Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
Abstract
We give a Lagrangian construction of forward self-similar solutions of the $2$D incompressible Euler equation. For every 1<α<2 and every real h∈ L2/α(𝕊1), the construction produces a flow with initial vorticity |x|−αh(arg x), without smallness or sign assumptions. The argument is based on a fixed-point formulation of backward particle transport from time t=1 to t=0. Mollifying the Biot--Savart velocity in self-similar variables preserves exact vorticity transport, while symmetrization yields the unforced Euler equation in velocity form even when the ordinary vorticity flux is not locally integrable. The limiting solution admits an almost-everywhere invertible, measure-preserving regular Lagrangian flow from time zero, with strong local continuity and preservation of signed value distributions and absolutely integrable Casimirs.
Related
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BibTeX
@misc{hou2026selfsimilar,
title = {{Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction}},
author = {Thomas Y. Hou and Xiang Qin and Peicong Song and Wenhan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25783},
eprint = {2610.25783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25783},
}
RIS
TY - UNPB
TI - Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
AU - Thomas Y. Hou
AU - Xiang Qin
AU - Peicong Song
AU - Wenhan Zhang
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25783
UR - https://arxiv.org/abs/2610.25783
AB - We give a Lagrangian construction of forward self-similar solutions of the $2$D incompressible Euler equation. For every 1<α<2 and every real h∈ L^2/α(𝕊^1), the construction produces a flow with initial vorticity |x|^−αh(arg x), without smallness or sign assumptions. The argument is based on a fixed-point formulation of backward particle transport from time t=1 to t=0. Mollifying the Biot--Savart velocity in self-similar variables preserves exact vorticity transport, while symmetrization yields the unforced Euler equation in velocity form even when the ordinary vorticity flux is not locally integrable. The limiting solution admits an almost-everywhere invertible, measure-preserving regular Lagrangian flow from time zero, with strong local continuity and preservation of signed value distributions and absolutely integrable Casimirs.
ER -