发表机构
Yonsei University; Hanyang University（延世大学; 汉阳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文针对奇异压力缩放下的可压缩Euler-alignment系统，建立了统一正则性理论，证明了低马赫数极限收敛于带分数阶耗散的不可压缩Navier-Stokes系统。
AI 中文摘要
我们研究了在奇异压力缩放下的可压缩Euler-alignment系统，其中超奇异通信权重诱导了一个阶数为$2\alpha$（$0<\alpha<1$）的分数阶对齐算子。该缩放对应于长时间和小速度区域，并导致一个低马赫数问题，其中密度被迫保持在常数状态附近。我们的主要结果是该缩放压力系统的统一正则性理论。我们建立了关于缩放参数的统一估计，并在常数状态附近构造了全局强解。分析的一个关键特征是，在低阶分数阶区域$0<\alpha\le\frac12$中，估计在较低的Sobolev条件$s>\frac d2+1-2\alpha$下闭合，比直接使用分数阶对齐耗散所产生的阈值$s>\frac d2 + 1 - \alpha$多获得了$\alpha$阶导数。这是通过将奇异对齐算子的精细交换子估计与压力缩放引起的密度耗散相结合而实现的。作为统一估计的一个应用，我们证明了向具有分数阶耗散的不可压缩Navier--Stokes系统的低马赫数极限。对于一般小的、可能未准备好的初始数据，Helmholtz分解与声学分量的色散估计相结合，在时空局部产生了对极限系统分布解的序列强收敛。对于准备好的初始数据，相对能量论证进一步将极限确定为具有规定充分正则解，并在分数阶耗散范数中产生了全局强收敛。
英文摘要
We study the compressible Euler-alignment system with pressure under a singular pressure scaling, where the hypersingular communication weight induces a fractional alignment operator of order $2α$, $0<α<1$. The scaling corresponds to a large-time and small-velocity regime and leads to a low Mach number problem in which the density is forced to remain close to a constant state. Our main result is a uniform regularity theory for this scaled pressure system. We establish uniform estimates with respect to the scaling parameter and construct global strong solutions near the constant state. A key feature of the analysis is that, in the low-order fractional regime $0<α\le\frac12$, the estimates close under the lower Sobolev condition $s>\frac d2+1-2α$, gaining $α$ derivatives over the threshold $s>\frac d2 + 1 - α$ arising from a direct use of the fractional alignment dissipation. This is achieved by combining refined commutator estimates for the singular alignment operator with the density dissipation induced by the pressure scaling. As an application of the uniform estimates, we justify the low Mach number limit toward the incompressible Navier--Stokes system with fractional dissipation. For general small, possibly ill-prepared initial data, a Helmholtz decomposition combined with dispersive estimates for the acoustic component yields subsequential strong convergence locally in space-time to a distributional solution of the limiting system. For well-prepared initial data, a relative-energy argument further identifies the limit with a prescribed sufficiently regular solution and yields global strong convergence in the fractional dissipation norm.
