Craig, Walter

Distinguished University Professor, Senior Canada Research Chair, Ph.D. (Courant, N.Y.U.)

craigDept. of Mathematics & Statistics, McMaster University
1280 Main Street West
Hamilton, Ontario,
Canada L8S 4K1
905-525-9140, ext. 23422
905-522-0935 (fax)
Office: HH/418


Research Area: Analysis, Applied Mathematics, Fluids & Turbulence

Research Profile: Nonlinear partial differential equations and dynamical systems
My research interests are in the theory of differential equations, concerning particularly those which arise in problems in mathematical physics and applied mathematics. The principal focus of my work has been on nonlinear equations, and for the most part these involve questions of time evolution and dynamics. Questions in this area arise in many areas of the physical sciences, and some of the topics of my research include quantum mechanics, fluid dynamics and cosmology. It is particularly pleasing to me when a mathematical question in one area brings up a problem of substance in another completely different and surprising area. Some examples of this in my own work include (1) the connection between free surface waves, Hamiltonian dynamical systems, and Morse-Bott equivariant cohomology of singular varieties, (2) Hamiltonian dynamical systems, Hamiltonian PDE and unusual questions on diophantine approximation, (3) Riemannian invariants, completely integrable systems and the ergodic theory of negatively curved manifolds, and (4) the reheating problem for the inflationary model in cosmology and the theory of Anderson localization in condensed matter physics.


Recent Publications

W. Craig, A. Selvitella and Y. Wang,
Birkhoff normal form for the nonlinear Schrödinger equation.
Rendiconti Accad. Lincei 24 (2013) in press.

W. Craig, X. Huang and Y. Wang,
Global wellposedness for the 3D inhomogeneous incompressible Navier - Stokes equations.
JMFM (2013) DOI 10.1007/s00021-013-0133-6.

W. Craig, P. Guyenne and C. Sulem.
The surface signature of internal waves.
Journal of Fluid Mechanics 710 (2012) pp. 277-303.
published in electronic form Sept. 3 2012.

W. Craig, P. Guyenne and C. Sulem.
Hamiltonian higher-order nonlinear Schrödinger equations for broad-banded waves on deep water.
European J. Mech. B - Fluids 32 (2012) pp. 22-31.

W. Craig, D. Lannes and C. Sulem.
Water waves over a rough bottom in the shallow water regime.
Annales IHP - Analyse Nonlinéaire 29 (2012) pp. 233-259. 10.1016/j.anihpc.2011.10.004.

W. Craig, P. Guyenne and C. Sulem.
Coupling between internal and surface waves.
Natural Hazards 57 (2011) pp. 617-642. DOI 10.1007/s11069-010-9535-4.

A. Biryuk and W. Craig.
Bounds on Kolmogorov spectrum for the Navier – Stokes equations.
ArXiv-0807.4505 math-physics,
Physica D 241 (2011) 10.1016/j.physd.2011.10.013.

M. Arnold and W. Craig.
On the size of the Navier - Stokes singular set.
DCDS 28 no. 3 (2010) pp. 1165–1178.

W. Craig.
Sur l'ensemble singulier et l'ensemble de concentration d'énergie de Navier – Stokes.
X-EDP Éditions X, Publications de l'École Polytechnique (2010).

W. Craig, P. Guyenne and C. Sulem.
Water waves over a random bottom.
Journal of Fluid Mechanics 640 (2009), pp. 79-107.

W. Craig, S. Danworaphong and G. Diebold.
Laser induced thermal diffusion shock waves.
VDM Verlag, Saarbrücken (2008), 84~pp.

Hamiltonian dynamical systems and applications.
(W. Craig, editor),
Proceedings of the Advanced Study Institute on Hamiltonian Dynamical Systems and Applications,
NATO Science for Peace and Security Series B: Springer – Verlag, (2008) XVI, 441~pp.

Graduate Students

Currently supervising:
Nikolay Hristov

Adilbek Kairzhan

Past students:

Nick Rogers
Bilal Abbasi
Amanda Kamping-Carder




On Research Leave

Office Hours


Math 4FT3
Math 742