MATH 251B : Partial differential equations

  • 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 Canada on Mon Apr 24, and so will not give classes that day. For that week I will hold office hours Tue Apr 25 11-12am.

        (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.