非齐次不可压缩MHD方程组强解的适定性
Well-posedness of Strong Solutions to the Inhomogeneous Incompressible MHD Equations
蒋滨瑶 1王华桥1
作者信息
- 1. 重庆大学数学与统计学院，重庆市401331
- 折叠
摘要
\justifying 针对不可压缩粘性流体方程在真空状态下的全局强解与稳定性问题, 本文利用初始密度局部非真空特征, 得到了密度的局部正下界, 再结合二维 Fourier 变换推导出速度场对数增长的逐点估计. 通过 Calderón-Zygmund 理论及 Riesz 位势获得高阶估计, 并利用分数阶 Sobolev 插值与逼近完成了强解存在性的证明. 在稳定性分析中, 为克服对流项引起的导数损失, 引入拉格朗日坐标变换, 并利用 Helmholtz 分解构造辅助速度场以吸收流映射散度差的误差. 借助微积分基本定理将位移差转化为沿轨迹的参数积分, 结合 Hölder 与 Grönwall 不等式, 得到了 Euler 坐标系下解的 $L^2$ 稳定性.
Abstract
\justifying Addressing the problem of global strong solutions and stability for incompressible viscous fluid equations in the presence of vacuum states, this paper utilizes the local non-vacuum property of the initial density to establish a local positive lower bound for the density. This is then combined with the two-dimensional Fourier transform to derive pointwise estimates for the logarithmic growth of the velocity field. Higher-order a priori estimates are obtained via Calderón-Zygmund theory and Riesz potentials, and the proof of the existence of strong solutions is completed using fractional Sobolev interpolation and approximation. In the stability analysis, to overcome the derivative loss caused by the convective term, Lagrangian coordinate transformations are introduced, and a Helmholtz decomposition is utilized to construct an auxiliary velocity field to absorb the error from the divergence difference of the flow maps. With the aid of the fundamental theorem of calculus, the displacement difference is converted into a parametric integral along the trajectories. Finally, combining Hölder's and Grönwall's inequalities, the $L^2$ stability of the solutions in the Eulerian coordinate system is established.关键词
不可压缩MHD方程组/拉格朗日 坐标变换/整体强解/稳定性/能量方法Key words
Incompressible MHD equations/Lagrangian coordinate transformation/Global strong solution/Stability/Energy method引用本文复制引用
蒋滨瑶,王华桥.非齐次不可压缩MHD方程组强解的适定性[EB/OL].(2026-04-08)[2026-04-11].http://www.paper.edu.cn/releasepaper/content/202604-67.学科分类
数学/物理学
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