**UBC Mathematics Department**

*http://www.math.ubc.ca*

## Colloquium Abstract: Professor John Heywood,
Department of Mathematics, UBC

*On Xie's Conjecture*

A striking new inequality for the Laplacian was proven in 1990
by a student in our department, Wensheng Xie. The inequality and
its proof are highly original, very beautiful, and based on only
elementary prerequisites.

Xie had undertaken to prove this inequality as a model problem. His
ambition was to prove an analogous inequality for the hydrodynamic Stokes
operator, for what would be an extremely important application to the
Navier-Stokes equations. However, at one point in his proof he resigned
himself to the use of the maximum principle. The Stokes equations are an
elliptic system for which there is no maximum principle. Aside from this one
point, Xie showed that his proof carries over to the Stokes operator. The
desired application to the Navier-Stokes equations has also been worked out by
Xie and myself, and published in the context of the vector Burgers equation.

Xie used the maximum principle to infer that the L^2 norm of the
Green's function for the spectral Laplacian is less than the L^2 norm of the
corresponding fundamental singularity. ``Xie's conjecture" is that a similar
relation holds for the spectral Stokes operator.

I offer this lecture as an invitation to my friends and colleagues
here to give serious thought to one of my favorite problems. I think that
the heart of the matter can still be approached in the context of the
Laplacian, by restricting oneself to methods that can be expected to
generalize. Thus this problem may offer someone the chance of
making a big splash in the Navier-Stokes world, without even knowing the
equations.

I will begin this lecture with the statement of Xie's inequality,
along with the core of its proof, and an explanation of the difficulty in
carrying it over to the Stokes operator. Then, as time permits, I will
explain its importance to the Navier-Stokes theory, and also offer a second
equivalent conjecture that may provide an easier handle on the problem.

Return to this week's seminars