To the top

Page Manager: Webmaster
Last update: 9/11/2012 3:13 PM

Tell a friend about this page
Print version

Semi-invariant Riemannian… - University of Gothenburg, Sweden Till startsida
To content Read more about how we use cookies on

Semi-invariant Riemannian metrics in hydrodynamics

Journal article
Authors M. Bauer
Klas Modin
Published in Calculus of Variations and Partial Differential Equations
Volume 59
Issue 2
ISSN 0944-2669
Publication year 2020
Published at Department of Mathematical Sciences
Language en
Keywords 58b10, 35q31, shallow-water equation, fractional order, epdiff equation, well-posedness, geodesic-flow, geometry, model, theorem, space, Mathematics
Subject categories Mathematics


Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa-Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of diffeomorphisms with a right invariant Riemannian metric. By establishing regularity properties of the Riemannian spray one can then obtain local, and sometimes global, existence and uniqueness results. There are, however, many hydrodynamic-type equations, notably shallow water models and compressible Euler equations, where the underlying infinite-dimensional Riemannian structure is not fully right invariant, but still semi-invariant with respect to the subgroup of volume preserving diffeomorphisms. Here we study such metrics. For semi-invariant metrics of Sobolev Hk-type we give local and some global well-posedness results for the geodesic initial value problem. We also give results in the presence of a potential functional (corresponding to the fluid's internal energy). Our study reveals many pitfalls in going from fully right invariant to semi-invariant Sobolev metrics; the regularity requirements, for example, are higher. Nevertheless the key results, such as no loss or gain in regularity along geodesics, can be adopted.

Page Manager: Webmaster|Last update: 9/11/2012

The University of Gothenburg uses cookies to provide you with the best possible user experience. By continuing on this website, you approve of our use of cookies.  What are cookies?