Zobrazeno 1 - 10
of 84
pro vyhledávání: '"Misiolek, Gerard"'
The study of diffeomorphism groups and their applications to problems in analysis and geometry has a long history. In geometric hydrodynamics, pioneered by V.~Arnold in the 1960s, one considers an ideal fluid flow as the geodesic motion on the infini
Externí odkaz:
http://arxiv.org/abs/2411.03265
We investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on grou
Externí odkaz:
http://arxiv.org/abs/2312.04697
Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and ac
Externí odkaz:
http://arxiv.org/abs/2205.01143
Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of volume pres
Externí odkaz:
http://arxiv.org/abs/2105.11869
Publikováno v:
Bull. Amer. Math. Soc. 58, 377-442 (2021)
We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton's equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations
Externí odkaz:
http://arxiv.org/abs/2001.01143
We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along th
Externí odkaz:
http://arxiv.org/abs/1911.10302
Publikováno v:
Arch. Ration. Mech. Anal., 234(2):549-573, 2019
The Madelung transform is known to relate Schr\"odinger-type equations in quantum mechanics and the Euler equations for barotropic-type fluids. We prove that, more generally, the Madelung transform is a K\"ahler map (i.e. a symplectomorphism and an i
Externí odkaz:
http://arxiv.org/abs/1807.07172
Publikováno v:
PNAS, 115(24):6165-6170, 2018
We introduce a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in thi
Externí odkaz:
http://arxiv.org/abs/1711.00321
Akademický článek
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We prove that the Riemannian exponential map of the right-invariant $L^2$ metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.
Comment: 16 page
Comment: 16 page
Externí odkaz:
http://arxiv.org/abs/1611.09993