Mark Williams, Professor of Mathematics
Department of Mathematics, CB 3250
Chapel Hill, NC 27599

Research interests:
Nonlinear partial differential equations:

Propagation and spreading of singularities

Nonlinear waves:
-oscillations, nonlinear geometric optics
-boundary layers
-stability of multiD viscous shocks
-combustion, detonations

Recent publications and preprints:

1. Nonlinear geometric optics for reflecting and glancing oscillations, in Singularities and Oscillations, IMA volume 91, p.
    137-151, Springer-Verlag, 1997.

2.  Nonlinear geometric optics for hyperbolic boundary problems, Comm. PDE 21 (1996), 1829-1895.

3.  Nonlinear geometric optics for multidimensional shocks I: Profile Equations, C.R. Acad. Sci. Paris 325 (1997), 975-980.

4.  Nonlinear geometric optics for multidimensional shocks II:  Oscillatory exact shocks, C.R. Acad. Sci. Paris (1997), 981-986.

5.  Highly oscillatory multidimensional shocks, Comm. Pure Appl. Math., 52 (1999), 129-192.

6.  Boundary layers and glancing blow-up in nonlinear geometric optics, Ann. Scient. Ec. Norm. Sup. 33 (2000), 383-432.

7.  Singular pseudodifferential operators, symmetrizers, and oscillatory multidimensional shocks,  to appear in J. Functional Analysis, 77 pages.

8.   Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J., 51 (2002), 421-450 (with O. Gues).

9.    Multidimensional viscous shocks II: the small viscosity limit,  with G.Metivier, O. Gues, K. Zumbrun,  Comm. Pure Appl. Math., 57, 2004, 141-218.

10.     Boundary layer and long time stability for multiD viscous   shocks, with G. Metivier, O. Gues, K. Zumbrun, Discrete and Continuous Dynamical Systems A, 11. 2004, 131-160.

    Stability of multidimensional viscous shocks, CIME Lectures, Cetraro, Italy, to appear in Springer Lecture Notes, 60 pages.


12.      Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, with G. Metivier, O. Gues, K. Zumbrun, Archive Rational     Mechanics and Analysis, 175, 2004, 151-244.


13.    Equivalence of low frequency stability conditions for multidimensional detonations in three models of combustion, with H.K. Jenssen, G. Lyng, Indiana Univ. Math. J., 54, 2005, 1-64.



15.     Navier-Stokes regularization of multidimensional Euler  shocks}, with G. Metivier, O. Gues, K. Zumbrun, to appear in Ann.

Sci. Ecole Norm. Sup. (2006), 101 pages.


16.       Uniform stability estimates for constant-coefficient

  symmetric hyperbolic boundary value problems, with G. Metivier, O.

Gues, K. Zumbrun, to appear Comm. in PDE, (2006), 15 pages.


17.      Existence and stability of curved multidimensional

detonation fronts, with N. Costanzino, H. K. Jenssen, G. Lyng, to

appear in Indiana University Math. J., 48 pages.


18.      Viscous boundary value problems for symmetric systems with

variable multiplicities, with  G. Metivier, O. Gues, K. Zumbrun,

submitted, 90 pages.

19.   Nonclassical multidimensional viscous and inviscid shocks,

With G. Metivier, O. Gues, K. Zumbrun, submitted, 88



Math 522

1.     Homework 1 due Jan. 22 (revised).

2.     Homework 2 due Jan. 29

3.     Homework 3 due Feb. 5

4.     Homework 4 due Feb. 12

5.     Homework 5 due Feb. 19

6.     Homework 6 due Feb. 26

7.     Homework 7 due March 5

8.     Homework 8 due March 19

9.     Homework 9 due March 26

10.  Homework 10 due Tuesday, April 14

11.  Homework 11 due Thursday April 23



Selected Math 522 Solutions

Correction to Tuesday, Feb. 24 lecture



email address:
office phone:   919 962 9613 .