The SQG Equation as a Geodesic Equation

Abstract

We demonstrate that the surface quasi-geostrophic (SQG) equation given by $${\theta }_t+⟨u,\nabla \theta ⟩=0,\theta =\nabla \times (-\Delta {)}^{-1/2}u,$$ is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold $M$ in the right-invariant ${\dot{H}}^{-1/2}$ metric. We show by example, that the Riemannian exponential map is smooth and non-Fredholm, and that the sectional curvature at the identity is unbounded of both signs.

References in corpus (1)

• Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

Cited by in corpus (3)

• Vanishing distance phenomena and the geometric approach to SQG
• Fractional Sobolev metrics on spaces of immersions
• Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves