Functional Analysis

Nonlinear PDEs I

Winter 2011/2012

Lectures

Mondays and Thursdays 10-12, Room 1.008 (Endenicher Allee 60)
Prof. J. J. L. Velázquez

Classes

Wednesday 14-16, Room 2.040 (Endenicher Alle 60)
M. Helmers

Office hours

By appointment.

Examination

An oral examination will be held on 6 and 7 February 2012 in office 2.023. To sit the exam, students have to gain at least 40% of the overall marks of the problem sheets as requirement for passing the classes.

Course synopsis

In this course some of the main techniques available to study the well-posedness and the qualitative behaviour of Nonlinear Partial Differential Equations will be described. Some of the topics covered will include the Theory of Conservation Laws, Variational Methods for Elliptic Problems and the study of Reaction-Diffusion Systems. The course will combine the study of general methods of proving existence, uniqueness and general properties of the solutions with the study of specific examples that will illustrate how to obtain insight about the general results as well as the limits of validity of the general methods.

Programme

  • Nonlinear first order equations: characteristics, Hamilton-Jacobi equations, scalar conservation laws, one-dimensional systems of  conservation laws, shock dynamics, existence and uniqueness of entropy solutions;
  • Explicit solutions: travelling waves, similarity solutions;
  • Reaction-diffusion equations.

Literature

  • L. C. Evans. Partial Differential Equations. Graduate Studies in Mathematics vol. 19, American Mathematical Society, 1991.
  • D. Henry. Geometric theory of seminlinear parabolic equations. Lecture Notes in Mathematics vol. 840, Springer-Verlag, 1981.
  • J. Smoller. Shock Waves and Reaction-Diffusion Equations. Grundlehren der mathematischen Wissenschaften vol. 258, Springer-Verlag, 1982.
  • G. B. Whitham. Linear and Nonlinear Waves. John Wiley & Sons, 1974.

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