# George F. D. Duff

George Francis Denton Duff (29 July 1926 – 2 March 2001) was a Canadian mathematician, known for his research on partial differential equations and wave phenomena, including tidal energy. He was a plenary speaker at the International Congress of Mathematicians in 1974 in Vancouver.

## Quotes

* The theory of harmonic forms in Riemannian manifolds may be regarded as a generalization of potential theory. It is therefore natural that the boundary value problems of this theory which generalize the classical Dirichlet and Neumann problems should play an important role in the theory.
** (1952). "Differential Forms in Manifolds with Boundary". The Annals of Mathematics 56 (1): 115–127. ISSN 0003486X. DOI:10.2307/1969770.
* The propagation of elastic waves in a homogeneous solid is governed by a hyperbolic system of three linear second-order partial differential equations with constant coefficients. When the solid is also isotropic, the form of these equations is well known and provides the foundation of the conventional theory of elasticity (Love 1944). The explicit solution of the initial value, or Cauchy, problem for the isotropic case was found by Poisson, and in a different way by Stokes (1883). If the initial disturbance is sharp and concentrated, the resulting disturbance at a field point will consist of an initial sharp pressure wave, a continuous wave for a certain period, and a final sharp shear wave. The disturbance then ceases.
** (February 1960)"The Cauchy problem for elastic waves in an anisotropic medium". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 252 (1010): 249–273. ISSN 0080-4614. DOI:10.1098/rsta.1960.0006.
* The magnitude of Fundy tides may be seen as having been reached by a balance between a dissipative mechanism, with assumed quadratic frictional forces, and an energy imparting mechanism in the deep ocean where work done by the tide raising force is proportional to distance travelled and hence to the first power of amplitude. Further, it now appears that the second and third North Atlantic modes are those primarily stimulated by the Fundian resonance. To represent these processes within one model both the continental shelf shallows and oceanic areas must be included, as well as their zone of interaction across the continental shelf.
** "Mathematical Problems of Tidal Energy". Proceedings of the International Congress of Mathematicians, Vancouver, 1974. vol. 1. pp. 87–94. (quote from p. 90)
* The mathematical theory of the Navier-Stokes equations has centered upon basic questions of the existence, uniqueness, and regularity of solutions of the initial value problem for fluid motions in all of space or in a subdomain of finite or infinite extent. Such solutions, when they can be constructed or shown to exist, represent flows of a viscous incompressible fluid. In two space dimensions the theorem of existence, uniqueness and regularity was essentially completed thirty years ago by the work of Leray ..., Lions ... and Ladyzhenskaya ... who showed that a smooth solution of the initial value problem exists for arbitrary square-integrable initial data. For viscous, incompressible fluid motions in three space dimensions, ... the theorem of existence uniqueness and regularity has been proved only for sufficiently small initial data or in special cases such as cylindrical symmetry that essentially reduce the problem to two space dimensions in some sense.
** (1990). "Derivative estimates for the Navier-Stokes equations in a three dimensional region". Acta Mathematica 164: 145–210. ISSN 0001-5962. DOI:10.1007/BF02392753. (quote from pp. 145–146)