Math 632 Partial Differential Equations II
Fall 2018

Course Information


Instructor:
  • Jonas Lührmann
  • Email: luehrmann (at) math.jhu.edu
  • Office: Krieger 219
  • Office hours: M 14:00 - 15:00
Lectures:
  • MW 12:00 - 13:15 at Olin 305
Textbook:
  • Partial Differential Equations, Lawrence C. Evans

This course deals with parabolic and hyperbolic evolution equations. In the first half of the course we will develop a general weak theory for linear parabolic and hyperbolic equations. This part of the course is based on Chapter 7 of the book by Evans. In the second half of the course I would like to turn to studying nonlinear wave equations. We will first develop more techniques such as decay estimates and Strichartz estimates for the linear wave equation. Then we will turn to proving local well-posedness for semlinear wave equations as well as global existence and scattering for the defocusing energy-critical nonlinear wave equation. The second half of the course will be mainly based on my own lecture notes, but I will provide further references as we go.



Prerequisites:

It would be desirable to have taken the course Math631 Partial Differential Equations I, but it is not a requirement. (Math631 basically covers Chapters 1, 2, 5, 6 and parts of Chapters 3 and 4 from Evans' textbook). However, some knowledge of Sobolev spaces as in Chapter 5 of Evans' textbook will be assumed. The course will start off with a brief review of Chapter 6 (second order elliptic equations) from Evans' textbook.



Grade Policy:

The course grade will be based solely on the homework. Homework assignments will be posted on blackboard.

Tentative Course Contents:

Linear parabolic and hyperbolic evolutions:

  • Analysis tools: Bochner integral, time-dependent Sobolev spaces
  • Weak theory of linear parabolic equations
  • Weak theory of linear hyperbolic equations
  • Hyperbolic systems of first-order equations
  • Operator semigroups

Nonlinear wave equations:

  • Littlewood-Paley theory
  • Decay estimates for the wave equation
  • Strichartz estimates for the wave equation
  • Local well-posedness for semilinear wave equations
  • Conservation laws and Morawetz estimates
  • Global existence and scattering for the defocusing energy-critical nonlinear wave equation


JHU Ethics Statement:

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Cheating is wrong. Cheating hurts our community by undermining academic integrity, creating mistrust, and fostering unfair competition. The university will punish cheaters with failure on an assignment, failure in a course, permanent transcript notation, suspension, and/or expulsion. Offenses may be reported to medical, law, or other professional or graduate schools when a cheater applies.

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In this course, as in many math courses, working in groups to study particular problems and discuss theory is strongly encouraged. Your ability to talk mathematics is of particular importance to your general understanding of mathematics. You should collaborate with other students in this course on the general construction of homework assignment problems. However, you must write up the solutions to these homework problems individually and separately. If there is any question as to what this statement means, please ask the instructor.