Global regularity and infinite Prandtl-number limit for 2D Boussinesq temperature patches in Wk,∞ and Ck,γ: lower-order case
Abstract
We study global regularity and the infinite Prandtl number limit for temperature patches of the two-dimensional nondiffusive Boussinesq system on ℝ2, with velocity dissipation Λ2α, 1/2<α≤1. We establish global well-posedness for θ0∈ L1∩ L∞ and divergence-free u0∈ W1,p, 2≤ p<∞. Under suitable profile and striated regularity assumptions, we prove global persistence of C1,γ and C2,γ patch boundaries for every 0<γ≤1, allowing nonconstant profiles and including the Lipschitz endpoints W2,∞ and W3,∞. The velocity gradient and boundary regularity estimates are uniform in Pr∈[1,∞) on each finite time interval. Uniform global Lr velocity bounds hold for suitable finite r when α<1, and at α=1 for zero-mean initial temperatures with finite first absolute moment. Such bounds fail at α=1 for compactly supported nonzero-mean data. For temperature patches, we justify the infinite Prandtl number limit in the original frame when α<1 or the initial temperature has zero mean, and in a moving frame otherwise. The limit is the unique normalized global weak solution of the (fractional) Stokes-transport system and, importantly, preserves the patch structure and boundary regularity.
Related
- Persistence of C2 regularity for aggregation patches
- Asymptotic stability of the 2D Boussinesq equations around Couette flow in infinite channel
- Global Well-posedness and Regularity of the Dynamical Prandtl Equation
- Stability threshold for 3D Boussinesq equations with rotation near the Couette flow and vertical stratified temperature
- Regularity Aspects of Temperature in Heat-Conducting Fluid Systems via De Giorgi Methods
BibTeX
@misc{lazar2026global,
title = {{Global regularity and infinite Prandtl-number limit for 2D Boussinesq temperature patches in $W^k,\infty$ and $C^k,γ$: lower-order case}},
author = {Omar Lazar and Liutang Xue and Jiakun Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12253},
eprint = {2610.12253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12253},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Global regularity and infinite Prandtl-number limit for 2D Boussinesq temperature patches in W^k,∞ and C^k,γ: lower-order case
AU - Omar Lazar
AU - Liutang Xue
AU - Jiakun Yang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12253
UR - https://arxiv.org/abs/2610.12253
AB - We study global regularity and the infinite Prandtl number limit for temperature patches of the two-dimensional nondiffusive Boussinesq system on ℝ^2, with velocity dissipation Λ^2α, 1/2<α≤1. We establish global well-posedness for θ_0∈ L^1∩ L^∞ and divergence-free u_0∈ W^1,p, 2≤ p<∞. Under suitable profile and striated regularity assumptions, we prove global persistence of C^1,γ and C^2,γ patch boundaries for every 0<γ≤1, allowing nonconstant profiles and including the Lipschitz endpoints W^2,∞ and W^3,∞. The velocity gradient and boundary regularity estimates are uniform in Pr∈[1,∞) on each finite time interval. Uniform global L^r velocity bounds hold for suitable finite r when α<1, and at α=1 for zero-mean initial temperatures with finite first absolute moment. Such bounds fail at α=1 for compactly supported nonzero-mean data. For temperature patches, we justify the infinite Prandtl number limit in the original frame when α<1 or the initial temperature has zero mean, and in a moving frame otherwise. The limit is the unique normalized global weak solution of the (fractional) Stokes-transport system and, importantly, preserves the patch structure and boundary regularity.
ER -