  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 $H^k$-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.
