The hidden null structure of the compressible Euler equations and a prelude to applications

TL;DR

This paper uncovers a new formulation of the compressible Euler equations with strong null structures, enabling advanced geometric analysis and paving the way for understanding shock formation in smooth solutions.

Contribution

It introduces a novel formulation of the Euler equations with null structures, facilitating the use of geometric vectorfield methods to analyze shock formation.

Findings

New formulation exhibits strong null structures.

Null structures are crucial for shock formation analysis.

Framework allows solutions to remain smooth up to shock time.

Abstract

We derive a new formulation of the compressible Euler equations exhibiting remarkable structures, including surprisingly good null structures. The new formulation comprises covariant wave equations for the Cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity, defined to be vorticity divided by density. The equations allow one to use the full power of the geometric vectorfield method in treating the "wave part" of the system. A crucial feature of the new formulation is that all derivative-quadratic inhomogeneous terms verify the strong null condition. The latter is a nonlinear condition signifying the complete absence of nonlinear interactions involving more than one differentiation in a direction transversal to the acoustic characteristics. Moreover, the same good structures are found in the equations verified by the Euclidean divergence and curl of the specific vorticity. This is important because one needs to combine estimates for the divergence and curl with elliptic estimates to obtain sufficient regularity for the specific vorticity, whose derivatives appears as inhomogeneous terms in the wave equations. The above structures collectively open the door for our forthcoming results: exhibiting a stable regime in which initially smooth solutions develop a shock singularity (in particular the first Cartesian coordinate partial derivatives of the velocity and density blow up) while, relative to a system of geometric coordinates adapted to the acoustic characteristics, the solution (including the vorticity) remains many times differentiable, all the way up to the shock. The good null structures, which are often associated with global solutions, are in fact key to proving that the shock singularity forms. Our secondary goal in this article is to overview the central role that the structures play in the proof.