tao, tao@math.ucla.edu, x64844, MS 5622

·
**Lecture: **MWF 12-12:50, MS5138

·
**Quiz section: **None

·
**Office Hours**: Mon 4-5.

·
**Textbook**: I will use my own book on the
subject. The first half of the course
will be devoted to covering Chapter 3, which is available
here. The second half will cover at
least part of Chapters 4, 5, and Appendix B; I will distribute material from
these chapters as appropriate.

·
**Prerequisite**: Must have passed 251A from
winter quarter, as we shall presume strong familiarity with the material
covered in that quarter (i.e. Chapters 1, 2, and the first part of Chapter
3).

·
**Grading**: Anyone who has passed 251A from
winter quarter is guaranteed a pass for attending this course. The letter grade will however depend on the
solution to approximately seven or eight homework problems, which will be
announced on this web page throughout the course. There will be no final exam.

·
**Reading**** Assignment**: It is strongly recommended
that you read the book concurrently with the course.

·
**Homework**: There will be three homework
assignments, assigned from the text. A 10% penalty is imposed for each day past
the due date that the homework is turned in.

1. First
homework (Due date Fri Apr 21 – note revised due date): Exercise A.8
(Relationship between Sobolev and isoperimetric inequality), Exercise A.13
(Fractional chain rule). **Correction to Exercise A.8**: In the
endpoint Sobolev inequality, both appearances of the exponent d should be
replaced by d/(d-1). **Simplification to Exercise A.13**: The
actual estimate is quite hard – in order to sum the Littlewood-Paley
pieces back together again one needs some nontrivial harmonic analysis, such as
the Hardy-Littlewood maximal inequality and the Littlewood-Paley square
function inequality. I will accept the
weaker estimate in which one only controls P_N F(f) rather than F(f) for each
N, i.e. || P_N F(f) ||_{W^{s,q}}
<~ || f ||_{W^{s,r}}^p. Solutions can be
found here.

2. Second homework (Due Fri May 5): Exercise 3.34 (H^1 bounds for 1D cubic NLS). Everybody followed the hint and did things correctly, so I won’t post a solution.

3. Third homework (Due Mon May 22): Exercise 3.28 (H^s growth for 1D cubic NLS), Exercise 3.45 (alternate derivation of interaction Morawetz inequality). Solutions available here.