二维可压缩微极流体方程的大体黏度与小角黏度组合极限

中文摘要 AI 辅助

本文研究二维可压缩微极流体方程的大体黏度与小角黏度组合极限。我们首先在初始速度满足$\u221a\nu \\|{\rm div} u_0\\|_{L^2}$一致有界的良备条件下，建立了大体黏度情形下柯西问题强解的全局适定性。所获得的全局先验估计对体黏度$\nu$和角黏度$\varepsilon$是一致的。基于这些一致估计，我们进而证明了当$\nu\to\infty$且$\varepsilon\to0$时的组合极限，并表明可压缩微极系统的解收敛到黏性且非扩散的不可压缩微极系统的解。我们还推导了不可压缩极限和角黏度消失极限的收敛速率。特别地，不可压缩极限的收敛速率一般被证明为$\nu^{-1/4}$阶，在初始密度满足额外衰减条件下可改进为$\nu^{-1/2}$阶。与以往工作相比，角黏度消失极限的建立无需额外的技术假设$\lim_{\varepsilon\to0+}\zeta/\varepsilon^\alpha<\infty$（其中$1/2\leq \alpha<\infty$），这将对相关结果从不可压缩情形推广到了可压缩情形。作为副产品，当体黏度足够大时，我们获得了带或不带角黏度的二维可压缩微极方程大解的全局正则性。

英文摘要

This paper is concerned with the combined limits of large bulk viscosity and small angular viscosity for the two-dimensional compressible micropolar fluid equations. We first establish the global well-posedness of strong solutions to the Cauchy problem with large bulk viscosity, when the initial velocity is well-prepared in the sense that $\sqrtν\|{\rm div} u_0\|_{L^2}$ is uniformly bounded. The global a priori estimates obtained are uniform with respect to both the bulk viscosity $ν$ and the angular viscosity $\varepsilon$. Based on these uniform estimates, we then justify the combined limits as $ν\to\infty$ and $\varepsilon\to0$, and show that the solutions of the compressible micropolar system converge to those of the viscous and non-diffusive incompressible micropolar system. The convergence rates for the incompressible limit and the vanishing angular viscosity limit are also derived. In particular, the convergence rate for the incompressible limit is shown to be of order $ν^{-1/4}$ in general, and can be improved to $ν^{-1/2}$ under an additional decay condition on the initial density. Compared with previous works, the vanishing angular viscosity limit is established without the extra technical assumption $\lim_{\varepsilon\to0+}ζ/\varepsilon^α<\infty$ for some $1/2\leq α<\infty$, which extends the relevant results from the incompressible case to the compressible setting. As a by-product, the global regularity of large solutions for the 2D compressible micropolar equations with/without angular viscosity is obtained, provided the bulk viscosity is large enough.

发表机构

• Xiamen University（厦门大学）
• Liaoning University（辽宁大学）
• Jimei University（集美大学）

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