Year: 2026
Type: preprint
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: $B^s_{3,\infty}$-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.
Source: arXiv (Cornell University)
Authors: Liu, Rongchang, Lu, Kening, Wang, Zexin
Institution: Sichuan University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G2388347233, G3411950497, G3701778458
OpenAlex ID: W7166053809