三维可压缩非牛顿MHD方程组局部强解的存在唯一性

Existence and Uniqueness of Local Strong Solutions to the 3D Compressible Non-Newtonian MHD Equations

摘要: 本文研究三维可压缩非牛顿磁流体动力学(MHD)方程组局部强解的适定性。在初始数据满足自然相容性条件且允许存在初始真空的情形下，证明了该方程组局部强解的存在性与唯一性。本研究的主要困难在于处理非牛顿应力张量的高度非线性，以及由真空导致的方程退化。基于速度场算子满足$W^{2,p}$ 椭圆正则性的假设，本文通过建立高阶先验估计成功克服了上述困难。该结果将现有可压缩牛顿型MHD方程组的适定性理论推广至更一般的非牛顿流体情形。研究结果表明，在速度场算子满足$W^{2,p}$ 椭圆正则性的前提下，该非牛顿MHD系统是局部适定的。这一结论明确了非牛顿特性对应力张量正则性的具体要求。

Abstract: This paper investigates the well-posedness of local strong solutions to the three-dimensional (3D) compressible non-Newtonian magnetohydrodynamic (MHD) system. Provided that the initial data satisfy natural compatibility conditions and the initial vacuum is allowed, we establish the existence and uniqueness of local strong solutions to the system. The main difficulties lie in handling the high nonlinearity of the non-Newtonian stress tensor and the degeneracy of the equations induced by the vacuum. Based on the assumption that the velocity field operator satisfies the $W^{2,p}$ elliptic regularity, we successfully overcome the aforementioned difficulties by establishing higher-order a priori estimates. This result extends the existing well-posedness theory for the compressible Newtonian MHD system to the more general non-Newtonian fluid setting. The research results demonstrate that the non-Newtonian MHD system is locally well-posed, provided that the velocity field operator satisfies the $W^{2,p}$ elliptic regularity. This conclusion clarifies the specific requirements of non-Newtonian characteristics for the regularity of the stress tensor.