[物理学与PDEs]第2章第5节 一维流体力学方程组的 Lagrange 形式 5.3 一维理想流体力学方程组的 Lagrange 形式

1. 一维理想流体力学方程组 $$\beex \bea \cfrac{\p\rho}{\p t}+\cfrac{\p}{\p x}(\rho u)&=0,\\ \cfrac{\p u}{\p t}+u\cfrac{\p u}{\p x}+\cfrac{1}{\rho }\cfrac{\p p}{\p x}&=F,\\ \cfrac{\p S}{\p t}+u\cfrac{\p S}{\p x}&=0 \eea \eeex$$ 的 Lagrange 形式为 $$\beex \bea \cfrac{\p \tau}{\p t'}-\cfrac{\p u}{\p m}&=0,\\ \cfrac{\p u}{\p t'}+\cfrac{\p p}{\p m}&=F,\\ \cfrac{\p S}{\p t'}&=0. \eea \eeex$$ 为方便起见, 仍将 $(t',m)$ 记为 $(t,x)$, 则 $$\bee\label{2_5_3_Lag} \bea \cfrac{\p \tau}{\p t}-\cfrac{\p u}{\p x}&=0,\\ \cfrac{\p u}{\p t}+\cfrac{\p p}{\p x}&=F,\\ \cfrac{\p S}{\p t}&=0. \eea \eee$$ $\eqref{2_5_3_Lag}_3$ 可由初值 $S_0(x)$ 立即求出. 而 $\eqref{2_5_3_Lag}_{1,2}$ 化为 $$\beex \bea \cfrac{\p \tau}{\p t}-\cfrac{\p u}{\p x}&=0,\\ \cfrac{\p u}{\p t}+\cfrac{\p }{\p x}p(\tau, S_0(x))&=F. \eea \eeex$$ 特别对于均熵流, 我们有 $p$ - 方程组 $$\beex \bea \cfrac{\p \tau}{\p t}-\cfrac{\p u}{\p x}&=0,\\ \cfrac{\p u}{\p t}+\cfrac{\p }{\p x}p(\tau)&=F. \eea \eeex$$