发表机构
American University of Sharjah; Beijing Normal University（沙迦美国大学; 北京师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二维非扩散Boussinesq系统的温度斑块，研究其在$W^{k,\infty}$和$C^{k,\gamma}$中的整体正则性与无限普朗特数极限，建立了相关全局适定性，证明了斑块边界的整体持久性，还验证了无限普朗特数极限的存在性并保持斑块结构与边界正则性。
AI 中文摘要
我们研究二维非扩散Boussinesq系统在$\mathbb R^2$上温度斑块的整体正则性与无限普朗特数极限，其中速度耗散为$\Lambda^{2\alpha}$、$\frac12<\alpha\leq1$。我们建立了$\theta_0\in L^1\cap L^\infty$及无散度$u_0\in W^{1,p}$、$2\leq p<\infty$的整体适定性。在适当的剖面和条纹正则性假设下，我们对每个$0<\gamma\leq1$证明了$C^{1,\gamma}$和$C^{2,\gamma}$斑块边界的整体持久性，允许非恒定剖面并包含Lipschitz端点$W^{2,\infty}$和$W^{3,\infty}$。速度梯度和边界正则性估计在每个有限时间区间上关于$\mathrm{Pr}\in[1,\infty)$是一致的。当$\alpha<1$时，对于适当的有限$r$存在一致的整体$L^r$速度界；当初始温度为零均值且具有有限一阶绝对矩时，在$\alpha=1$处也成立。对于紧支非零均值数据，在$\alpha=1$处此类界不成立。对于温度斑块，当$\alpha<1$或初始温度具有零均值时，我们在原坐标系中证明无限普朗特数极限；否则在运动坐标系中证明。该极限是（分数阶）Stokes-输运系统唯一的归一化整体弱解，且重要的是，它保持斑块结构和边界正则性。
英文摘要
We study global regularity and the infinite Prandtl number limit for temperature patches of the two-dimensional nondiffusive Boussinesq system on $\mathbb R^2$, with velocity dissipation $Λ^{2α}$, $\frac12<α\leq1$. We establish global well-posedness for $θ_0\in L^1\cap L^\infty$ and divergence-free $u_0\in W^{1,p}$, $2\leq p<\infty$. Under suitable profile and striated regularity assumptions, we prove global persistence of $C^{1,γ}$ and $C^{2,γ}$ patch boundaries for every $0<γ\leq1$, allowing nonconstant profiles and including the Lipschitz endpoints $W^{2,\infty}$ and $W^{3,\infty}$. The velocity gradient and boundary regularity estimates are uniform in $\mathrm{Pr}\in[1,\infty)$ 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.
Comments70 pages
