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Sharp Regularity Thresholds for Viscous Katz--Pavlović Dyadic Models
Abstract
We study the unforced viscous Katz--Pavlović dyadic model with non-negative smooth initial data for two shell settings. For super-exponential scales Nn=N0bn, with N0>1 and 1<b<2, we prove that α=1/(b+2) is the sharp dissipation threshold: every such datum produces a globally smooth solution at and above this value, whereas suitable compactly supported data lose regularity in finite time below it. For geometric scales Nn=N0Λn, global regularity holds when α≥1/3 and 4α−1>(log2)/(2logΛ). In particular, for Λ>23/2, the result reaches α=1/3 and, together with the known classical blow-up theorem, identifies the sharp threshold in classical shell ratios.
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BibTeX
@misc{zhang2026sharp,
title = {{Sharp Regularity Thresholds for Viscous Katz--Pavlović Dyadic Models}},
author = {Ziqian Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21312},
eprint = {2609.21312},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21312},
}
RIS
TY - UNPB
TI - Sharp Regularity Thresholds for Viscous Katz--Pavlović Dyadic Models
AU - Ziqian Zhang
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21312
UR - https://arxiv.org/abs/2609.21312
AB - We study the unforced viscous Katz--Pavlović dyadic model with non-negative smooth initial data for two shell settings. For super-exponential scales N_n=N_0^b^n, with N_0>1 and 1<b<2, we prove that α=1/(b+2) is the sharp dissipation threshold: every such datum produces a globally smooth solution at and above this value, whereas suitable compactly supported data lose regularity in finite time below it. For geometric scales N_n=N_0Λ^n, global regularity holds when α≥1/3 and 4α−1>(log2)/(2logΛ). In particular, for Λ>2^3/2, the result reaches α=1/3 and, together with the known classical blow-up theorem, identifies the sharp threshold in classical shell ratios.
ER -