On Incompressible Averaged Lagrangian Hydrodynamics

This paper is devoted to the geometric analysis of the incompressible averaged Euler equations on compact Riemannian manifolds with boundary. The equation also coincides with the model for a second-grade non-Newtonian fluid. We study the analytical and geometrical properties of the Lagrangian flow map. We prove existence and uniqueness of smooth-in-time solutions for initial data in $H^s$, $s > n/2 +1$ by establishing the existence of smooth geodesics of a new weak right invariant metric on new subgroups of the volume-preserving diffeomorphism group. We establish smooth limits of zero viscosity for the second-grade fluids equations even on manifolds with boundary. We prove that the weak curvature operator of the weak invariant metric is continuous in the $H^s$ topology for $s> n/2+2$, thus proving existence and uniqueness for the Jacobi equation. We show that this new metric stabilizes the Lagrangian flow of the original Euler equations by changing the sign of the sectional curvature.