The Formation of Shockwaves

Lecturer: Prof. Speck
Dates: June 6-7 and June 15-17
Course description:
We will discuss stable blow-up results for solutions to quasilinear wave equations, including the irrotational compressible Euler equations of fluid mechanics. The singularities are of "wave-breaking" type, which means that the solution remains bounded even though its first derivatives blow up. The first general blow-up results for these and related equations were derived in 1964 by P. Lax, who treated a class of systems in one space dimension. In the first part of the course, we will revisit Lax’s results from a modern perspective. Our modern approach will allow us to sharpen Lax’s results in a way that is essential for proving blow-up in more than one space dimension. In the second part of the course, we will overview how modern techniques in the geometric analysis of wave equations can be used to extend Lax’s results to the case of two or more space dimensions. In the last part of the course, we will survey the state of the field and discuss open problems.

Optional Background Reading

  1. Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations, J. Speck; Chapters 1 and 2 only
  2. Shock formation in small-data solutions to 3D quasilinear wave equations: An Overview, G. Holzegel, S. Klainerman, J. Speck, and W. Wong
  3. Stable shock formation for nearly plane symmetric waves, G. Holzegel, J. Luk, J. Speck, and W. Wong
  4. Riemannian Geometry, M. do Carmo and F. Flaherty, first few chapters (for the basics of differential geometry)

Waves on Black Hole Backgrounds

Lecturer: Prof. Holzegel
Dates: June 8-10 and June 13-14
Course description: 
We will begin by introducing the notion of a black hole and discussing the Schwarzschild spacetime as the simplest example of a black hole geometry. Afterwards, our main objective will be to establish global (decay) estimates for solutions to the linear wave equation on the black hole exterior of a fixed Schwarzschild spacetime. Here a version of the vectorfield method introduced in Dr. Shlapentokh-Rothman’s course will play an important role, both to prove the desired bounds and to understand the interplay between the black hole geometry and the estimates that we shall prove. The techniques developed will also allow us to consider more complicated black hole backgrounds, which will be discussed as time permits. The course will end with an outlook to the black hole stability problem including some topics of current research.

Optional Background Reading

  1. Lectures on black holes and linear waves, M. Dafermos and I. Rodnianski
  2. Riemannian Geometry, M. do Carmo and F. Flaherty, first few chapters (for the basics of differential geometry)

Geometric Methods for Kinetic Equations

Lecturer: Prof. Smulevici
Dates: June 13-17
Course description:
The aim of the course is to introduce some new tools for the analysis of geometric systems of Kinetic and wave equations such as the Vlasov-Maxwell or the Einstein-Vlasov systems. In particular, we will present a vector field approach for decay of velocity averages together with some applications to small data global existence for some
simplified models. Side topics will include bilinear forms for the interaction between massless Kinetic and scalar fields, an example of propagation of moments for a non-linear system and a discussion of current research in the field.

Optional Background Reading

  1. Small data solutions of the Vlasov-Poisson system and the vector field method
  2. The Cauchy Problem in Kinetic Theory, B. Glassey

Introduction to the Vector Field Method

Lecturer: Dr. Shlapentokh-Rothman
Dates: June 6-10 
Course description:
The goal of this course will be to introduce the so called "vector field method" as applied to the study of nonlinear wave equations. We will start with an introduction to geometric energy estimates and use this to establish basic (local) existence and uniqueness results for linear (nonlinear) wave equations. Next, we will show how the conformal estimate of Morawetz and Klainerman-Sobolev inequalities may be used to establish decay for the linear wave equation using energy methods. Finally, we will apply these decay results to establish some classical small data global existence theorems for nonlinear wave equations, i.e., global existence for small data when n >= 4 and, when n>= 3, global existence for small data when the equations satisfy Klainerman's null condition.

Optional Background Reading

  1. Lectures on black holes and linear waves, M. Dafermos and I. Rodnianski
  2. Lectures on Nonlinear Wave Equations, second edition, C. Sogge
  3. Lectures on Nonlinear Wave Equations by Q. Wang
  4. Geometric Analysis of Hyperbolic Equations: An Introduction, S. Alinhac
  5. Hyperbolic Partial Differential Equations, S. Alinhac