Unconditional regularity and trace results for the isentropic Euler equations with $\gamma = 3$

TL;DR

This paper investigates the regularity of solutions to the isentropic Euler equations with γ=3, revealing new trace theorems and partial regularity results that suggest near-classical behavior for bounded entropy solutions.

Contribution

It introduces novel trace and regularity theorems for the isentropic Euler equations with γ=3, demonstrating regularizing effects in a nonlinear hyperbolic system not of Temple class.

Findings

Density is almost everywhere upper semicontinuous away from vacuum.

First example of nonlinear hyperbolic system with near classical regularity.

Regularity results limit applicability of convex integration methods.

Abstract

In this paper, we study the regularity properties of bounded entropy solutions to the isentropic Euler equations with $\gamma = 3$. First, we use a blow-up technique to obtain a new trace theorem for all such solutions. Second, we use a modified De Giorgi type iteration on the kinetic formulation to show a new partial regularity result on the Riemann invariants. We are able to conclude that in fact for any bounded entropy solution $u$, the density $\rho$ is almost everywhere upper semicontinuous away from vacuum. To our knowledge, this is the first example of a nonlinear hyperbolic system, which fails to be Temple class, but has the property that generic $L^\infty$ initial data give rise to bounded entropy solutions with a form of near classical regularity. This provides one example that $2\times 2$ hyperbolic systems can possess some of the more striking regularizing effects known to hold generically in the genuinely nonlinear, multidimensional scalar setting. While we are not able to use our regularity results to show unconditional uniqueness, the results substantially lower the likelihood that current methods of convex integration can be used in this setting.