MATH 251A : Introduction to PDE

  • Course description: Introduction to nonlinear evolution equations, both ODE and PDE. Basic tools such as Picard iteration, Gronwall’s inequality, continuity bootstrap methods, and use of conservation laws and integrability will be developed. The Fourier-analytic theory of constant-coefficient linear dispersive equations will be introduced and used to give the basic theory of semilinear Schrodinger, wave, and Korteweg de Vries equations. In the next quarter, the more advanced theory of such equations will be developed.

Announcements:

        (May 21) I will accept the final until 5pm on Wednesday, May 22.

        (May 15) Math 251B will continue this class next quarter. I will be traveling until Apr 6, so the classes will begin on Fri Apr 7.

        (May 10) There is a slight typo in the first homework assignment: the Laplacians Delta should be replaced by Delta throughout. Also a conjugation symbol is missing in the definition of v(0).

        (May 9) I will move my Tue Mar 14 office hours to Friday 1-2.

        (Feb 22) Note that Assignment 6 is due two weeks after Assignment 5.

        (Feb 17) I am now instituting a late policy for homeworks; while I do accept late homeworks, I will now impose a cumulative 10% penalty for each day past the due date, thus for instance 20% will be deducted from a homework which is two days late. On the other hand, due to President’s day, I am extending the deadline for HW5 by one day.

        (Feb 15) Class notes updated slightly to correct Lorentz transform issues. Exercise numbering is unchanged.

        (Feb 7) Class notes updated; all exercise numbering now refers to the most recent version. Solutions to recent HW will be available shortly.

        (Feb 7) Monday office hours moved to Mon 4-5 to avoid a conflict.

        (Jan 6) The first four weeks of lecture will be given by Jim Ralston; his page for the course can be found here.

        (Dec 31) The lecture time and place has reverted to MS 5217 at 12pm, as the move had caused a schedule conflict. It also seems likely that I will be unable to attend the first lecture on Jan 9.

        (Dec 16) The lecture time and place has changed to MS 6201 at 1pm, to avoid conflict with 245C.


 

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

        Lecture: MWF 12-12:50, MS5217

        Quiz section: None

        Office Hours: Mon 4-5, Tu 11-12. I will also check the virtual office hours.

        Textbook: I will use my own book on the subject. This will be printed and distributed in class. In the meantime, you may look at the first three sample chapters. (Here is the postscript version). I hope to cover Chapters 1-3 and the appendix on harmonic analysis in this quarter; chapters 4-6 will be covered next quarter. We will omit Sections 1.7 (integrable systems) and Section 2.6 (X^{s,b} spaces).

        Prerequisite: Math 245AB is highly recommended, as is some exposure to ODE and PDE in undergraduate classes. In particular, you should be comfortable with normed vector spaces, the Fourier transform on R^n, and weak convergence versus strong convergence. Knowledge of distributions and Sobolev spaces is not strictly necessary (we will cover as needed in the class) but would be very useful.

        Grading: Homework (50%), Final (50%).

        Exams: The take-home exam consists of the following five questions, and is due in Wednesday, March 22. I will be traveling on Thursday evening, and cannot accept any finals beyond this date – so don’t procrastinate! You are free to discuss the problems with other students or with myself, but your solution must be in your own words; solutions that are written identically (and have identical errors) will be penalized.

1.      Exercise 1.48 (Viriel identity). There is a typo in the law of gravitation; the direction vector (x_j(t)-x_i(t))/|x_j(t)-x_i(t)| is missing from the right-hand side. Note the previous correction itself had a sign error – it modeled a law of antigravity rather than gravity!

2.      Exercise 1.55 (Stable manifold).

3.      Exercise 2.52 (Weighted Sobolev spaces).

4.      Exercise 3.9. (Localised blowup of NLW). Important typo: “defocusing” should be “focusing”

5.      Exercise 3.18. (L^2 critical wellposedness of NLS). Note: “n” should be “d” throughout, e.g. 2(n+2)/n should be 2(d+2)/d.

Final solutions are available here.

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

        Homework: There will be seven homework assignments. Each assignment will consist of two to three exercises from the text. The numbering is with respect to the latest versions of the notes. Some of the earlier copies of the notes contain errors and the numbering is also somewhat distinct. A 10% penalty is imposed for each day past the due date that the homework is turned in.

1.      First homework (Due Friday, Jan 20): Ex 1.2 [Contraction mapping theorem] or Ex 1.7 [Comparison principle]; Ex 1.9 [Sturm comparison principle]

2.      Second homework (Due Friday, Jan 27): Ex 1.16 [Levinson’s theorem], Ex 1.37 [integrals of motion]; The bootstrap part of Ex 1.23 [compactness method]

3.      Third homework (Due Monday, Feb 6): Ex 1.50 [Stability], Ex 1.52 [Second order Duhamel]. Solutions available here.

4.      Fourth homework (Due Monday, Feb 13): Ex 2.23 [Conformal energy], Ex 2.26 [Gaussian integrals] . Solutions available here. Note: the Lorentz invariance component of the conformal energy question has been stricken, due to (a) being more difficult than originally anticipated, and (b) an incorrect description of the Lorentz transform in the notes.

5.      Fifth homework (Due Tuesday, Feb 21): Ex 2.34 [Asymptotic L^p behaviour of Schrodinger], Ex 2.50 [Local near-conservation of mass], Ex 2.51 [Local near-conservation of mass II]. Note: for Ex 2.34, the decay of t^{-N} in the first part of the problem is correct, but will not be sufficient to get upper bounds on the L^p norm for p < 2. One will have to improve that t^{-N} decay to an |x|^{-N} t^{-N} decay first before you can get L^p bounds. In Exercises 2.50 and 2.51, the constant C is allowed to depend on the dimension d. Solutions available here.

6.      Sixth homework (Due Monday, Mar 6 – note the extension by one week): Ex 2.55 [Local smoothing for Airy equation], Ex 2.64 [Morawetz inequality for wave equation]. Solutions available here.

7.      Seventh homework (Due Friday, Mar 17): Ex 2.28 [Pseudoconformal transformation] (ignore the “additional challenge” and the assertion that the pc transformation is its own inverse. To show that v solves the Schrodinger equation at t=0, verify this for nonzero t first, and also show that v is continuous at t=0 (hint: use the fundamental solution to compute u(1/t, x/t)). At this point there is a slick way to proceed using uniqueness of Schwartz solutions to the Schrodinger equation, without having to do any further differentiation under the integral sign. Correction: Delta should be Delta in the pseudoconformal identity. Also, the Fourier transform of u_0 should instead be the Fourier transform of bar{u_0} (or equivalently, the bar of the Fourier transform of u_0, applied to –x).), Ex 3.14 [Algebra property for H^{k,k}]. (For a definition of the H^{k,k} norm, see Ex 2.52. Hint: you may want to try the d=1,k=1 case first as a warmup.) Solutions available here.