Non-Linear Wave equations - Einzelansicht

This course is an introduction to the theory of non-linear wave equations. The plan is to cover the following topics:

(1) The linear wave equation in Minkowski space: Representation formula, Fourier techniques, energy estimates, vectorfield method of Klainerman, local existence for general linear equations

(2) Local well-posedness theory non-linear wave equations.

(3) Small data global existence for semi-linear equations with null condition.

(4) Classical blow-up results for semi-linear wave equations

(5) Shock formation results for quasi-linear wave equations (in symmetry)

Some applications to the Einstein equations of general relativity will also be discussed.

[1] J. Luk, Lecture Notes on Non-Linear wave equations; Available at https://web.stanford.edu/~jluk/NWnotes.pdf<https://web.stanford.edu/%7Ejluk/NWnotes.pdf>

[2] C. Sogge, Lectures on Non-linear wave equations, International Press (2008)

[3] F. John, Blow‐up for quasi‐linear wave equations in three space dimensions. Comm. Pure Appl. Math., 34: 29-51 (1981)

[4] G. Holzegel, S. Klainerman, J. Speck, W. Wong, Small-data shock formation in solutions to 3D quasilinear wave equations: An overview, Journal of Hyperbolic Differential EquationsVol. 13, No. 01, pp. 1-105 (2016)

Grundkenntnisse sollten Einführung in die Funktionalanalysis oder Partielle Differentialgleichungen I sein.

Aus der Vorlesung können sich auch schon Masterarbeiten ergeben.

Unabdingbare Voraussetzung für die Vorlesung ist ein gutes Verständnis der Differentialgeometrie mindestens im Umfang der Vorlesung Differentialgeometrie I.