Generic Singularities for 2D Pressureless Flow

TL;DR

This paper analyzes the formation and structure of singularities in 2D pressureless gas flows, focusing on the behavior of singular curves, their interactions, and the initial conditions leading to singularity formation.

Contribution

It introduces a detailed study of singular curves in 2D pressureless flow, including their non-hyperbolic nature, local solutions via Cauchy-Kovalevskaya, and interaction dynamics.

Findings

Singular curves are not governed by hyperbolic equations.

Local solutions exist for analytic initial data.

Interactions of singular curves are characterized in generic positions.

Abstract

In this paper, we consider the Cauchy problem for pressureless gases in two space dimensions with generic smooth initial data (density and velocity). These equations give rise to singular curves, where the mass has positive density w.r.t.~1-dimensional Hausdorff measure. We observe that the system of equations describing these singular curves is not hyperbolic. For analytic data, local solutions are constructed using a version of the Cauchy-Kovalevskaya theorem. We then study the interaction of two singular curves, in generic position. Finally, for a generic initial velocity field, we investigate the asymptotic structure of the smooth solution up to the first time when a singularity is formed.