[物理学与PDEs]第3章习题4 理想磁流体的能量守恒方程

试证明: 对理想磁流体, 能量守恒方程 (4. 14) 可以写为如下形式: $$\beex \bea \cfrac{\p}{\p t}&\sex{\rho e+\cfrac{1}{2}\rho u^2 +\cfrac{1}{2}\mu_0H^2}\\ +\sum_{k=1}^3 \cfrac{\p}{\p x_k}&\sed{ \rho u_k\sex{e+\cfrac{1}{2}u^2-\cfrac{p}{\rho}} +\mu_0u_kH^2-\mu_0H_k({\bf u}\cdot{\bf H}) }=\rho{\bf F}\cdot{\bf u}. \eea \eeex$$

 

证明: 仅须注意到 $$\bex ({\bf u}\times{\bf H})\times{\bf H}=({\bf u}\cdot {\bf H}){\bf H}-H^2{\bf u}. \eex$$