Course Description (Preliminary)
- Linear and nonlinear hyperbolic waves
Energy estimates for symmetric hyperbolic systems
Existence for linear systems
Local existence and finite time breakdown of quasilinear systems
Lecture notes: Linear time dependent problems [ pdf file ]
Lecture notes: Stuff we need on matrics[ pdf file ]
Assignment #1 [
pdf file ]
- Scalar conservation laws
Example of traffic flow
Shocks rarefactions and weak solutions
Viscosity, entropy and monotonicity
Numerical approximations
Lecture notes: Nonlinear conservation laws -- the traffic model [ pdf file ]
Assignment #2 [
pdf file ]
- Systems of conservations laws
Where do they come from?
Shocks rarefactions and the Riemann problems
2x2 systems; Riemann Invariants
Gas dynamics
Numerical methods; Godunov schemes
Lecture notes: Finite time breakdown for 2x2 systems [ pdf file ]
Assignment #3 [
pdf file ]
- Convection diffusion equations
Examples: chemotaxis, porous medium equation, aggregation with long term interaction
Global existence and blow up mechanism
Navier-Stokes equations: weak and strong solutions. Galerkin method and L2 contraction approach
Euler equations: finite time blow up. The BKM criterion
Detour: Sobolev spaces, compensated compactness and weak solutions
"On the motion of a viscous liquid filling space"
J. Leray, 1934 [ pdf file ]
"Euler eqs: local existence and singularity formation"
M. Lopes, H. Nussenzveig and Y. Zheng, [ pdf file ]
"A short note on Besov spaces"
[ pdf file ]
- Nonlinear diffusion
Maximum principle, Comparison Principle, Large time behavior
Global existence, Asymptotic decay and blow up mechanism
Applications: Image processing, Navier-Stokes equations
- The nonlinear wave equation and related problems
Linear and semi-linear wave equations; Schrödinger and Klein-Gordon equations
Assignment #4+Final [
pdf file ]
References
(Partial List)
Lawrence C. Evans,
Partial Differential Equations (Graduate Studies in Mathematics, V. 19), AMS
Robert McOwen, Partial Differential Equations, Methods and Applications,
Chapters 10-12
Heinz-Otto Kreiss and Jens Lorenz,
Initial-Boundary Value Problems and the Navier-Stokes Equations
Randy LeVeque,
Numerical Methods for Hyperbolic Conservation Laws
Peter Lax, Hyperbolic
Conservation Laws and the Mathematical Theory of Shock Waves, CBMS, 1972
Joel Smoller,
Shock Waves and Reaction-Diffusion Equations, Springer, 1994
Constantine Dafermos,
Hyperbolic Conservation Laws in Continuum Mechanics, Springer, 2005
Richtmyer & Morton,
Finite Difference Methods for Initial Value Problems, 1967
DeVore, & Lorentz,
Constructive approximation, Springer-Verlag, Berlin, 1993
Lecture Notes
(Partial List)
Luc Tartar,
An introduction to Navier-Stokes and Oceanography
Luc Tartar,
An introduction to Sobolev spaces and interpolation spaces
Luc Tartar,
Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics:
Heriot-Watt Symposium, Vol. IV, pp. 136--212, Res. Notes in Math., 39, Pitman, Boston, Mass.-London, 1979
Eitan Tadmor,
Approximate solutions of nonlinear conservation laws, Lecture notes in Math.
1697, Springer (1998) 1-149
Ami Harten, Computational Fluid Dynamics
