Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows
Abstract
We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) \textbf{58} (2025), no.~4, 943--970], which treats m-fold symmetry solutions with m≥4. We prove that every nonconstant W1,p solution satisfies |gθ(t)|Lp→∞ as t→±∞ for 1<p≤∞. For p=1, the total variation is conserved, but the Llog L modular tends to infinity whenever it is initially finite. If Dθg0 is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in Wα,r for αr<1. For such data, every weak L2 infinite-time limit generates a complete L2-precompact orbit. This structural assumption on Dθg0 is automatic for C1 data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. \textbf{10} (2023), no.~1, 1--100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for m≥4, they cover the full well-posed scale-invariant range m≥3.
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BibTeX
@misc{cao2026gradient,
title = {{Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows}},
author = {Daomin Cao and Junhong Fan and Guolin Qin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16755},
eprint = {2608.16755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16755},
}
RIS
TY - UNPB
TI - Gradient growth and relaxation to jump profiles for 3-fold symmetric scale-invariant Euler flows
AU - Daomin Cao
AU - Junhong Fan
AU - Guolin Qin
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.16755
UR - https://arxiv.org/abs/2608.16755
AB - We consider long-time behavior of the zero-homogeneous solutions with 3-fold symmetry to the two-dimensional Euler equation. This is the remaining case in the relaxation theory of Said, Elgindi, and Murray [Ann. Sci. Éc. Norm. Supér. (4) \textbf{58} (2025), no.~4, 943--970], which treats m-fold symmetry solutions with m≥4. We prove that every nonconstant W^1,p solution satisfies |g_θ(t)|_L^p→∞ as t→±∞ for 1<p≤∞. For p=1, the total variation is conserved, but the Llog L modular tends to infinity whenever it is initially finite. If D_θg_0 is a summable sum of non-atomic one-sign components and atoms, every profile in the two omega-limit sets is a jump profile, and each half-orbit approaches its omega-limit set in W^α,r for αr<1. For such data, every weak L^2 infinite-time limit generates a complete L^2-precompact orbit. This structural assumption on D_θg_0 is automatic for C^1 data. In particular, these results answer the question concerning small-scale creation and compact orbit raised by Drivas and Elgindi [EMS Surv. Math. Sci. \textbf{10} (2023), no.~1, 1--100, Problem~5] for all nonconstant smooth 3-fold symmetric scale-invariant flows. Together with the known theory for m≥4, they cover the full well-posed scale-invariant range m≥3.
ER -