Semi-invariant Riemannian metrics in hydrodynamics
Journal article, 2020

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.

Author

M. Bauer

Florida State University

Klas Modin

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Calculus of Variations and Partial Differential Equations

0944-2669 (ISSN) 1432-0835 (eISSN)

Vol. 59 2 65

Subject Categories

Computational Mathematics

Geometry

Mathematical Analysis

DOI

10.1007/s00526-020-1722-x

More information

Latest update

3/30/2020