Math 690 Topics in PDEs

Spring 2011@ISU

• Instructor: Hailiang Liu
• Office : Carver 434; Tel:  515-294-0392;  E-mail : hliu@iastate.edu
• Office hours:   by appointment
• Homepage http://www.math.iastate.edu/hliu/teaching/690.1.11s/
  
• Time and Location:   TR 12:40P -- 02:00P,  Carver Hall 124
• Meeting Dates: 01/10/11—05/06/11
• Problem Assignments  an up-to-date list of problem assignments
  
  

• Course Description:
  This three-credit course will provide students with basic techniques for doing a priori estimates of solutions to some significant partial differential equations, including hyperbolic and parabolic equations. Such estimates, showing how solutions depend on the given data (initial data and/or boundary data), are of fundamental importance in both development of theory of PDEs and construction of numerical approximations of their solutions.  
  
  
• Topics to be covered in Math 690: classified by methods, including
  1. Methods of invariant region    
  2. Energy methods
  3. Entropy methods
  4. Critical threshold and large time dynamics
  

• Grading: 
  1. Homework problems involving various estimates for applied PDEs will be assigned and graded.  
  2. Students are to work alone on homework assignments.
  3. There will be no written exams during the semester and there will be a written final examination at the end of the semester.
  4. Homework: 40%+ Midterm: 20% + Final: 40%. An appropriate scaling may be applied at the end of the semester to determine the final grade.
     
     
• Reference Texts: the material will be based on a collection of book chapters/ papers and lectures notes.
  
  Here is a partial list of books:
  
  [Sm] J. Smoller, Shock Wave and Reaction-Diffusion Equations
  
  [Jo] F. John, Partial Differential Equations
  
  [Ev] Lawrence C. Evans, Partial Differential Equations
  
  [Da] Constantine M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics
  
  [Te] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1983.
  
  [MB] Andrew J. Majda and Andrea L. Bertozzi, Vorticity and Incompressible Flow.
  
  Here is an updated list of papers:
  
  [LCO] Liu, Hailiang; Cheng, Li-Tien; Osher, Stanley A level set framework for capturing multi-valued solutions of nonlinear first-order equations. J. Sci. Comput. 29 (2006), no. 3, 353–373. [ on level set formulation for first order PDEs]
  
  [Na1958] Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 1958 931–954.