From entropy waves to vortex sheets for the Euler equations
Abstract
We construct piecewise smooth solutions of the full compressible Euler equations that evolve from entropy waves into vortex sheets while retaining a strictly positive Taylor jump. For every integer κ≥1, we obtain solutions satisfying initial material compatibility for pressure and velocity through order κ, whose tangential velocity jump has a nonzero leading term of order tκ+1. These solutions are periodic in the tangential variables and admit a smooth realization in a slab with impermeable walls. We also construct solutions on a local nonperiodic tangential patch that satisfy initial material compatibility at every order, yet have a tangential velocity jump that is nonzero for every sufficiently small positive time and flat at t=0. The finite-order construction uses a normal Cauchy-Kowalevski problem for velocity potentials, whereas the all-order construction uses a homogeneous reduction to a smooth hyperbolic Cauchy problem.
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BibTeX
@misc{trakhinin2026from,
title = {{From entropy waves to vortex sheets for the Euler equations}},
author = {Yuri Trakhinin and Wei Wang and Zhifei Zhang and Wenbin Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32893},
eprint = {2610.32893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32893},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - From entropy waves to vortex sheets for the Euler equations
AU - Yuri Trakhinin
AU - Wei Wang
AU - Zhifei Zhang
AU - Wenbin Zhao
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32893
UR - https://arxiv.org/abs/2610.32893
AB - We construct piecewise smooth solutions of the full compressible Euler equations that evolve from entropy waves into vortex sheets while retaining a strictly positive Taylor jump. For every integer κ≥1, we obtain solutions satisfying initial material compatibility for pressure and velocity through order κ, whose tangential velocity jump has a nonzero leading term of order t^κ+1. These solutions are periodic in the tangential variables and admit a smooth realization in a slab with impermeable walls. We also construct solutions on a local nonperiodic tangential patch that satisfy initial material compatibility at every order, yet have a tangential velocity jump that is nonzero for every sufficiently small positive time and flat at t=0. The finite-order construction uses a normal Cauchy-Kowalevski problem for velocity potentials, whereas the all-order construction uses a homogeneous reduction to a smooth hyperbolic Cauchy problem.
ER -