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

This workshop, sponsored by AIM and the NSF, 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:

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

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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