On blow-up shock waves for a nonlinear PDE associated with Euler equations

TL;DR

This paper derives a second-order PDE from Euler's equations, demonstrating the existence of shock and rarefaction waves, and shows that a point gradient blow-up can be uniquely extended, resolving key issues in such equations.

Contribution

It introduces a novel second-order PDE from Euler's equations and establishes a unique similarity extension after blow-up, addressing entropy and uniqueness concerns.

Findings

Existence of shock and rarefaction waves in the PDE

Unique similarity extension after point gradient blow-up

Resolution of entropy and uniqueness issues

Abstract

A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that settles uniqueness/entropy issues for such equations.