Curvature and instability of flows of ideal incompressible fluid (06rit311)
Alexander Shnirelman (CRC, Concordia University, Montreal, Canada)
The stability theory of flows of ideal incompressible fluid is a testfield of all new ideas and methods of mathematical fluid dynamics. One of this approaches which has lead already to deep results is the study of fluid flows from the viewpoint of infinite-dimensional differential geometry. Ideal incompressible fluid inside a bounded domain M is an example of an infinite-dimensional Lagrangian system. Without external forces it moves along geodesics on the group SDiff(M) of volume-preserving diffeomorphisms of M, which is an infinite-dimensional Riemannian manifold. Hence, the stability of the flows should be connected with the Riemannian curvature of SDiff(M), as it was pointed out by V.Arnold. However, the question is far from certain. For example, it was found by V.Yudovich that for any parallel flow in a channel the curvature in any 2-d direction containing velocity itself is negative (this result was considerably extended by A.M.Lukatsky, G. Misiolek and S. Preston). However, stability properties of such flows depend on the velocity profile and are quite different for different profiles.
The differential geometry of SDiff(M) is complicated enough. The curvature can assume either signs and be zero, depending on the direction. The space SDiff(M) is homogeneous, but it is not symmetric. To the contrary, it is extremely far from symmetry. Therefore our ''symmetric'' intuition may be misleading, and in some situations negative curvature can stabilize the flow, if it varies in certain way. Likewise, positive curvature can be destabilizing. Directions of zero curvature and asymptotically flat geodesic subspaces should play important role in the stability and in the long time behaviour of the flows.
The goal of the proposed program is the discussion and joint work on the interconnections between the stability of fluid flows and the curvature and, possibly, other differential-geometric properties of the configuration space SDiff(M) of ideal incompressible fluid.
Another goal is the further study of the exponential map on the group SDiff(M), especially its singularities. We expect that the Fredholm structure of this map enables to use topological methods (like the degree theory) in the study of the Euler equations. The interesting question is, whether the variant of condition C of Palais-Smale holds for the energy functional on $SDiff$ in the 2D case. If true, this could lead to a Morse theory of geodesics in this context. In the 3D case, we would also like to study the reverse question: whether the pathological singularities of the exponential map imply anything about its global properties.
We hope that this meeting will result in better understanding of this important and still unclear domain.