#
The 3D Euler and 2D surface quasi-geostrophic equations

April 6 to April 10, 2009
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Peter Constantin,
Diego Cordoba,
and Jiahong Wu

## Original Announcement

This workshop will be devoted to
the 3D Euler equations of incompressible fluids and the 2D surface
quasi-geostrophic (QG) equation of geophysical flows.
These equations are proposed for simultaneous study because of the
striking analogies between them. The 2D inviscid surface QG equation
serves as a lower dimensional model of the 3D Euler equations in
additional to its geophysical relevance. These equations have been
rather intensely studied by quite a number of researchers and in the
past few years there has been important progress on various topics
related to the fundamental issue of global existence and smoothness.
As recently shown by Kiselev, Nazarov and Volberg for the periodic
case and by Caffarelli and Vasseur for the whole space case, the 2D
surface QG equation with critical dissipation possesses a unique
global smooth solution for any smooth data. The works of D. Cordoba
and of D. Cordoba and Fefferman have ruled out several finite-time
singularity scenarios for the Euler and the inviscid QG equations,
such as the simple hyperbolic type. The geometric criteria of
Constantin, Majda and Fefferman relating the regularity of the
direction of the vorticity field to global smoothness of solutions
for the 3D Euler equations have recently been improved by T. Hou
and his collaborators. Analytic regularity criteria for these
equations have been further developed and pushed to weak functional
settings such as Besov spaces with negative indices. A few new
one-dimensional models retaining crucial features of the 3D Euler
equations have newly been constructed. New computational results to
the 3D Euler and the 2D surface QG equation have also appeared in the
past few years.

These recent accomplishments have inspired us to look for a complete
solution of the fundamental issue on global existence and smoothness
for these equations. More precisely, this workshop will focus on
three major topics:

- The global existence of classical solutions to the dissipative
surface QG equation with a supercritical index;
- The global existence of classical solutions to the inviscid
surface QG equation; and
- The global existence and regularity problem for the 3D
incompressible Euler equations.

Given the current trend that harmonic analysis and numerical
computations play more and more important roles in the study of
these equations, ideas and methods from different perspectives
will be exchanged, discussed and analyzed during this workshop.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: