Shock formation for 2D Isentropic Euler equations with self-similar variables

TL;DR

This paper demonstrates shock formation in 2D isentropic Euler equations with ideal gas law, showing that smooth initial data can lead to self-similar point shocks with non-zero vorticity, using a modulation method based on Burgers profiles.

Contribution

It extends shock formation analysis to 2D Euler equations with non-zero vorticity, establishing self-similar shock profiles and employing a novel modulation approach.

Findings

Shock formation occurs at a single point near planar symmetry.

Solutions exhibit uniform 1/3-Hölder regularity over time.

Shocks are self-similar and share the same profile as 2D Burgers solutions.

Abstract

We study the 2D isentropic Euler equations with the ideal gas law. We exhibit a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions are associated with non-zero vorticity at the shock and have uniform-in-time 1/3-H\"older bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. Our proof, following the 3D shock formation result of Buckmaster, Shkoller and Vicol, is based on the stable 2D self-similar Burgers profile and the modulation method.