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Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation
Abstract
We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on ℝ2. More precisely, for every 1<s<2, T>0 and ε>0, we construct initial data θ0∈ Hs(ℝ2) with |θ0|Hs≤ε that generate a solution θ on [0,T] such that θ∈ L∞([0,T];H1+γ(ℝ2)), θ(t)∉ Hσ(ℝ2) for every t∈(0,T], σ>σ*(s), where γ>0 and σ*(s) satisfy 1+γ<σ*(s):= (s(3−s))/(1+2s−s2)<s. The solution is unique in a determined family of classical solutions with initial datum θ0.
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BibTeX
@misc{chen2026instantaneous,
title = {{Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation}},
author = {Qionglei Chen and Changxing Miao and Yaowei Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12011},
eprint = {2610.12011},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12011},
}
RIS
TY - UNPB
TI - Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation
AU - Qionglei Chen
AU - Changxing Miao
AU - Yaowei Xie
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12011
UR - https://arxiv.org/abs/2610.12011
AB - We prove instantaneous loss of Sobolev regularity across a fixed gap for the inviscid surface quasi-geostrophic equation on ℝ^2. More precisely, for every 1<s<2, T>0 and ε>0, we construct initial data θ_0∈ H^s(ℝ^2) with |θ_0|_H^s≤ε that generate a solution θ on [0,T] such that θ∈ L^∞([0,T];H^1+γ(ℝ^2)), θ(t)∉ H^σ(ℝ^2) for every t∈(0,T], σ>σ_*(s), where γ>0 and σ_*(s) satisfy 1+γ<σ_*(s):= (s(3−s))/(1+2s−s^2)<s. The solution is unique in a determined family of classical solutions with initial datum θ_0.
ER -