Geometric investigations of a vorticity model equation

Abstract

Abstract. This article consists of a detailed geometric study of the one-dimensional vorticity model equationwhich is a particular case of the generalized Constantin-Lax-Majda equation. Wunsch showed that this equation is the Euler-Arnold equation on Diff(S 1 ) when the latter is endowed with the rightinvariant homogeneousḢ 1/2 -metric. In this article we prove that the exponential map of this Riemannian metric is not Fredholm and that the sectional curvature is locally unbounded. Furthermore, we prove a Beale-Kato-Majda-type blow-up criterion, which we then use to demonstrate a link to our non-Fredholmness result. Finally, we extend a blow-up result of Castro-Córdoba to the periodic case and to a much wider class of initial conditions, using a new generalization of an inequality for Hilbert transforms due to Córdoba-Córdoba.

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).…”

Fractional Sobolev metrics on spaces of immersions

“…Many prominent partial differential equations (PDEs) in hydrodynamics admit variational formulations as geodesic equations on an infinite-dimensional manifold of mappings. These include the incompressible Euler [2], Burger [35], modified Constantin-Lax-Majda [19,60,14], Camassa-Holm [17,39], Hunter-Saxton [30,43], surface quasi-geostrophic [20,59] and Korteweg-de Vries [50] equations of fluid dynamics as well as the governing equation of ideal magneto-hydrodynamics [58,44]. This serves as a strong motivation for the study of Riemannian geometry on mapping space.…”

Small Scales and Singularity Formation in Fluid Dynamics

“…Recently, global regularity near a manifold of equilibria as well as other interesting features of the solutions of (4.2) have been shown in [45]. Variants of (4.2) and other related models appear in for example [2,8,31,32,74], where further references can be found. Already in [56], Hou and Luo proposed a simplified one-dimensional model specifically designed to gain insight into the singularity formation process in the scenario described in Section 2.…”