On pressureless Euler equation with external force

TL;DR

This paper derives hodograph equations for the n-dimensional pressureless Euler equations with linear external forces, providing implicit solutions and insights into gradient catastrophes, with special cases including periodic solutions and Coriolis forces.

Contribution

It introduces a new hodograph formulation for pressureless Euler equations with external forces, revealing solution behaviors and periodicity in even dimensions.

Findings

Solutions are implicit and can indicate gradient catastrophes.

In even dimensions, solutions can be periodic under certain external forces.

Includes specific examples like Coriolis force in various dimensions.

Abstract

Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force.