On Euler equation for incoherent fluid in curved spaces

TL;DR

The paper develops a hodograph-based approach to the $n$-dimensional Euler equation with constant pressure on curved spaces, treating characteristics as geodesics. By exploiting integrals of geodesic motion on specific geometries—most notably 2D surfaces of revolution, the cone, and spheres $S^2$ and $S^3$—the authors construct hodograph equations that yield broad families of solutions, including stationary flows, and they analyze blow-up (derivative singularity) conditions via the hodograph Jacobian. The work extends to higher dimensions by leveraging the large sets of conserved quantities arising from symmetry groups (e.g., $SO(4)$), enabling stationary solutions parametrized by arbitrary functions and outlining a program for further exploration of Euler flows on $S^n$ with $n\ge 4$. Overall, the paper provides a systematic framework for generating and analyzing nonlinear fluid solutions on curved manifolds and highlights the rich integrability structure induced by curvature and symmetry.

Abstract

Hodograph equations for the Euler equation in curved spaces with constant pressure are discussed. It is shown that the use of known results concerning geodesics and associated integrals allows to construct several types of hodograph equations. These hodograph equations provide us with various classes of solutions of the Euler equation, including stationary solutions. Particular cases of cone and sphere in the 3-dimensional Eulidean space are analysed in detail. Euler equation on the sphere in the 4-dimensional Euclidean space is considered too.