Topics in Nonlinear PDEs
AMSC 698E, Fall 2004

Lectures: 12:30 - 1:45, Tuesdays and Thursdays in CSIC 4122

Instructor: Professor Eitan Tadmor
 Office:  CSCAMM, 4119 CSCIC Building #406
Phone:  301-405-0652
Email:
Office Hours: By Appointment (cgray@cscamm.umd.edu)

Course Outline. We will discuss the basic regularity spaces & other related topics which are beyond the "straightforward" techniques of integration by parts, maximum principle, etc. The purpose is to have a self-contained discussion emphasizing "classical" techniques of interpolation & oscillations of function spaces, which in turn are used in the context of PDEs - mostly nonlinear time-dependent problems.

Following an introductory discussion on technical tools of Sobolev imbeddings, real interpolation, Fourier multipliers, ... we will turn to topics nonlinear PDEs. Among the issues to be covered are:

  • Regularity of Navier-Stokes equations

  • Compensated compactness

  • Averaging lemma applications to kinetic formulation of nonlinear conservation laws

  • Concentration-cancellation phenomena of approximate Euler solutions

  1. Beyond L2
    . . . Oscillations of functions: Hardy-Littlewood,...
    . . . Sobolev spaces; imbeddings etc
    . . . Besov spaces
    . . . Interpolation
    . . . Fourier multipliers
     

  2. Euler and related equations
    . . . Regularity
    . . . Concentration-cancelation
     

  3. Navier-Stokes and related equations
    . . . ...
    . . . ...
     

  4. Nonlinear conservation laws
    . . . Shock discontinuities and BV regularity
    . . . Entropies and compensated compactness
    . . . Kinetic formulation and averaging lemma
     

  5. ...

Final project [ pdf file ]
Eitan Tadmor