AI 中文总结
研究大普朗特数下三维非扩散布辛涅斯克系统强解全局适定性问题，通过证明相关强解全局存在唯一，建立非恒定温度斑块初始数据下的边界正则性全局持续性，还证明了普朗特数趋于无穷时的极限及收敛情况。
AI 中文摘要
到目前为止，具有大初始数据的三维非扩散布辛涅斯克系统强解的全局适定性仍然是一个显著的开放问题。本文在大普朗特数 regime 中解决了这个问题。具体而言，我们证明了与初始数据\((u_0,\theta_0)\in H^{\frac{1}{2}}(\mathbb{R}^3) \times (L^1\cap L^s(\mathbb{R}^3))\)（\(s>3\)）相关的此三维布辛涅斯克系统强解的全局存在性和唯一性，前提是普朗特数足够大（阈值仅取决于\((u_0,\theta_0)\)的尺度不变范数）；此外，对于非恒定温度斑块初始数据，我们建立了演化温度斑块的\(C^{1,\gamma}\)、\(W^{2,\infty}\)和\(C^{2,\gamma}\)（\(0<\gamma<1\)）边界正则性的全局持续性，且在大普朗特数 regime 中具有相应的一致估计。此外，我们严格证明了普朗特数趋于无穷时的极限，并表明三维布辛涅斯克系统的斑块解收敛到三维斯托克斯 - 输运系统的唯一斑块解，并且\(C^{1,\gamma}\)、\(W^{2,\infty}\)和\(C^{2,\gamma}\)中的斑块边界正则性在时间上全局保持。特别是，我们关于三维斯托克斯 - 输运系统的结果可视为与二维斯托克斯 - 输运系统相关的 Grayer II [ARMA 2023] 主要结果的三维类似物。
英文摘要
So far the global well-posedness of strong solutions for the 3D non-diffusive Boussinesq system with large initial data remains a remarkable open problem. In this paper, we solve this problem in the regime of large Prandtl number. More precisely, we prove the global existence and uniqueness of strong solution for this 3D Boussinesq system associated with initial data $(u_0,θ_0)\in H^{\frac{1}{2}}(\mathbb{R}^3) \times (L^1\cap L^s(\mathbb{R}^3))$ with $s>3$, provided that the Prandtl number is sufficiently large (the threshold depends only on a scale-invariant norm of $(u_0,θ_0)$); moreover, for the non-constant temperature patch initial data, we establish the global persistence of $C^{1,γ}$, $W^{2,\infty}$, and $C^{2,γ}$ ($0<γ<1$) boundary regularity of the evolved temperature patch, with corresponding estimates uniform in the large Prandtl number regime. Furthermore, we rigorously justify the limit as the Prandtl number tends to infinity and show that the patch solution of the 3D Boussinesq system converges to the unique patch solution of the 3D Stokes-transport system, and that the patch boundary regularity in $C^{1,γ}$, $W^{2,\infty}$, and $C^{2,γ}$ is preserved globally in time. In particular, our result for the 3D Stokes-transport system can be viewed as the 3D analogue of the main result in Grayer II [ARMA 2023] concerning 2D Stokes-transport system.
Comments31 pages
