Advanced PDEs: Nonlinear Time Dependent Problems

MATH/AMSC 698T, Spring 2008

Course Information

Lecture4122 CSIC Bldg. #406; TuTh 2-3:15pm
InstructorProfessor Eitan Tadmor
Contacttel.: x5-0648   Email:
Office Hours By appointment 4119 CSIC Bldg. #406
tel.: x5-0652   Email:
Final TBA, 4122 CSIC Bldg. #406
Grading50% Homework, 50% Final


    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



Eitan Tadmor