Seminar in Applied Mathematics:

Sobolev spaces and related topics in applications

Math 285, Winter 1997, UCLA

 
• Contents
• References
• Nonlinear PDEs - Time-Dependent Examples
  • Navier-Stokes equations
  • Hyperbolic conservation laws and related equations
• Distributions
  • and its dual
  • Calculus of distributions - convolutions, derivatives,..
  • Distributions as generalized functions; mollifiers
  • Examples
  • Restrictions, limitations and warnings
  • Tempered distributions and their FT
• Regularity Spaces of Functions
  • Sobolev spaces
    • -spaces of integer order
    • Moduli of smoothness
    • Spaces of fractional order
  • Besov and related spaces
    • Characterization by dyadic Fourier blocks
    • Characterization by divided differences
    • Orlicz spaces
  • Imbeddings and compact imbeddings
    • Sobolev imbedding theorems
    • Other useful inequalities
  • *Oscillation of functions
    • Hardy-Littlewood maximal function
    • Weak and Lorentz spaces
    • Sharp function; BMO...
    • Hardy spaces
• Interpolation (of Operators)
  • Complex interpolation
    • The Risez-Thorin theorem
    • Rearrangement invariant spaces
  • The K-functional and real interpolation
  • Interpolation between Sobolev and Besov spaces
• Fourier Multipliers
  • Marcinkiewicz theorem
  • Hörmander theorem
• Nonlinear PDEs revisited - Applications
  • Navier-Stokes equations
    • from kinetic to fluid models
  • Hyperbolic conservation laws
  • Compensated compactness, averaging lemma...
• About this document ...