A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy

Isaac Neal; Steve Shkoller; Vlad Vicol; Isaac Neal; Steve Shkoller; Vlad Vicol

doi:10.3934/cam.2025009

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 188-236. doi: 10.3934/cam.2025009

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Research article

A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy

1.
Department of Mathematics, University of California Davis, One Shields Avenue, Davis, CA 95616, USA
2.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer St, New York, NY 10012, USA

Received: 16 December 2024 Revised: 03 January 2025 Accepted: 10 February 2025 Published: 04 March 2025
35Q31, 35L67

We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of the Euler equations form a first singularity or gradient blow-up. This first singularity is termed a Hölder $ C^{\frac{1}{3}} $ pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to ^[1] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.
- Euler equations,
- shock formation,
- singularity formation,
- characteristics
Citation: Isaac Neal, Steve Shkoller, Vlad Vicol. A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy[J]. Communications in Analysis and Mechanics, 2025, 17(1): 188-236. doi: 10.3934/cam.2025009

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Abstract

We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of the Euler equations form a first singularity or gradient blow-up. This first singularity is termed a Hölder $ C^{\frac{1}{3}} $ pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to ^[1] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.

References

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