Microglobal Analysis of the Euler Equations

Abstract.

The flow of the ideal incompressible fluid can be regarded as the motion along a geodesic on the group of volume preserving diffeomorphisms of the flow domain. Thus, we can define the exponential map transferring the initial velocity (an incompressible vector field tangent to the boundary) into the fluid configuration at the time one. In the 2-d case this map is defined globally (for all initial velocities) and smooth. Recently Ebin, Misiołlek and Preston [EMP] have proved that the exponential map is Fredholm. In this paper a stronger result is proved: the exponential map is Fredholm quasiruled (FQR), i.e. has a rigid global structure. The proof uses some simplified tools of paradifferential calculus.