The Initial Stages of a Generic Singularity for a 2D Pressureless Gas

TL;DR

This paper analyzes the formation and evolution of singularities in 2D pressureless gas equations, providing an asymptotic description beyond singularity time and constructing local solutions using analytic data and advanced PDE techniques.

Contribution

It offers a novel asymptotic analysis of singularity development and extends solutions beyond singularity formation for pressureless gases in two dimensions.

Findings

Singular curve formation with positive density

Non-hyperbolic behavior of the system beyond singularity

Existence of local solutions via Cauchy-Kovalevskaya theorem

Abstract

We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where both the the density and the velocity gradient become unbounded. Aim of this paper is to provide an asymptotic description of the solution beyond the time of singularity formation. For $t>t_0$ we show that a singular curve is formed, where the mass has positive density w.r.t.1-dimensional Hausdorff measure. The system of equations describing the behavior of the singular curve is not hyperbolic. Working within a class of analytic data, local solutions can be constructed using a version of the Cauchy-Kovalevskaya theorem. For this purpose, by a suitable change of variables we rewrite the evolution equations as a first order system of Briot-Bouquet type, to which a general existence-uniqueness theorem can then be applied.