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Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler

Isaac Neal, Steve Shkoller, Vlad Vicol

TL;DR

The paper proves that smooth solutions to the 1D Euler equations with non-constant entropy can experience an infinite hierarchy of finite-time gradient catastrophes, each corresponding to a cusp-type pre-shock with Hölder exponent $1/(2n+1)$. It develops a differentiated-Riemann-variable framework and a fast-acoustic characteristic change of variables to obtain uniform high-regularity control up to the first singularity, then leverages an implicit-function- theorem-based construction to show finite-codimension stability of these singularities in $W^{2n+2,\infty}$. The main result identifies a codimension-$(2n-2)$ Banach manifold of initial data $\mathcal{M}_n$ for which the Euler solution forms the $C^{0,1/(2n+1)}$ pre-shock at a time $T_*=\tfrac{2}{1+\alpha}+\mathcal{O}(\varepsilon)$, with explicit cusp expansions for $u$ and $\sigma$ and near-Riemann data behavior for the entropy. The framework unifies prior $C^{0,1/3}$ results and known unstable higher-order cusps, providing a robust, characteristic-based method that extends to broader hyperbolic systems and clarifies the transition from smooth flow to gradient blowup and shock development.

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 $C^{0,\frac{1}{2n+1}}$ Hölder-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$.

Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler

TL;DR

The paper proves that smooth solutions to the 1D Euler equations with non-constant entropy can experience an infinite hierarchy of finite-time gradient catastrophes, each corresponding to a cusp-type pre-shock with Hölder exponent . It develops a differentiated-Riemann-variable framework and a fast-acoustic characteristic change of variables to obtain uniform high-regularity control up to the first singularity, then leverages an implicit-function- theorem-based construction to show finite-codimension stability of these singularities in . The main result identifies a codimension- Banach manifold of initial data for which the Euler solution forms the pre-shock at a time , with explicit cusp expansions for and and near-Riemann data behavior for the entropy. The framework unifies prior results and known unstable higher-order cusps, providing a robust, characteristic-based method that extends to broader hyperbolic systems and clarifies the transition from smooth flow to gradient blowup and shock development.

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 , 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 Hölder-regularity. We show that such Euler solutions are codimension- stable in the Sobolev space .
Paper Structure (34 sections, 39 theorems, 430 equations)

This paper contains 34 sections, 39 theorems, 430 equations.

Table of Contents

  1. Introduction
  2. The 1D Euler equations
  3. Prior results
  4. Main results
  5. Outline and new ideas
  6. Notation
  7. Intuition from the Burgers equation
  8. The differentiated Riemann variables, the initial data, and a blowup criterion
  9. Classical and differentiated Riemann variables
  10. Assumptions on the initial data
  11. Local well-posedness and the Eulerian blowup criterion
  12. Initial estimates along the fast acoustic characteristic
  13. The system in Lagrangian variables
  14. Initial estimates and determination of the blowup time
  15. Higher order estimates
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $n \geq 1$ be an integer, and fix the adiabatic exponent $\gamma > 1$. Then there exists a codimension $(2n-2)$ Banach submanifold $\mathcal{M}_n$ of $(W^{2n+2, \infty}(\mathbb T))^3$ which contains the point $(\overline u_0, \overline \sigma_0,\overline S_0):= \frac{1}{2} (\overline{w}_0, \over for all $y$ such that $|y-y_*|^{\frac{1}{2n+1}} \lesssim 1$ and a constant $\mathsf{b}$ with the as

Theorems & Definitions (71)

  • Theorem 1.1: Finite codimension-stable shock formation, abbreviated
  • Remark 1.2: The Riemann data and the infinitely unstable limit $n\to \infty$
  • Proposition 2.1: Stability and finite codimension-stability
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 5.1
  • proof
  • ...and 61 more