Gradient catastrophes and an infinite hierarchy of H\"older cusp-singularities for 1D Euler

TL;DR

This paper demonstrates an infinite hierarchy of finite-time gradient singularities in 1D Euler equations with non-constant entropy, revealing increasingly complex cusp-like behaviors characterized by specific H"older regularities.

Contribution

It establishes the existence of a hierarchy of gradient catastrophes with precise regularity properties and analyzes their stability in Sobolev spaces.

Findings

Existence of solutions with cusp-type singularities at specific H"older regularities.

Each singularity level has a codimension-$(2n-2)$ stability in Sobolev space.

Infinite hierarchy of finite-time gradient catastrophes proven for 1D Euler with entropy variations.

Abstract

We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ H\"older-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$.