401-3650-66L  Numerical Analysis Seminar: Measure Theoretic Tools for Analyzing and Approximating Nonlinear PDEs

SemesterHerbstsemester 2016
DozierendeF. Weber
Periodizitätjedes Semester wiederkehrende Veranstaltung
LehrspracheEnglisch
KommentarMaximale Teilnehmerzahl: 6


KurzbeschreibungThe seminar covers measure theoretic tools used for the analysis and approximation of nonlinear hyperbolic partial differential equations. In particular, we will discuss Young measures, compensated compactness, weak-strong uniqueness and algorithms for the approximation of measure-valued solutions. The participants will present individual topics based on the study of research papers.
Lernziel- To learn some measure theoretic tools for the analysis and approximation of nonlinear PDEs.

- To read and understand a research paper and present it in an understandable way to other students.
InhaltPartial differential equations can be used to model an abundance of natural and physical phenomena, as well as industrial processes. Many of the more sophisticated and more realistic models involve nonlinear PDEs, among others, PDEs in fluid dynamics, astrophysics, elasticity or weather modeling. The solutions to these often exhibit complex structures, such as shocks, oscillations, singularities that are difficult to deal with mathematically and numerically. In our seminar we aim to get a better understanding of the difficulties that arise when dealing with nonlinear PDEs. In particular, we will discuss problems related to the PDEs of fluid dynamics. Solutions to these equations may exhibit shocks and oscillations, and have less regularity than what the definition of a classical solution requires. Therefore, the solution concept has to be relaxed. One way of doing this, is to look for solutions in the space of measures instead of actual functions. Our goal in this seminar is to try to understand this concept better by studying research papers related to this issue.
Specifically, we will discuss weak convergence in general, the notion of Young measures as a means to represent weak limits of nonlinear functions, and its application to compensated compactness, existence of solutions to scalar hyperbolic conservation laws, Euler equations, turbulence and statistical solutions of Navier-Stokes equations. We will also discuss algorithms to approximate solutions in the space of measures.
We are open to extend the list of topics by others that are of special interests to the attending students.
LiteraturJ. M. Ball. A version of the fundamental theorem for Young measures (1989).

Yann Brenier, Camillo De Lellis, and László Szekelyhidi, Jr. Weak-strong uniqueness for measure-valued solutions (2011).

Camillo De Lellis and László Szekelyhidi, Jr. The Euler equations as a differential inclusion (2009).

Ronald J. DiPerna. Measure-valued solutions to conservation laws (1985).

Ronald J. DiPerna and Andrew J. Majda. Concentrations in regularizations for 2-D incompressible flow (1987).

Lawrence C. Evans. Weak convergence methods for nonlinear partial differential equations.

Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, and Eitan Tadmor. Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws (2015).

A. Szepessy. An existence result for scalar conservation laws using measure-valued solutions (1989).
Voraussetzungen / BesonderesGood knowledge of real/functional analysis required, knowledge of hyperbolic partial differential equations and/or numerical analysis of advantage.