Advanced PDE II: Hyperbolic PDE (MIGSAA Ph.D. course & SMSTC advanced course, Spring 2016, Spring 2017, Spring 2018)

 

Instructors:	Pieter Blue and Oana Pocovnicu
Syllabus & references 2018:	click here
Lecture notes from 2016:	Lecture 1 (Classification of PDEs, definition of hyperbolicity, method of characteristics, O.P. 2016), Lecture 2 (Shock formation for the inviscid Burger's equation, D'Alembert's formula for the linear wave equation on ℝ, O.P. 2016) Lecture 3 (Explicit formulas for the solutions of the linear wave equation on ℝn, brief presentation of the Cauchy-Kowalevski theorem, O.P. 2016) Lecture 4 (Cauchy-Kowalevski theorem, Fourier transform and the linear wave equation, O.P. 2016) Lecture 5 (part 1) (Sobolev spaces and embeddings, Gronwall's lemma, ...) Lecture 5 (part 2) (Well-posedness, Illustrative examples to prepare for the vector-field method, P.B. 2016) Lecture 6 (Introduction to tensor notation, energy-momentum tensor and its properties, $C^2$ uniqueness. Follows C. Sogge's book. P.B. 2016) 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. P.B. 2016) Lecture 8 (Existence and uniqueness of solutions to quasilinear equations using energy estimates. $s$-chain/product rule. Follows C. Sogge's book. P.B. 2016) Lecture 8b (Continuous dependence of solutions on initial data for quasilinear wave equations. P.B. 2016) 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. P.B. 2016) Lecture 10 (Global existence for quasilinear wave equations in high dimensions. A brief survey of null forms. Again, follows C. Sogge's book. P.B. 2016)
Some additional lecture notes from 2017:	Lectures 4, 5, 7 (Strichartz estimates for the wave equation, O.P. 2017),
Some additional lecture notes from 2018:	Lectures 4 (Local and global well-posedness in H1(ℝd) of the defocusing cubic nonlinear wave equation, O.P. 2018), Lectures 5 (Local well-posedness in H1(ℝd) of the energy-critical quintic nonlinear wave equation, O.P. 2018), Lectures 6 (Global well-posedness in H1(ℝd) of the energy-critical defocusing quintic nonlinear wave equation I, O.P. 2018), Lectures 7 (Global well-posedness in H1(ℝd) of the energy-critical defocusing quintic nonlinear wave equation II, O.P. 2018),
Assignments:	2016 HW 1, 2016 HW 2, 2016 HW 3 2017 HW 1, 2017 HW 2 2018 HW 1, 2018 HW 2