The geometry of a vorticity model equation

Abstract

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^k$ with $k\ge 2$.

24 pages

References in corpus (4)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• The Degasperis-Procesi equation as a non-metric Euler equation
• The periodic b-equation and Euler equations on the circle
• Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications

Cited by in corpus (23)

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• Local and Global Well-posedness of the fractional order EPDiff equation on ${\mathbb{R}}^d$
• The homogeneous Sobolev metric of order one on diffeomorphism groups on the real line
• Geodesic Completeness for Sobolev $H^s$-metrics on the Diffeomorphisms Group of the Circle
• Vanishing distance phenomena and the geometric approach to SQG
• On the De Gregorio modification of the Constantin-Lax-Majda Model
• Local well-posedness of the EPDiff equation: a survey
• On the Constantin-Lax-Majda Model with Convection
• Semi-invariant Riemannian metrics in hydrodynamics
• On a two-component $\pi$-Camassa--Holm system
• A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations
• Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
• Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• Fractional Sobolev metrics on spaces of immersed curves
• Completeness and geodesic distance properties for fractional Sobolev metrics on spaces of immersed curves
• Global Existence and Blowup for Geodesics in Universal Teichmüller Spaces
• Sub-Riemannian geometry on infinite-dimensional manifolds
• Lagrangian aspects of the axisymmetric Euler equation
• Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmueller space and curve