Instructors: | Pieter Blue and Oana Pocovnicu |
Lecture: |
Wednesday 9:00-11:00 (10 lectures, 11 Jan. 2016 - 18 Mar. 2016)
Room 107 (on the ground floor) in ICMS, 15 South College Street, Edinburgh |
SMSTC course website: | http://www.smstc.ac.uk/advanced_courses/advanced_pde_2 |
Syllabus & references: | click here |
Homework: |
HW 1 (to be submitted in class on February 3rd)
HW 2 (to be submitted in class on March 1st) HW 3 (posted on March 10th, to be submitted by March 25th) |
Lecture notes: |
Lecture
1 (Classification of PDEs, definition of hyperbolicity, method
of characteristics),
Lecture 2 (Shock formation for the inviscid Burger's equation, D'Alembert's formula for the linear wave equation on ℝ) Lecture 3 (Explicit formulas for the solutions of the linear wave equation on ℝ^{n}, brief presentation of the Cauchy-Kowalevski theorem) Lecture 4 (Cauchy-Kowalevski theorem, Fourier transform and the linear wave equation) Lecture 5 (Sobolev spaces and embeddings, Gronwall's lemma, ...) Lecture 5 (part 2) (Well-posedness, Illustrative examples to prepare for the vector-field method.) Lecture 6 (Introduction to tensor notation, energy-momentum tensor and its properties, $C^2$ uniqueness. Follows C. Sogge's book.) Lecture 7 (Existence of solutions to linear equations using energy estimates. $s$-chain/product rule. Hand-written draft to be replaced by type notes "soon". Follows C. Sogge's book.) Lecture 8 (Existence and uniqueness of solutions to quasilinear equations using energy estimates. $s$-chain/product rule. Follows C. Sogge's book. Gap in H^s product rule from lecture has now been corrected in the notes. Minor typoes fixed 2016-04-22.) Lecture 8b (Continuous dependence of solutions on initial data for quasilinear wave equations. This replaces an incorrect argument for continuity given in an earlier version of the notes. Minor typoes fixed 2016-04-22.) Lecture 9 (Symmetries for the wave equation, the Klainerman-Sobolev inequality, and decay for the wave equation using vector-field methods. Again, follows C. Sogge's book. Minor typoes fixed 2016-05-04.) Lecture 10 (Global existence for quasilinear wave equations in high dimensions. A brief survey of null forms. Again, follows C. Sogge's book. Powers of epsilon fixed 2016-07-08.) |