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


Research interests:
Nonlinear partial differential equations:

Propagation and spreading of singularities

Nonlinear waves:
-oscillations, nonlinear geometric optics
-shocks
-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.

 
11.    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.

 

14.       “Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability, with G. Metivier, O. Gues, K. Zumbrun, 73 pages, Journal of the A.M.S. 18 (2005), 61-120.

 

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

pages.




email address:  williams@math.unc.edu
office phone:   919 962 9613
 

Last update:  January 22, 2007