Singularities of PDE's: physical systems
October - December 2011, Friday 14:00-16:00 Prof. Jens Eggers.
Most continuum descriptions of physical systems are formulated
in terms of (nonlinear) partial differential equations (PDE's).
Even if initial or boundary conditions are smooth, these equations
will typically develop singularities, i.e. points in space (and time)
at which the solution or its derivatives blow up. Examples are
shock waves, drop formation, cracks in a solid, or caustics
in optics. In this course, we will
stress the physical interpretation of singularities
investigate the self-similar structure of singularities
explain the concept of universality
explain important techniques such as matched asymptotics
highlight connections between different physical problems
propose a "dictionary" of singularities
Syllabus
- Examples: what are singularities about?
- Blowup: ODEs
- Self-similar structure in space
- Continuation
- Continuum ideas
- Local expansions
- Drop breakup: similarity solutions and crossover
- Slow convergence: type-II self-similarity
- Free surface cusps on a fluid surface
- Caustics in optics
- Discrete self-similarity
Reading
- L. D. Landau and E. M. Lifshitz, "Hydrodynamics",
Pergamon:Oxford, 1984
- G.I. Barenblatt, "Similarity Self-Similarity and Intermedeate
Asymptotics",
Cambridge, 1996
- J. Eggers, "Singularities and Similarities", in: "Thin films of
soft matter", edited by S. Kalliadasis and U. Thiele, Springe, 2007
Course Assessment
This course will be assessed by an essay on a selected paper from
the research literature