正交曲线坐标系中三维不可压粘性MHD系统的大初值整体适定性

doi:10.12677/aam.2025.145245

期刊菜单

正交曲线坐标系中三维不可压粘性MHD系统的大初值整体适定性
Global Well-Posedness of the Three-Dimensional Incompressible Viscous MHD System in Orthogonal Curvilinear Coordinates with Large Initial Data

DOI: 10.12677/aam.2025.145245, PDF, HTML, XML, 科研立项经费支持
作者: 袁桂芳：广州大学数学与信息科学学院，广东广州
关键词: 整体适定性；三维不可压MHD系统；正交曲线坐标系；光滑解；Global Well-Posedness； Three-Dimensional Incompressible MHD System； Orthogonal Curvilinear Coordinates； Smooth Solution

摘要: 本文研究了正交曲线坐标系中三维不可压MHD系统在大光滑初值条件下的整体适定性。我们针对一类新的光滑大初值，建立了三维不可压粘性MHD系统Cauchy问题在正交曲线坐标系下的整体光滑解的存在性和唯一性。

Abstract: This paper investigates the global well-posedness of the three-dimensional incompressible MHD system in orthogonal curvilinear coordinates with large smooth initial data. We establish the global existence and uniqueness of the smooth solutions to the Cauchy problem for the three-dimensional incompressible viscous MHD system in orthogonal curvilinear coordinates for a new class of the smooth large initial data.

文章引用：袁桂芳. 正交曲线坐标系中三维不可压粘性MHD系统的大初值整体适定性[J]. 应用数学进展, 2025, 14(5): 162-174. https://doi.org/10.12677/aam.2025.145245

1. 引言

三维不可压缩粘性磁流体力学(MHD)方程是研究磁场和流体动力学相互作用的重要数学模型。在一般直角坐标系下，MHD系统的局部适定性和全局适定性已经得到了深入研究。然而，在一般正交曲线坐标系下，MHD方程的适定性问题仍然存在诸多数学和计算挑战。受这些研究的启发，本文主要考虑在正交曲线坐标系中的三维不可压粘性 $(ν > 0, η > 0)$ MHD方程Cauchy问题：

${\begin{array}{l} u_{t} + u \cdot \nabla u + \nabla P = ν Δ u + b \cdot \nabla b, x \in E, t > 0, \\ b_{t} + u \cdot \nabla b - b \cdot \nabla u = η Δ b, x \in E, t > 0, \\ div u = 0, div b = 0, x \in E, t > 0, \\ (u, b) (0, x) = (u_{0}, b_{0}) (x), with d i v u_{0} = 0, div b_{0} = 0, x \in E . \end{array}$

这里， $x = (x_{1}, x_{2}, x_{3})$ ， $E = ℝ^{3}, ℝ^{2} \times T_{3}, ℝ^{1} \times T_{2} \times T_{3}$ ，或者 $T_{1} \times T_{2} \times T_{3}$ 其中 $T_{i}$ 是 $ℝ^{1}$ 中周期为 $l_{i} > 0$ 的环面。未知量 $u = (u_{1} (t, x), u_{2} (t, x), u_{3} (t, x)) (t, x)$ 表示流体速度场，而 $b \in ℝ^{3}$ 表示磁场 $P = P (t, x)$ 是标量压力， $ν \geq 0$ 是粘性系数， $η > 0$ 是磁扩散系数。此外， $u_{0}$ 和 $b_{0}$ 是给定的初始函数，满足 $div u_{0} = 0$ 和 $div b_{0} = 0$ 。

2. 研究模型

2.1. 正交曲线坐标系中的算子

在实际问题中，我们在直角坐标系中研究磁流体动力学方程有时可能会非常复杂，而采用其他坐标系却更具优势。直角坐标系、柱坐标系和球坐标系都是正交曲线坐标系的特殊形式，因此，可以通过推广至更一般的正交曲线坐标系来更灵活地研究MHD系统。由于矢量形式基本方程组中的梯度算子、散度算子和旋度算子的定义与坐标系的选取无关，因此，只需要通过坐标变换推导出正交曲线坐标系下对应算子的具体表达式，就可以方便地得到该坐标系下的MHD方程的分量形式。为了表述我们的问题，我们引入正交曲线坐标系[1]和一些算子的表示方法[2]。

设 $ξ_{1}, ξ_{2}, ξ_{3}$ 为一组正交曲线坐标，单位向量 $e_{1}, e_{2}, e_{3}$ 分别与坐标线平行，并沿着 $ξ_{1}, ξ_{2}, ξ_{3}$ 递增的方向，而非负函数 $H_{1}, H_{2}, H_{3}$ 分别是与单位向量 $e_{1}, e_{2}, e_{3}$ 相关的拉梅系数。一般情况下，单位向量 $e_{1}, e_{2}, e_{3}$ 以及函数 $H_{1}, H_{2}, H_{3}$ 都是坐标 $ξ = (ξ_{1}, ξ_{2}, ξ_{3})$ 的函数。设 $x = x (ξ) = (x_{1}, x_{2}, x_{3}) (ξ)$ 为直角坐标系 $x$ 与正交曲线坐标系 $ξ$ 之间的坐标变换函数，假设该变换是一个从 $ξ \in \tilde{Ω}$ 到 $x \in Ω$ 的双射，其逆映射记作 $ξ = ξ (x) = (ξ_{1}, ξ_{2}, ξ_{3}) (x)$ 。众所周知，该变换映射、单位向量和拉梅系数之间满足以下关系：

$\frac{\partial x (ξ)}{\partial ξ_{i}} = H_{i} (ξ) e_{i}, i = 1, 2, 3.$

由于三组坐标线构成一个正交系统，单位向量 $e_{1}, e_{2}, e_{3}$ 的导数具有以下表达式：

$\begin{array}{l} \frac{\partial e_{1}}{\partial ξ_{1}} = - \frac{e_{2}}{H_{2}} \frac{\partial H_{1}}{\partial ξ_{2}} - \frac{e_{3}}{H_{3}} \frac{\partial H_{1}}{\partial ξ_{3}}, \frac{\partial e_{1}}{\partial ξ_{2}} = \frac{e_{2}}{H_{1}} \frac{\partial H_{2}}{\partial ξ_{1}}, \frac{\partial e_{1}}{\partial ξ_{3}} = \frac{e_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}}, \\ \frac{\partial e_{2}}{\partial ξ_{1}} = \frac{e_{1}}{H_{2}} \frac{\partial H_{1}}{\partial ξ_{2}}, \frac{\partial e_{2}}{\partial ξ_{2}} = - \frac{e_{3}}{H_{3}} \frac{\partial H_{2}}{\partial ξ_{3}} - \frac{e_{1}}{H_{1}} \frac{\partial H_{2}}{\partial ξ_{1}}, \frac{\partial e_{2}}{\partial ξ_{3}} = \frac{e_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}, \\ \frac{\partial e_{3}}{\partial ξ_{1}} = \frac{e_{1}}{H_{3}} \frac{\partial H_{1}}{\partial ξ_{3}}, \frac{\partial e_{3}}{\partial ξ_{2}} = \frac{e_{2}}{H_{3}} \frac{\partial H_{2}}{\partial ξ_{3}}, \frac{\partial e_{3}}{\partial ξ_{3}} = - \frac{e_{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} - \frac{e_{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}} . \end{array}$

对于一个标量函数 $V$ ，其梯度算子 $\nabla$ 和拉普拉斯算子 $Δ$ 具有如下表达式：

$\begin{array}{l} \nabla V = e_{1} \frac{1}{H_{1}} \frac{\partial V}{\partial ξ_{1}} + e_{2} \frac{1}{H_{2}} \frac{\partial V}{\partial ξ_{2}} + e_{3} \frac{1}{H_{3}} \frac{\partial V}{\partial ξ_{3}}, \\ Δ V = \frac{1}{H_{1} H_{2} H_{3}} [\frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial V}{\partial ξ_{1}}) + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial V}{\partial ξ_{2}}) + \frac{\partial}{\partial ξ_{3}} (\frac{H_{1} H_{2}}{H_{3}} \frac{\partial V}{\partial ξ_{3}})] . \end{array}$

对于矢量 $A = A_{1} (ξ) e_{1} + A_{2} (ξ) e_{2} + A_{3} (ξ) e_{3}$ ，其散度和旋度分别表示为：

$\begin{array}{l} div A = \nabla \cdot A = \frac{1}{H_{1} H_{2} H_{3}} [\frac{\partial}{\partial ξ_{1}} (H_{2} H_{3} A_{1}) + \frac{\partial}{\partial ξ_{2}} (H_{1} H_{3} A_{2}) + \frac{\partial}{\partial ξ_{3}} (H_{1} H_{2} A_{3})], \\ rot A = \nabla \times A = \frac{1}{H_{2} H_{3}} [\frac{\partial}{\partial ξ_{2}} (H_{3} A_{3}) - \frac{\partial}{\partial ξ_{3}} (H_{2} A_{2})] e_{1} + \frac{1}{H_{1} H_{3}} [\frac{\partial}{\partial ξ_{3}} (H_{1} A_{1}) - \frac{\partial}{\partial ξ_{1}} (H_{3} A_{3})] e_{2} \\ + \frac{1}{H_{1} H_{2}} [\frac{\partial}{\partial ξ_{1}} (H_{2} A_{2}) - \frac{\partial}{\partial ξ_{2}} (H_{1} A_{1})] e_{3} . \end{array}$

2.2. 正交曲线坐标系中三维不可压粘性MHD方程分量形式的一般表达式

本文主要考虑问题的解和初始数据为以下形式：

$\begin{array}{l} u (t, x) = u^{1} (t, ξ_{1}, ξ_{2}) e_{1} + u^{2} (t, ξ_{1}, ξ_{2}) e_{2} + u^{3} (t, ξ_{1}, ξ_{2}) e_{3}, \\ b (t, x) = b^{1} (t, ξ_{1}, ξ_{2}) e_{1} + b^{2} (t, ξ_{1}, ξ_{2}) e_{2} + b^{3} (t, ξ_{1}, ξ_{2}) e_{3}, P (t, x) = P (t, ξ_{1}, ξ_{2}) . \\ u_{0} (x) = u_{0}^{1} (ξ_{1}, ξ_{2}) e_{1} + u_{0}^{2} (ξ_{1}, ξ_{2}) e_{2} + u_{0}^{3} (ξ_{1}, ξ_{2}) e_{3}, \\ b_{0} (x) = b_{0}^{1} (ξ_{1}, ξ_{2}) e_{1} + b_{0}^{2} (ξ_{1}, ξ_{2}) e_{2} + b_{0}^{3} (ξ_{1}, ξ_{2}) e_{3} . \end{array}$

我们假设坐标变换 $x = x (ξ)$ 满足以下性质：

1) $x = x (ξ)$ 是一个双射，将 $ξ \in \tilde{Ω}$ 变换到 $x \in Ω \subset E$ ，其中 $\tilde{Ω} = D \times [A, B] \subset ℝ^{3}$ 或 $\tilde{Ω} = D \times {\hat{T}}_{3} \subset ℝ^{3}$ 。

在这种情况下，区域 $Ω$ 可以表示为：

$Ω = {(x_{1}, x_{2}, x_{3}) = (x_{1}, x_{2}, x_{3}) (ξ) \in E : (ξ_{1}, ξ_{2}) \in D \subset \hat{E}, - \infty < A \leq ξ_{3} \leq B < \infty 或 ξ_{3} \in {\hat{T}}_{3}} .$

其中， ${\hat{T}}_{1} \times {\hat{T}}_{2}, {\hat{T}}_{i}, i = 1, 2, 3$ ， ${\hat{T}}_{i}$ 是 $ℝ^{1}$ 中周期为 $L_{i} > 0$ 的环面，区域 $D$ 是 $\hat{E}$ 上的有界或无界区域，若 $D$ 是有界且非周期的，则其边界 $\partial D$ 是光滑的。常数 $A$ 和 $B$ 满足 $- \infty < A \leq ξ_{3} \leq B < \infty$ 。

2) 单位向量 $e_{1}, e_{2}, e_{3}$ 依赖于坐标 $ξ = (ξ_{1}, ξ_{2}, ξ_{3})$ ，而拉梅系数 $H_{1}, H_{2}, H_{3}$ 仅依赖于 $ξ_{1}, ξ_{2}$ ，即：

$\begin{array}{l} e_{1} = e_{1} (ξ_{1}, ξ_{2}, ξ_{3}), e_{2} = e_{2} (ξ_{1}, ξ_{2}, ξ_{3}), e_{3} = e_{3} (ξ_{1}, ξ_{2}, ξ_{3}), \\ H_{1} = H_{1} (ξ_{1}, ξ_{2}), H_{2} = H_{2} (ξ_{1}, ξ_{2}), H_{3} = H_{3} (ξ_{1}, ξ_{2}) . \end{array}$

并且，集合 ${(ξ_{1}, ξ_{2}) \in D : (H_{1} H_{2} H_{3}) (ξ_{1}, ξ_{2}) = 0}$ 在 $ℝ^{2}$ 的Lebesgue测度意义下的测度为零。值得注意的是，这里的变换 $x (ξ)$ 可以看作是一个从 $\tilde{Ω}$ 映射到 $Ω$ 的一一映射，其定义域可替换为 $\tilde{\tilde{Ω}} = \tilde{Ω} \ S$ ，其中 $S$ 在 $ℝ^{3}$ 上的测度为零。

基于以上的假设，我们重新计算可得：

$\tilde{\nabla} = e_{1} \frac{1}{H_{1}} \frac{\partial}{\partial ξ_{1}} e_{2} \frac{1}{H_{2}} \frac{\partial}{\partial ξ_{2}}, \tilde{Δ} = \frac{1}{H_{1} H_{2} H_{3}} [\frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial}{\partial ξ_{1}}) + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial}{\partial ξ_{2}})] .$

根据以上对解和拉梅系数的假设，计算可得到在正交曲线坐标系中三维不可压粘性MHD方程分量形式的一般表达式：

其中：

$\begin{array}{l} L_{1} = \tilde{Δ} - \frac{1}{{(H_{1} H_{2})}^{2}} [{(\frac{\partial H_{1}}{\partial ξ_{2}})}^{2} + {(\frac{\partial H_{2}}{\partial ξ_{1}})}^{2}], \\ L_{2} = \frac{1}{H_{1} H_{2} H_{3}} [2 \frac{H_{3}}{H_{1}} \frac{\partial H_{1}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} + \frac{\partial}{\partial ξ_{1}} (\frac{H_{3}}{H_{1}} \frac{\partial H_{1}}{\partial ξ_{2}}) - 2 \frac{H_{3}}{H_{2}} \frac{\partial H_{2}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}} - \frac{\partial}{\partial ξ_{2}} (\frac{H_{3}}{H_{2}} \frac{\partial H_{2}}{\partial ξ_{1}})], \\ L_{3} = \tilde{Δ} - \frac{1}{{(H_{3})}^{2}} [{(\frac{1}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + {(\frac{1}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})}^{2}] . \end{array}$

所以，根据给出的初始条件或边界值条件，我们可以完全确定三维不可压粘性MHD方程在正交曲线坐标系中的演化方程。

这里，我们取初始条件为 $u_{0}^{3} = b_{0}^{1} = b_{0}^{2} = 0$ ，且 $(ξ_{1}, ξ_{2}) \in D \subset ℝ^{2}, t > 0$ 。并且，若域 $D$ 是有界的或者具有部分有界的边界，边界条件 ${u |}_{\partial Ω} = {b |}_{\partial Ω} = 0$ ， $t \geq 0$ 等价于 ${(u^{1}, u^{2}, u^{3}, b^{1}, b^{2}, b^{3}) |}_{\partial Ω} = 0$ ， $t \geq 0$ 。由于MHD系统的局部经典解唯一性，若初始条件为 $u_{0}^{3} = b_{0}^{1} = b_{0}^{2} = 0$ ，则在所有后续时间 $t$ 中都有 $u^{3} (t, ξ_{1}, ξ_{2}) = b^{1} (t, ξ_{1}, ξ_{2}) = b^{2} (t, ξ_{1}, ξ_{2}) = 0$ 。在这种情况下，得到我们要研究的演化方程：

$\begin{array}{l} \partial_{t} u^{1} + \frac{u^{1}}{H_{1}} \frac{\partial u^{1}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial u^{1}}{\partial ξ_{2}} + \frac{1}{H_{1}} \frac{\partial P}{\partial ξ_{1}} = ν [L_{1} u^{1} - \frac{1}{{(H_{1} H_{3})}^{2}} {(\frac{\partial H_{3}}{\partial ξ_{1}})}^{2} u^{1} + L_{2} u^{2} - \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial H_{3}}{\partial ξ_{2}} u^{2}] \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} - \frac{u^{1} u^{2}}{H_{1} H_{2}} \frac{\partial H_{1}}{\partial ξ_{2}} + \frac{{(u^{2})}^{2}}{H_{1} H_{2}} \frac{\partial H_{2}}{\partial ξ_{1}} - \frac{{(b^{3})}^{2}}{H_{1} H_{3}} \frac{\partial H_{3}}{\partial ξ_{1}}, \\ \partial_{t} u^{2} + \frac{u^{1}}{H_{1}} \frac{\partial u^{2}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial u^{2}}{\partial ξ_{2}} + \frac{1}{H_{2}} \frac{\partial P}{\partial ξ_{2}} = ν [L_{1} u^{2} - \frac{1}{{(H_{2} H_{3})}^{2}} {(\frac{\partial H_{3}}{\partial ξ_{2}})}^{2} u^{2} - L_{2} u^{1} - \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial H_{3}}{\partial ξ_{2}} u^{1}] \\ \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} - \frac{u^{1} u^{2}}{H_{1} H_{2}} \frac{\partial H_{2}}{\partial ξ_{1}} + \frac{{(u^{1})}^{2}}{H_{1} H_{2}} \frac{\partial H_{1}}{\partial ξ_{2}} - \frac{{(b^{3})}^{2}}{H_{2} H_{3}} \frac{\partial H_{3}}{\partial ξ_{2}}, \\ \partial_{t} b^{3} + \frac{u^{1}}{H_{1}} \frac{\partial b^{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial b^{3}}{\partial ξ_{2}} = η L_{3} b^{3} + \frac{b^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}), \\ \frac{\partial}{\partial ξ_{1}} (H_{2} H_{3} u^{1}) + \frac{\partial}{\partial ξ_{2}} (H_{1} H_{3} u^{2}) = 0. \end{array}$

由速度场 $u = u^{1} (t, ξ_{1}, ξ_{2}) e_{1} + u^{2} (t, ξ_{1}, ξ_{2}) e_{2}$ 和磁场 $b = b^{3} (t, ξ_{1}, ξ_{2}) e_{3}$ ，我们可以得到涡量 $ω = \nabla \times u$ 和电流密度 $j = \nabla \times b$ 的表达式，它们在正交曲线坐标系中的表示如下：

$ω (t, x) = ω^{3} (t, ξ_{1}, ξ_{2}) e_{3}, j (t, x) = j^{1} (t, ξ_{1}, ξ_{2}) e_{1} + j^{2} (t, ξ_{1}, ξ_{2}) e_{2},$

初始涡量为 $ω_{0} = ω (t = 0, x) = ω_{0}^{1} (ξ_{1}, ξ_{2}) e_{1} + ω_{0}^{2} (ξ_{1}, ξ_{2}) e_{2} + ω_{0}^{3} (ξ_{1}, ξ_{2}) e_{3}$ ，其中

$ω^{3} = \frac{1}{H_{1} H_{2}} (\frac{\partial}{\partial ξ_{1}} (H_{2} u^{2}) - \frac{\partial}{\partial ξ_{2}} (H_{1} u^{1})), j^{1} = \frac{1}{H_{2} H_{3}} \frac{\partial}{\partial ξ_{2}} (H_{3} b^{3}), j^{2} = - \frac{1}{H_{1} H_{3}} \frac{\partial}{\partial ξ_{1}} (H_{3} b^{3}) .$

显然有： $div ω = \frac{\partial}{\partial ξ_{1}} (H_{2} H_{3} ω^{1}) + \frac{\partial}{\partial ξ_{2}} (H_{1} H_{3} ω^{2}) \equiv 0$ 。此外，我们可以推导出 $ω^{3}$ 的方程：

$\begin{array}{l} \frac{\partial ω^{3}}{\partial t} + \frac{u^{1}}{H_{1}} \frac{\partial ω^{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial ω^{3}}{\partial ξ_{2}} = ν L_{3} ω^{3} + \frac{ω^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) + \frac{b^{1}}{H_{1}} \frac{\partial j}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial j}{\partial ξ_{2}} \\ - \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} [\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial {(H_{3} b^{3})}^{2}}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial {(H_{3} b^{3})}^{2}}{\partial ξ_{2}}] - \frac{j}{H_{3}} (\frac{b^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) . \end{array}$

3. 三维不可压粘性MHD方程Cauchy问题的整体光滑解

本章主要讨论正交曲线坐标系中三维不可压粘性MHD方程组Cauchy问题的整体光滑解问题。结构如下：第1节，为了完成本章定理的估计，我们介绍了几个基本引理；第2节，我们考虑具有特殊几何结构的MHD方程组在正交曲线坐标系中的结论并给出证明。

3.1. 一些预备引理

引理3.1 [3] 设 $u \in W^{1, p} (E)$ 为无散度的速度场，且涡量为 $ω$ ，则不等式

${‖ \nabla u ‖}_{L^{p}} \leq C (p) {‖ ω ‖}_{L^{p}} .$

对于任意 $p \in (1, \infty)$ 都成立，其中常数 $C (p)$ 仅依赖于 $p$ 。

引理3.2 [4] [5]假设初始数据 $u_{0} \in H^{2} (ℝ^{3})$ 满足 $div u_{0} = 0$ ，那么三维不可压粘性MHD方程的任意弱解 $(u, b, P)$ 满足

$u \in L^{p} (0, T; L^{q} (ℝ^{3})), \frac{2}{p} + \frac{3}{q} \leq 1, 3 < q \leq \infty,$

$\nabla u \in L^{p} (0, T; L^{q} (ℝ^{3})), \frac{2}{p} + \frac{3}{q} \leq 2, \frac{3}{2} < q \leq \infty .$

则解在 $(0, T] \times ℝ^{3}$ 上也是光滑的。

3.2. 定理及其证明

定理1.1 取 $ν > 0, η > 0$ ，若三维不可压粘性MHD方程Cauchy问题的初值 $(u_{0}, b_{0}) \in H^{s} (E)$ ， $s \geq 2$ ，由2.2中给出，满足 $div u_{0} = div b_{0} = 0, u_{0}^{3} = b_{0}^{1} = b_{0}^{2} = 0$ 。假设 ${‖ \frac{b_{0}^{3}}{H_{3}} ‖}_{L^{p} (E)} < \infty, 2 \leq p \leq \infty$ ， ${‖ \frac{ω_{0}^{3}}{H_{3}} ‖}_{L^{6} (E)} < \infty$ ，并且在 $H_{1} H_{2} H_{3} = 0$ 的根集上满足适当的相容性条件。此外，令拉梅系数 $H_{1} H_{2} H_{3}$ 满足：

$| \frac{1}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} | + | \frac{1}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}} | \leq C < \infty,$

则三维不可压粘性MHD方程Cauchy问题存在唯一的全局光滑解，并且满足 $(u, b) \in L^{\infty} (0, + \infty; H^{s} (E))$ 。此外，若初始数据是光滑的，并且满足适当的相容性条件，则该解在时间上全局光滑。

进一步地，若 $H_{3}$ 为常数，对于任意的光滑初始数据 $u_{0}^{3} \equiv 0$ ，则问题存在唯一的全局光滑解 $(u, b) (t, x)$ ，其中 $u^{3} (t, ξ_{1}, ξ_{2}) \equiv 0$ ，并且该解在所有 $t \in [0, \infty)$ 上成立。

以下是证明以上定理的过程：

首先，从三维不可压粘性 $(ν > 0, η > 0)$ MHD方程组中，我们可以得到基本的能量不等式，对任何的 $t > 0$ ，

${‖ (u, b) (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} + 2 \int_{0}^{t} {‖ (\sqrt{ν} \nabla u, \sqrt{η} \nabla b) (s, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} d s \leq {‖ (u_{0}, b_{0}) (\cdot) ‖}_{L^{2} (ℝ^{3})}^{2} .$

关于 $ω^{3}$ 和 $b^{3}$ 的方程可以写成下面的形式：

$\begin{array}{l} \frac{\partial ω^{3}}{\partial t} + \frac{u^{1}}{H_{1}} \frac{\partial ω^{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial ω^{3}}{\partial ξ_{2}} = ν L_{3} ω^{3} + \frac{ω^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) - \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} [\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial {(H_{3} b^{3})}^{2}}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial {(H_{3} b^{3})}^{2}}{\partial ξ_{2}}] . \\ \frac{\partial b^{3}}{\partial t} + \frac{u^{1}}{H_{1}} \frac{\partial b^{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial b^{3}}{\partial ξ_{2}} = η L_{3} b^{3} - \frac{u^{3}}{H_{3}} (\frac{b^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) + \frac{b^{1}}{H_{1}} \frac{\partial u^{3}}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial u^{3}}{\partial ξ_{2}} + \frac{b^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) . \end{array}$

我们引入表达式： $g_{1} (t, ξ_{1}, ξ_{2}) = \frac{b^{3} (t, ξ_{1}, ξ_{2})}{H_{3}}$ ， $g_{2} (t, ξ_{1}, ξ_{2}) = \frac{ω^{3} (t, ξ_{1}, ξ_{2})}{H_{3}}$ 。

其次，为了推导 $ω$ 在 $L^{\infty} ((0, t]; L^{2} (ℝ^{3}))$ 范数下的估计，我们首先建立对 $g_{1}, g_{2}$ 的 $L^{\infty}$ 估计。利用MHD系统的光滑解的局部适定性，可以直接得到： $b_{ξ_{1} = 0}^{3} = b_{ξ_{2} = 0}^{3} = ω_{ξ_{1} = 0}^{3} = ω_{ξ_{2} = 0}^{3} = {g_{1} |}_{ξ_{1} = 0} = {g_{1} |}_{ξ_{2} = 0} = {g_{2} |}_{ξ_{1} = 0} = {g_{2} |}_{ξ_{2} = 0} = 0$ 。

直接计算，我们得到 $(g_{1}, g_{2}) (t, ξ_{1}, ξ_{2})$ 的不等式：

$\begin{array}{l} \partial_{t} g_{1} + (\frac{u^{1}}{H_{1}} \frac{\partial}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial}{\partial ξ_{2}}) g_{1} = η (L_{3} g_{1} + L_{4} g_{1}), \\ \partial_{t} g_{2} + (\frac{u^{1}}{H_{1}} \frac{\partial}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial}{\partial ξ_{2}}) g_{2} = ν (L_{3} g_{2} + L_{4} g_{2}) - \frac{1}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}}) {(g_{1})}^{2} . \end{array}$

其中， $L_{4} = \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} [\frac{2 H_{2} H_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{1}} + \frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}}) + \frac{2 H_{1} H_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{2}} + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})]$ 。

将第一个方程两边乘以 ${| g_{1} |}^{p - 1} g_{1}$ (其中 $p \geq 1$ )，并且对所得方程在 $ℝ^{3}$ 上积分，我们得到：

$\begin{array}{l} \frac{1}{p + 1} \frac{d}{d t} \int_{ℝ^{3}} {| g_{1} |}^{p + 1} d x + \frac{1}{p + 1} \int_{ℝ^{3}} (\frac{u^{1}}{H_{1}} \frac{\partial}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial}{\partial ξ_{2}}) {| g_{1} |}^{p + 1} d x \\ = η \int_{ℝ^{3}} \tilde{Δ} g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d x + η \int_{ℝ^{3}} L_{4} g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d x \\ - η \int_{ℝ^{3}} \frac{g_{1}}{{(H_{3})}^{2}} [{(\frac{1}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + {(\frac{1}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})}^{2}] \cdot {| g_{1} |}^{p - 1} g_{1} d x . \end{array}$

利用不可压条件 $\frac{\partial}{\partial ξ_{1}} (H_{2} H_{3} u^{1}) + \frac{\partial}{\partial ξ_{2}} (H_{1} H_{3} u^{2}) = 0$ 以及 $d x = H_{1} H_{2} H_{3} d ξ_{1} d ξ_{2} d ξ_{3}$ ，我们计算等式左边第二项得到：

通过计算得到 ${| \tilde{\nabla} g_{1} (t, ξ_{1}, ξ_{2}) |}^{2} = {| \frac{1}{H_{1}} \frac{\partial g_{1}}{\partial ξ_{1}} |}^{2} + {| \frac{1}{H_{2}} \frac{\partial g_{1}}{\partial ξ_{2}} |}^{2}$ ，代入到等式右边第二项得到：

$\begin{matrix} I_{2} = η \int_{ℝ^{3}} \tilde{Δ} g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d x \\ = η \int_{Ω} \frac{1}{H_{1} H_{2} H_{3}} [\frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial}{\partial ξ_{1}}) + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{1}} \frac{\partial}{\partial ξ_{2}})] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d x \\ = (B - A) η \int_{D} [\frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial}{\partial ξ_{1}}) + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{1}} \frac{\partial}{\partial ξ_{2}})] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d ξ_{1} d ξ_{2} \\ = - p η (B - A) \int_{D} [\frac{H_{2} H_{3}}{H_{1}} {(\frac{\partial g_{1}}{\partial ξ_{1}})}^{2} + \frac{H_{1} H_{3}}{H_{1}} {(\frac{\partial g_{1}}{\partial ξ_{2}})}^{2}] {| g_{1} |}^{p - 1} d ξ_{1} d ξ_{2} \\ = - p η \int_{ℝ^{3}} [\frac{1}{{(H_{1})}^{2}} {(\frac{\partial g_{1}}{\partial ξ_{1}})}^{2} + \frac{1}{{(H_{2})}^{2}} {(\frac{\partial g_{1}}{\partial ξ_{2}})}^{2}] {| g_{1} |}^{p - 1} d x \\ = - p η \int_{ℝ^{3}} {| g_{1} |}^{p - 1} {| \tilde{\nabla} g_{1} |}^{2} d x . \end{matrix}$

$\begin{matrix} I_{3} = η \int_{ℝ^{3}} - \frac{1}{{(H_{3})}^{2}} [{(\frac{1}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + {(\frac{1}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})}^{2}] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} + \frac{1}{H_{1} H_{2} {(H_{3})}^{2}} [\frac{2 H_{2} H_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial g_{1}}{\partial ξ_{1}} \\ + \frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}}) g_{1} + \frac{2 H_{1} H_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial g_{1}}{\partial ξ_{2}} + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) g_{1}] \cdot {| g_{1} |}^{p - 1} g_{1} d x \\ = η (B - A) \int_{D} [- (\frac{H_{2}}{H_{3} H_{1}} {(\frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + \frac{H_{1}}{H_{3} H_{2}} {(\frac{\partial H_{3}}{\partial ξ_{2}})}^{2}) g_{1} + \frac{2 H_{2}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial g_{1}}{\partial ξ_{1}} + \frac{2 H_{1}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial g_{1}}{\partial ξ_{2}} \\ + \frac{1}{H_{3}} (\frac{\partial}{\partial ξ_{1}} (\frac{H_{2} H_{3}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}}) + \frac{\partial}{\partial ξ_{2}} (\frac{H_{1} H_{3}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}))] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d ξ_{1} d ξ_{2} \\ = η (B - A) \int_{D} - [\frac{H_{2}}{H_{3} H_{1}} {(\frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + \frac{H_{1}}{H_{3} H_{2}} {(\frac{\partial H_{3}}{\partial ξ_{2}})}^{2}] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} \\ + [\frac{1}{H_{3}} (\frac{H_{2}}{H_{1}} {(\frac{\partial H_{3}}{\partial ξ_{1}})}^{2} + \frac{H_{1}}{H_{2}} {(\frac{\partial H_{3}}{\partial ξ_{1}})}^{2})] g_{1} \cdot {| g_{1} |}^{p - 1} g_{1} d ξ_{1} d ξ_{2} \\ = 0. \end{matrix}$ (1)

将 $I_{1}, I_{2}, I_{3}$ 代入第一个方程中得到：

$\frac{1}{p + 1} \frac{d}{d t} \int_{ℝ^{3}} {| g_{1} |}^{p + 1} d x + p η \int_{ℝ^{3}} {| g_{1} |}^{p - 1} {| \tilde{\nabla} g_{1} |}^{2} d x = 0.$

则对任何 $0 \leq t \leq \infty$ ，得到 $g_{1} (t, ξ_{1}, ξ_{2})$ 估计

${‖ g_{1} (t, ξ_{1}, ξ_{2}) ‖}_{L^{p + 1} (ℝ^{3})} \leq {‖ g_{1} (0, ξ_{1}, ξ_{2}) ‖}_{L^{p + 1} (ℝ^{3})} = {‖ \frac{b_{0}^{3} (ξ_{1}, ξ_{2})}{H_{3}} ‖}_{L^{p + 1} (ℝ^{3})} \leq C < \infty, 1 \leq p \leq \infty .$

同样，对于第二个等式进行同样的计算，对于任何 $1 \leq p < \infty$ ，

$\begin{array}{l} \frac{1}{p + 1} \frac{d}{d t} \int_{ℝ^{3}} {| g_{2} |}^{p + 1} d x + p ν \int_{ℝ^{3}} {| g_{2} |}^{p - 1} {| \tilde{\nabla} g_{2} |}^{2} d x \\ = - \int_{ℝ^{3}} \frac{1}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}}) {(g_{2})}^{2} \cdot {| g_{2} |}^{p - 1} g_{2} d x \\ = I_{4} . \end{array}$

这里，通过分部积分和Young不等式[6]，结合等式 ${| \tilde{\nabla} g_{2} (t, ξ_{1}, ξ_{2}) |}^{2} = {| \frac{1}{H_{1}} \frac{\partial g_{2}}{\partial ξ_{1}} |}^{2} + {| \frac{1}{H_{2}} \frac{\partial g_{2}}{\partial ξ_{2}} |}^{2}$ ， $I_{4}$ 可以放缩为：

$\begin{matrix} I_{4} = - \int_{ℝ^{3}} \frac{1}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}}) {(g_{2})}^{2} \cdot {| g_{2} |}^{p - 1} g_{2} d x \\ = - (B - A) \int_{D} \frac{1}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}}) {(g_{2})}^{2} \cdot {| g_{2} |}^{p - 1} g_{2} H_{1} H_{2} H_{3} d ξ_{1} d ξ_{2} \\ = (B - A) \int_{D} \frac{p g_{1}^{2}}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} \frac{\partial}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} \frac{\partial}{\partial ξ_{2}}) g_{2} \cdot {| g_{2} |}^{p - 1} g_{2} H_{1} H_{2} H_{3} d ξ_{1} d ξ_{2} \\ = \int_{ℝ^{3}} p g_{1}^{2} [\frac{1}{H_{1} H_{2}} (\frac{\partial H_{3}}{\partial ξ_{2}} {| g_{2} |}^{p - 1} \frac{\partial g_{2}}{\partial ξ_{1}} - \frac{\partial H_{3}}{\partial ξ_{1}} {| g_{2} |}^{p - 1} \frac{\partial g_{2}}{\partial ξ_{2}})] d x \\ \leq p \frac{ν}{2} \int_{ℝ^{3}} {| g_{2} |}^{p - 1} ({| \tilde{\nabla} g_{2} |}^{2}) d x + p \frac{1}{2 ν} \int_{ℝ^{3}} {| g_{2} |}^{p - 1} {| g_{1} |}^{4} d x \\ \leq p \frac{ν}{2} \int_{ℝ^{3}} {| g_{2} |}^{p - 1} ({| \tilde{\nabla} g_{2} |}^{2}) d x + p \frac{1}{2 ν} \int_{ℝ^{3}} (\frac{p - 1}{p + 1} {| g_{2} |}^{p + 1} + \frac{2}{p + 1} {| g_{1} |}^{2 (p + 1)}) d x . \end{matrix}$

结合上面两个式子可以得到：

$\frac{1}{p + 1} \frac{d}{d t} \int_{ℝ^{3}} {| g_{2} |}^{p + 1} d x + \frac{p ν}{2} \int_{ℝ^{3}} {| g_{2} |}^{p - 1} {| \tilde{\nabla} g_{2} |}^{2} d x \leq \frac{(p - 1) p}{2 ν (p + 1)} \int_{ℝ^{3}} {| g_{2} |}^{p + 1} d x + \frac{p}{ν (p + 1)} \int_{ℝ^{3}} {| g_{1} |}^{2 (p + 1)} d x .$

将 $g_{1} (t, ξ_{1}, ξ_{2})$ 的估计代入上式，得到对于 $0 \leq t \leq \infty$ ，

$\begin{array}{l} \int_{ℝ^{3}} {| g_{2} (t, ξ_{1}, ξ_{2}) |}^{p + 1} d x \\ \leq e^{\frac{p (p - 1) t}{2 ν}} (\int_{ℝ^{3}} {| g_{2} (t = 0, ξ_{1}, ξ_{2}) |}^{p + 1} d x + \frac{p}{ν} \int_{0}^{t} e^{- \frac{p (p - 1) t}{2 ν}} \int_{ℝ^{3}} {| g_{1} (s, ξ_{1}, ξ_{2}) |}^{2 (p + 1)} d x d s) \\ \leq C (p) e^{B (p) t} . \end{array}$

将 $p = 5$ 代入，得到 $\int_{ℝ^{3}} {| g_{2} (t, ξ_{1}, ξ_{2}) |}^{6} d x \leq C e^{B t}$ 和

$\int_{ℝ^{3}} {| g_{2} (t, ξ_{1}, ξ_{2}) |}^{6} d x + 15 ν \int_{0}^{t} \int_{ℝ^{3}} {| g_{2} (s, ξ_{1}, ξ_{2}) |}^{4} {| \tilde{\nabla} g_{2} |}^{2} d x d s \leq C e^{B t} .$

接下来，我们利用解 $(u, b)$ 的特殊几何结构，对速度场 $u$ 的涡量 $ω (t, x) = ω (t, ξ_{1}, ξ_{2}, ξ_{3})$ 进行 $L^{\infty} (0, t; L^{2} (ℝ^{3}))$ 估计。由三维不可压粘性MHD系统的局部光滑解的存在唯一性以及对初始数据的假设 $u_{0}^{3} = b_{0}^{1} = b_{0}^{2} = 0$ ，可以容易得到：对于所有 $t \geq 0$ ， $u^{3} (t, ξ_{1}, ξ_{2}) = b^{1} (t, ξ_{1}, ξ_{2}) = b^{2} (t, ξ_{1}, ξ_{2}) = 0$ 。因此，我们有：

$u (t, x) = u^{1} (t, ξ_{1}, ξ_{2}) e_{1} + u^{2} (t, ξ_{1}, ξ_{2}) e_{2}, b (t, x) = b^{3} (t, ξ_{1}, ξ_{2}) e_{3}, ω (t, x) = ω^{3} (t, ξ_{1}, ξ_{2}) e_{3} .$

众所周知，在三维不可压粘性MHD系统中，涡量方程 $ω = \nabla \times u$ 的表达式如下：

$\partial_{t} ω + (u \cdot \nabla) ω - ν Δ ω = ω \cdot \nabla u + \nabla \times (b \cdot \nabla b) .$

将方程两边同时乘以 $ω$ ，并对所得方程在 $ℝ^{3}$ 上积分，对于任意 $0 \leq t \leq \infty$ ，可得：

$\frac{1}{2} \frac{d}{d t} {‖ ω (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} + ν {‖ \nabla ω ‖}_{L^{2} (ℝ^{3})}^{2} = \int_{ℝ^{3}} (ω \cdot \nabla) u \cdot ω d x + \int_{ℝ^{3}} \nabla \times (b \cdot \nabla b) \cdot ω d x .$

我们估计等式右侧的项，已知 $j = \nabla \times b = \frac{1}{H_{2} H_{3}} \frac{\partial}{\partial ξ_{2}} (H_{3} b^{3}) e_{1} - \frac{1}{H_{1} H_{3}} \frac{\partial}{\partial ξ_{1}} (H_{3} b^{3}) e_{2}$ 和 $div u = div b = 0$ 。方程右侧的项 $ω \cdot \nabla u$ 和 $\nabla \times (b \cdot \nabla b)$ 在正交曲线坐标系中的表达式为：

和

$\begin{matrix} ω \cdot \nabla u = [ω^{3} e_{3} \cdot (\frac{e_{1}}{H_{1}} \frac{\partial}{\partial ξ_{1}} + \frac{e_{2}}{H_{2}} \frac{\partial}{\partial ξ_{2}} + \frac{e_{3}}{H_{3}} \frac{\partial}{\partial ξ_{3}})] (u^{1} e_{1} + u^{2} e_{2}) \\ = \frac{ω^{3}}{H_{3}} \frac{\partial}{\partial ξ_{3}} (u^{1} e_{1} + u^{2} e_{2}) \\ = \frac{ω^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) e_{3} . \end{matrix}$

利用 $ω (t, x) = ω^{3} (t, ξ_{1}, ξ_{2}) e_{3}$ ，我们有：

${| \nabla ω (t, x) |}^{2} = {| \frac{1}{H_{1}} \frac{\partial ω^{3}}{\partial ξ_{1}} |}^{2} + {| \frac{1}{H_{2}} \frac{\partial ω^{3}}{\partial ξ_{2}} |}^{2}$

然后，方程右侧的各项可以按照以下方式进行估计。

利用所得结果，结合引理2.2，借助Hölder不等式[7]、Gagliardo-Nirenberg不等式[8]和Young不等式[6]，可得：

$\begin{matrix} I_{5} = \int_{ℝ^{2}} (ω \cdot \tilde{\nabla}) u \cdot ω d x \\ = \int_{ℝ^{3}} [\frac{ω^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})] \cdot ω^{3} d x \\ = \int_{ℝ^{3}} ω^{3} (\frac{1}{H_{1} H_{3}} u^{1} ω^{3} \frac{\partial H_{3}}{\partial ξ_{1}}) d x + \int_{R^{3}} ω^{3} (\frac{1}{H_{3} H_{2}} u^{2} ω^{3} \frac{\partial H_{3}}{\partial ξ_{2}}) d x \end{matrix}$

$\begin{matrix} = \int_{ℝ^{3}} u^{1} g_{2} ω^{3} (\frac{1}{H_{1}} \frac{\partial H_{1}}{\partial ξ_{1}}) d x + \int_{ℝ^{3}} u^{2} g_{2} ω^{3} (\frac{1}{H_{1}} \frac{\partial H_{2}}{\partial ξ_{2}}) d x \\ \leq C (\int_{ℝ^{3}} | u^{1} g_{2} ω^{3} | d x + \int_{ℝ^{3}} | u^{2} g_{2} ω^{3} | d x) \\ \leq C {‖ ω^{3} ‖}_{L^{2} (ℝ^{3})} ({(\int_{ℝ^{3}} {| u^{1} g_{2} |}^{2} d x)}^{\frac{1}{2}} + {(\int_{ℝ^{3}} {| u^{2} g_{2} |}^{2} d x)}^{\frac{1}{2}}) \\ \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})} ({‖ u^{1} ‖}_{L^{3} (ℝ^{3})} + {‖ u^{2} ‖}_{L^{3} (ℝ^{3})}) {‖ g_{2} ‖}_{L^{6} (ℝ^{3})} \\ \leq 2 C {‖ ω ‖}_{L^{2} (ℝ^{3})} {‖ u ‖}_{L^{3} (ℝ^{3})} {‖ g_{2} ‖}_{L^{6} (ℝ^{3})} \\ \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})} {‖ u ‖}_{L^{2} (ℝ^{3})}^{\frac{1}{2}} {‖ \nabla u ‖}_{L^{2} (ℝ^{3})}^{\frac{1}{2}} {‖ g_{2} ‖}_{L^{6} (ℝ^{3})} \\ \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})}^{\frac{3}{2}} {‖ u ‖}_{L^{2} (ℝ^{3})}^{\frac{1}{2}} {‖ g_{2} ‖}_{L^{6} (ℝ^{3})} \\ \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})}^{2} + {‖ u ‖}_{L^{2} (ℝ^{3})}^{2} {‖ g_{2} ‖}_{L^{6} (ℝ^{3})}^{4} \\ \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})}^{2} + C e^{B t} . \end{matrix}$

和

结合上面的结果，我们得到：

$\frac{1}{2} \frac{d}{d t} {‖ ω (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} + \frac{ν}{2} {‖ \nabla ω ‖}_{L^{2} (ℝ^{3})}^{2} \leq C {‖ ω ‖}_{L^{2} (ℝ^{3})}^{2} + C e^{B t} .$

由Grönwall不等式[9]可得，对于任意 $t > 0$ ，有

${‖ ω (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} + ν \int_{0}^{t} {‖ \nabla ω (s, \cdot) ‖}_{L^{2} (ℝ^{3})}^{2} d s \leq C e^{B t} .$

利用引理2.1，得到：

${‖ \nabla u (t, \cdot) ‖}_{L^{2} (ℝ^{3})} \leq C e^{B t}$ 或者 ${‖ u (t, \cdot) ‖}_{L^{q} (ℝ^{3})} \leq C e^{B t}, 2 \leq q \leq 6.$

根据Gagliardo-Nirenberg不等式[8]，有：

${‖ u (t, \cdot) ‖}_{L^{q} (ℝ^{3})} \leq C {‖ \nabla u (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{A} {‖ u (t, \cdot) ‖}_{L^{2} (ℝ^{3})}^{1 - A}, A = \frac{3 q - 6}{2 q}, 2 \leq q \leq 6.$

由此，我们得到三维不可压粘性MHD系统所需的正则性估计，因此，应用引理2.2，在 $u_{0}^{3} \equiv 0$ 的情况下可以得定理1.1中所陈述的结果。

当 $H_{3}$ 为常数时，显然 $H_{3}$ 与 $ξ_{1}, ξ_{2}$ 无关。因为 $H_{3}$ 是常数的假设，将2.2中给出的正交曲线坐标系中MHD方程组涉及 $u_{}^{3} \neq 0$ 的项变成零，即：

$ν L_{3} u^{3} = 0, \frac{u^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) = 0.$

这种情况下，2.2中给出的正交曲线坐标系中MHD方程组中涉及 $(u^{1}, u^{2}, b^{1}, b^{2}, b^{3})$ 的方程与 $u^{3}$ 无关。因此，与本文考虑 $u^{3} = b^{1} = b^{2} = 0$ 的情况一样，我们可以构造新的速度场，关于 $u_{0}^{3} \equiv 0$ 时的解的构造与正则性：

在定理1.1中，若假设 $H_{3}$ 为常数，则三维不可压MHD系统在正交曲线坐标下的第三分量 $u^{3}$ 满足如下方程：

$\partial_{t} u^{3} + \frac{u^{1}}{H_{1}} \frac{\partial u^{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial u^{3}}{\partial ξ_{2}} = ν L_{3} u^{3} - \frac{u^{3}}{H_{3}} (\frac{u^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{u^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}}) + \frac{b^{1}}{H_{1}} \frac{\partial b^{3}}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial b^{3}}{\partial ξ_{2}} + \frac{b^{3}}{H_{3}} (\frac{b^{1}}{H_{1}} \frac{\partial H_{3}}{\partial ξ_{1}} + \frac{b^{2}}{H_{2}} \frac{\partial H_{3}}{\partial ξ_{2}})$

在初始数据满足 $u_{0}^{3} \in H^{2} (Ω)$ 且 $u_{0}^{3} \equiv 0$ 的情况下，由局部解唯一性及上述方程形式可知 $u^{3}$ 不会退化为零。重复以上证明步骤，通过对 $u^{3}$ 的能量估计，结合Grönwall不等式[9]可证明 $u^{3} (t)$ 的正则性可随时间保持。因此，在 $H_{3}$ 为常数时，存在一类具有 $u^{3} \equiv 0$ 的解族，并可由相同方法延拓到全局强解。对于任何给定的光滑初值 $u_{0}^{3}$ ，可以在时间上求解 $u^{3}$ 。

注：关于解的长时间行为与指数衰减的补充。

Duvaut和Lions (1972) [10]最早建立了MHD全局小数据下的光滑解与能量耗散。Lei (2015) [11]和Wang等(2019) [12]在具有特殊对称性或特殊坐标(球坐标)下，建立了大数据情形的全局解存在性及其指数衰减。Deng和Zhang (2018) [13]和Ke等(2021) [14]分别讨论了在三维MHD系统中，平衡态附近的解在范数下的最优时间衰减速率，证明了解的长时间行为和耗散结构。基于已有的结果，我们对系统能量

估计，依然有 $\frac{d}{d t} ({‖ u ‖}_{L^{2}}^{2} + {‖ b ‖}_{L^{2}}^{2}) + 2 \min (ν, η) ({‖ \nabla u ‖}^{2} + {‖ \nabla b ‖}^{2}) \leq 0$ 。由于 $Ω$ 有界，Poincaré不等式[9]成立： ${‖ u ‖}_{L^{2}} \leq C_{P} {‖ \nabla u ‖}_{L^{2}}$ ， ${‖ b ‖}_{L^{2}} \leq C_{P} {‖ \nabla b ‖}_{L^{2}}$ ，从而得到 $\frac{d}{d t} E (t) + 2 λ E (t) \leq 0$ ，其中 $λ = \frac{\min (ν, η)}{C_{P}^{2}}$ ，故 $E (t) \leq E (0) e^{- 2 λ t}$ ，即解具有指数衰减性。进而，高阶能量也满足：

${‖ u (t) ‖}_{H^{1}} + {‖ b (t) ‖}_{H^{1}} \leq C e^{- κ t}, 0 < κ \leq λ .$

至此，定理1.1的证明完成。

基金项目

广东省基础与应用基础研究基金(No. 2022A1515010566)。

参考文献

[1]	Bachelor, G.K. (1967) An Introduction to Fluid Dynamics. Cambridge University Press.
[2]	谢树艺. 矢量分析与场论[M]. 北京: 高等教育出版社, 1978.
[3]	Chae, D. and Kim, N. (1997) Axisymmetric Weak Solutions of the 3-D Euler Equations for Incompressible Fluid Flows. Nonlinear Analysis: Theory, Methods & Applications, 29, 1393-1404. https://doi.org/10.1016/s0362-546x(96)00186-1
[4]	He, C. and Xin, Z. (2005) On the Regularity of Weak Solutions to the Magnetohydrodynamic Equations. Journal of Differential Equations, 213, 235-254. https://doi.org/10.1016/j.jde.2004.07.002
[5]	Zhou, Y. (2005) Remarks on Regularities for the 3D MHD Equations. Discrete & Continuous Dynamical Systems, 12, 881-886. https://doi.org/10.3934/dcds.2005.12.881
[6]	王术. Sobolev空间与偏微分方程引论[M]. 北京: 科学出版社, 2008.
[7]	Sermange, M. and Temam, R. (1983) Some Mathematical Questions Related to the Mhd Equations. Communications on Pure and Applied Mathematics, 36, 635-664. https://doi.org/10.1002/cpa.3160360506
[8]	Nirenberg, L. (1959) On Elliptic Partial Differential Equations. The Annali della Scuola Normale Superiore di Pisa, 13, 115-162.
[9]	Evans, L.C. (2010) Partial Differential Equations (Graduate Studies in Mathematics). 2nd Edition, American Mathematical Society.
[10]	Duvaut, G. and Lions, J.L. (1972) Inéquations en thermoélasticité et magnétohydrodynamique. Archive for Rational Mechanics and Analysis, 46, 241-279. https://doi.org/10.1007/bf00250512
[11]	Lei, Z. (2015) Global Well-Posedness of Incompressible Magnetohydrodynamic Equations with Minimal Regularity. Communications on Pure and Applied Mathematics, 68, 1839-1891.
[12]	Wang, Y. and Wang, M. (2019) Global Smooth Solutions to the 3D Incompressible Navier-Stokes Equations with Large Initial Data in Spherical Coordinates. Science China Mathematics, 62, 1743-1758.
[13]	Deng, W. and Zhang, P. (2018) Large Time Behavior of Solutions to 3-D MHD System with Initial Data near Equilibrium. Archive for Rational Mechanics and Analysis, 230, 1017-1102. https://doi.org/10.1007/s00205-018-1265-x
[14]	Ke, X., Yuan, B. and Xiao, Y. (2021) A Stability Problem for the 3D Magnetohydrodynamic Equations near Equilibrium. Acta Mathematica Scientia, 41, 1107-1118. https://doi.org/10.1007/s10473-021-0405-9

为你推荐

友情链接