Math 632. Partial Differential Equations II: Variable coefficient and nonlinear equations
- Fall 12 - Hans Lindblad

Description: 632 deals with existence for elliptic and evolution equations, variable coefficent and nonlinear, e.g. Laplace equation, Wave equations, Heat equations and Hyperbolic systems. We will start with giving elements of the general theory following Chapters 6 and 7 in Evan's book. Then my plan is to go on to existence for Euler's equations for fluids, starting with material from Taylor's PDE books. The course does not depend on haven taken 631 but some knowledge of Sobolev spaces as in Evans's chapter 5 is needed. We start by reviewing the last part of 631, from Chapter 6 in Evans.
Lectures: MW 1.30-2.45 in Maryland 201. Instructor Hans Lindblad, lindblad@math.ucsd.edu. Office hour: M 3-4 Krieger 406.
TA: Xin Yu, xyu@math.jhu.edu, Office hour: Tue 10.30-12.30 in Krieger 211
Text: Evans Partial Differential Equations
Syllabus:
Preliminary schedule:
wk  date  Monday  Wednesday  Homework due following Wed.
  1  1/31 Elliptic PDE: 6.1-2 Weak solutions 6.2 Existence 6.6:2,3,4,Riesz Repr. Theorem
  2  2/7 6.2, 6.5.1, D.5 Eigenfunctions 6.3 Interior & Boundary Regularity 6.6:7,13, 5.10:5,6,12, Spectral Th. (for compact op.)
  3  2/14  6.3.2 Boundary Regularity 7.1 Parabolic PDE, 1-D example 6.6:11, 7.5:2,4
  4  2/21 holiday 7.1.2-3 Existence-Regularity 7.5:1,5,9
  5  2/28     7.5:7,10
  6  3/7 7.2.1-3 Hyperbolic PDE 7.3 Hyperbolic Systems  
  7  3/14  7.3 Hyperbolic Systems 7.4 Semigroup theory 7.5:13,14,15,16?
  8  3/21 break break  
  9  3/28 13.1-3 H^k-L^\infty est. 13.3, 16.1 Exist. Symm. hyp. syst. 13.1:1, 13.3:1,6, 16.1:11,12,13
10  4/4 16.2 Euler's compr. eq.   16.5 Euler's compr. eq.   16.2:1  
11  4/11 17.1,2 Exist Euler's incomp. eq.   no class    
12  4/18 13.5-6   13.8(Ch7)    
13  4/25  13.8(Ch7) 5.8 Hodge decomp., 17.2    
14  5/2 17.1 Vorticity bound   17.2 or 17.3-4    
The first 7 weeks refer to Chapters from Evans book and the following weeks after the break to Chapters from Taylor's 3 volume PDE book.