Applied Partial Differential Equations
Math 266C, Spring 2002, UCLA

Instructor: Professor E. Tadmor

    MS 5147  MW 4:00-5:15


Textbooks.

Partial Differential Equations,  Methods and Applications
by Robert McOwen

Partial Differential Equations. Lecture Notes.
by Craig Evans

Tentative Course Plan (with reading and homework assignments from McOwen's book).

  • 04/01 Overview. Examples of linear and nonlinear applied PDEs

    • No Assignment
 
  • 04/03  Linear functionals and weak solutions of elliptic equations
    •  

    Hilbert space, Lp spaces, Sobolev spaces, Risez representation, 
    weak solutions of second-order equations, Lax-Milgram lemma

    Reading: 6.1 a-b
    Homework: pp. 160, #3, #5, #8.
     

  • 04/08 Regularity 

          Compactness I: Arzela-Ascoli, Rellich; Compactness II: Fredholm theory

    Reading 6.3 a-c
    Homework: pp. 175 #3

 

  • 04/10  Sobolev Inequalities

    • Reading: 6.4 a-c
    Homework: pp. 183 #3, #6
     

  • 04/15 Smooth approximation and mollifiers
    •  

    Weak=strong.

    Reading: 6.5 a-c
    Homework: pp 197 #4, #5
     

  • 04/17   Minimizers
    •  

    Critical points, coercivity, convexity

    Reading: 7.1 a-c
    Homework: pp. 207 #1, #2*, #3*, #4
     

  • 04/22  Linear and nonlinear hyperbolic waves

     Energy estimates for symmetric  hyperbolic systems, Existence for linear system

    Reading:  12.1 a-b
    Homework: pp348 #1, #2, # 4, #5
     
  • 04/24 Systems of conservation law

          Local existence

    Reading: 10.1 a-b,  12.1 c-d
    Homework: pp. 287 #5, pp348 #8
     
  • 04/29  Shocks rarefactions and  entropy condition

    • Reading: 10.2 a-c
    Homework: page 300 #1, #2
     
  • 05/01  TBA
    Reading: lecture notes

 

 

  • 05/08  Riemann problems

    • Reading:  10.2 b-d
    Homework: pp. 300 #4 #8
     
  • 05/13  Riemann Invariants, Gas dynamics
    •  


    Reading: 10.3 a-c
    Homework: pp. 306 #2
     

  • 05/15  Kinetic formulations

    •  
  • 05/20  Nonlinear diffusion 

          Maximum principle, Comparison Principle

    Reading:  11.1 a-c
    Homework:  pp. 313, #5, #6

     

  • 05/22  Large time behavior

          Global existence, asymptotic decay and blow up mechanism

    Reading:  11.3 a-c
    Homework:  pp 326, #1, #2, #3, #8

 

  • 05/27  Holiday
  • 05/29  Navier-Stokes equations  

          Reading:  11.4 a-d
          Homework:  pp 335, #7

 

  • 06/03  The wave equation

          Liner and semi-linear wave equations; Schrödinger and Klein-Gordon equations

    Reading:  12.2 a-d 
    Homework:  pp 355 #6, #7
     
     
  • 06/05  Review
Final grade: 25% homework, 25% midterm exam, 50% final exam