The Navier-Stokes equations are nonlinear partial differential equations
thought to describe the flow of fluids including turbulent flow. Although
they have been the object of intensive mathematical study for many years,
fundamental problems remain unsolved. For example, it is not known whether
solutions to the Navier-Stokes equations in three space dimensions (the
important case!) develop singularities in finite time, and hence whether
or not they really provide a mathematically sound model for
three-dimensional flows! On the other hand they seem to provide reasonably
reliable results in computational experiments although computation in
turbulent flows of real engineering interest remains difficult.
Together with colleagues Russell Brown and Zhongwei Shen (Mathematics) and
James McDonough (Mechanical Engineering), we are studying mathematical and
computational questions related to the Navier-Stokes equations, including:
multi-scale computational schemes, existence and dimension of attractors,
and existence of solutions in certain special three-dimensional
geometries.
Geometric Scattering Theory
One of the most interesting areas of modern pure mathematics is geometric
analysis, where techniques of analysis and partial differential equations
are applied to obtain new geometric data attached to Riemannian manifolds.
Riemannian manifolds are natural generalizations of Euclidean space
endowed with a notion of distance and geometric notions such as curvature.
They play a key role both in Einstein's geometric theory of gravity and
in more modern areas of theoretical physics such as string theory.
A basic mathematical problem in Riemannian geometry is the discovery of
invariants that help classify Riemannian manifolds, and many natural
analytic invariants have been derived by studying the Laplacian, a natural
geometric operator, and its associated spectral theory. Scattering theory
studies the propagation of waves on Riemannian manifolds and geometric
scattering theory tries to link these propagation properties to the
underlying geometry.
A very interesting "test bed" for geometric scattering theory is the class
of Riemannian manifolds known as hyperbolic manifolds. These manifolds are
"locally" like the non-Euclidean space discovered by Lobachevskii, Bolyai,
and others which was one of the first concrete models of non-Euclidean
geometry. Selberg's celebrated trace formula relates geometric data of
a compact Riemann surface to the eigenvalues of the Laplacian. More recent
generalizations of Selberg's formula relate so-called scattering
resonances to similar geometric data of infinite volume Riemannian
manifolds. Together with postdoctoral fellow Ruth Gornet and colleagues
Robert Brooks (Technion Institute, Haifa, Israel) and S. J. Patterson
(University of Goettingen) we are investigating the geometric content of
scattering data for a large class of infinite-volume hyperbolic manifolds.
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