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 $k\ge 2$.Comment: 24 page

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

“…For the bicentury of this achievement, Arnold has extended this geometric framework to hydrodynamics and recast the equations of motion of a perfect fluid (with fixed boundary) as the geodesic flow on the volume-preserving diffeomorphisms group of the domain. Since, then a similar geometric formulation has been found for several important PDEs in mathematical physics, including in particular the Camassa-Holm equation [17,46,42], the modified Constantin-Lax-Majda equation [21,60,29,10] or the SQG-equation [22,59,9], see [56,40] for further examples and references. From a geometrical view-point, this theory can be reduced to the study of right-invariant Riemannian metrics on the diffeomorphism group of a manifold M (or one of its subgroup like SDiff ∞ µ (M ), the group of diffeomorphism which preserve a volume form µ).…”

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…For the homogeneousḢ 1/2 -metric this result was proven in [47] and the estimates were then extended to cover general metrics given via Fourier multipliers in [46].…”

Geodesic completeness of the $$H^{3/2}$$ metric on $$\mathrm {Diff}(S^{1})$$

“…The interest in (fractional) order metrics on diffeomorphism groups is fuelled by their relations to various prominent PDEs of mathematical physics: In the seminal article [1], Arnold showed in 1965 that Euler's equations for the motion of an incompressible, ideal fluid have a geometric interpretation as the geodesic equations on the group of volume preserving diffeomorphisms. Since then, an analogous result has been found for a whole variety of PDEs, including the inviscid Burgers equation, the Hunter-Saxton equation, the Camassa-Holm equation [6,19], and the modified Constantin-Lax-Majda (mCLM) equation [10,14]. Building on the pioneering work of Ebin and Marsden [11], these geometric interpretations have been used to obtain rigourous well-posedness and stability results for the corresponding PDEs [9,26,25,23,3,15,21].…”