Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation
Abstract
We study dual-cascade flux laws for the stochastic forced--dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and a mixed flux law for the inverse cascade of the Hamiltonian. We further establish complementary scaling laws: a cubic small-scale law for the Hamiltonian structure function and a large-scale vanishing law for the SPE structure function. In particular, both Hamiltonian laws appear to be new for SQG even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: Bs3,∞-regularity above the Onsager threshold 1/3 rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.
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BibTeX
@misc{liu2026mixed,
title = {{Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation}},
author = {Rongchang Liu and Kening Lu and Zexin Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.26788},
eprint = {2606.26788},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26788},
}
RIS
TY - UNPB
TI - Mixed Third-Order Flux Laws for Dual Cascade in the Stochastic SQG Equation
AU - Rongchang Liu
AU - Kening Lu
AU - Zexin Wang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2606.26788
UR - https://arxiv.org/abs/2606.26788
AB - We study dual-cascade flux laws for the stochastic forced--dissipative surface quasi-geostrophic (SQG) equation on a large periodic box. For statistically stationary solutions, under a weak anomalous dissipation assumption, we derive rigorous mixed third-order structure-function laws for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy (SPE) and a mixed flux law for the inverse cascade of the Hamiltonian. We further establish complementary scaling laws: a cubic small-scale law for the Hamiltonian structure function and a large-scale vanishing law for the SPE structure function. In particular, both Hamiltonian laws appear to be new for SQG even as an explicit third-order structure-function relation. We also prove Onsager-type obstruction results showing that sufficiently regular stationary families cannot sustain the corresponding non-zero fluxes: B^s_3,∞-regularity above the Onsager threshold 1/3 rules out the direct SPE flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux. These results provide a rigorous formulation of the SQG dual-cascade phenomenology in a stochastic stationary setting.
ER -