MTH 628, Winter, 2008

MTH 628: Partial Differential Equations, Winter, 2008

LECTURE: MWF 1100 -- 1150 Kidder 280 CRN 27932
Instructor: R.E. Showalter Kidder 298B show@math.oregonstate.edu

Prerequisite: MTH 627 or consent of instructor.
Final Exam: March 18: 1400
Textbook: Hilbert Space Methods for Partial Differential Equations.
The textbook is available online at this site courtesy of the Electronic Journal of Differential Equations.
Elements of Hilbert Space : Expansion in Eigenfunctions.
Distributions and Sobolev Spaces : Compactness.
Boundary Value Problems : Closed operators, adjoints and eigenfunction expansions.
First Order Evolution Equations : Introduction, The Cauchy Problem, Generation of Semigroups, Accretive Operators, two examples, Generation of Groups, a wave equation, Analytic Semigroups, Parabolic Equations.
Implicit Evolution Equations : Introduction, Regular Equations, Pseudoparabolic Equations, Degenerate Equations, Examples.
Second Order Evolution Equations : Introduction, Regular Equations, Sobolev Equations, Degenerate Equations, Examples.
We shall introduce various expansion or variational methods to construct solutions. These will be applied to solve initial-boundary-value problems for time-dependent partial differential equations as evolution equations of the form u'(t) + A(u(t)) = f(t) or u"(t) + A(u(t)) = f(t). Major objectives are to characterize those operators for which the preceding problems are solvable and to understand the distinction between parabolic and hyperbolic problems and the properties of their solutions. These ideas will be extended to systems describing fluids (Stokes), elasticity (Navier), and porous media (Darcy).

Schedule:
1. Expansion in Eigenfunctions I.7.1 pp. 24 - 25. (1/7/08)
2. I.7.2 pp. 25 - 27.
3. Boundary-Value Problems III.1-2. pp. 59 - 65.
4. Eigenfunction Expansions for Elliptic PDE III.7.5 pp. 86 - 89.(01/14)
5. Compactness II.5.2 pp. 53- 55. (01/14)
6. Fredholm Alternative for Elliptic BVPs
7. Initial-BVP for Diffusion Equation, I (01/23)
8. Initial-BVP for Diffusion Equation, II
9. Wave Equation. (01/28)
10. The Cauchy Problem IV.1 and IV.2.
11. The Semi-group Generator IV.3. pp. 99-102.
12. No class today (02/04)
13. Generation Theorem (Necessary conditions) IV.3. pp. 102 - 103.
14. Generation Theorem (Sufficient conditions) IV.3. pp. 103 - 104.
15. Two Examples IV.4. pp. 107 - 109. (02/11)
16. Accretive operators IV.4. pp. 105-106.
17. Groups, Conservative operators IV.5. pp. 109-112. Exercises 1,2.
18. Parabolic regularity and m-accretive operators (02/18)
19. Non-homogeneous and semi-linear equations
20. Implicit Evolution Equations V.1,2. pp. 127-129.
21. Regular equations: examples, I V.2. pp. 130-131. (02/25)
22. examples, II V.2. p. 131.
23. examples, III V.5. pp. 139-140.
24. evolution boundary conditions pp. 141-142. (03/03)
25. Second order evolution equations VI.1,2
26. Exercises 3,4 due 03/10.
27. Wave equation . (03/10)
28. Lions' projection theorem .
29. evolution equations .