1.  一维理想流体力学方程组 $$\beex \bea \cfrac{\p\rho}{\p t}+\cfrac{\p}{\p x}(\rho u)&=0,\\ \cfrac{\p}{\p t}(\rho u) +\cfrac{\p}{\p x}(\rho u^2+p)&=\rho F,\\ \cfrac{\p}{\p t}\sex{\rho e+\cfrac{1}{2}\rho u^2} +\cfrac{\p}{\p x}\sez{\sex{ \rho e+\cfrac{1}{2}\rho u^2+p }u}&=\rho Fu; \eea \eeex$$ 或 $$\beex \bea \cfrac{\p\rho}{\p t}+\cfrac{\p}{\p x}(\rho u)&=0,\\ \cfrac{\p u}{\p t}+u\cfrac{\p u}{\p x}+\cfrac{1}{\rho }\cfrac{\p p}{\p x}&=F,\\ \cfrac{\p S}{\p t}+u\cfrac{\p S}{\p x}&=0; \eea \eeex$$ 再或 $$\beex \bea A(t,x,U)\cfrac{\p U}{\p t}+B(t,x,U)\cfrac{\p U}{\p x} =F(t,x,U), \eea \eeex$$ 其中 $$\bex A(t,x,U)=I,\quad B=\sex{\ba{ccc} u&\rho&0\\ \cfrac{c^2}{\rho}&u&\cfrac{p_S}{\rho}\\ 0&0&u \ea},\quad F=\sex{\ba{c}0\\F\\0 \ea}. \eex$$

2.  一阶拟线性双曲组

(1)   对一阶拟线性 PDE $$\bee\label{2_1_sq} A(t,x,U)\cfrac{\p U}{\p t}+B(t,x,U)\cfrac{\p U}{\p x} =F(t,x,U), \eee$$ 若对 $\forall\ (t,x,U)$, 特征方程 $$\bex |B-\lm A|=0 \eex$$ 有 $n$ 个实根 $$\bex \lm_1(t,x,U),\cdots,\lm_n(t,x,U), \eex$$ 且相应的广义左特征向量 $$\bex \eta^i:\ \eta^iB=\lm_i\eta^iA \eex$$ 构成完全组 $(|\eta^i_j|\neq 0)$. 则称 \eqref{2_1_sq} 为双曲型方程组.

(2)   若 $$\bex \lm_1(t,x,U)<\lm_2(t,x,U)<\cdots<\lm_n(t,x,U), \eex$$ 则称 \eqref{2_1_sq} 为严格双曲型方程组.

(3)   若曲线 $x=x(t)$ 满足 $$\bex \sev{B-\cfrac{\rd x}{\rd t}A}=0, \eex$$ 则称其为特征曲线.

(4)   例: 在非真空区域, 一维理想流体力学方程组为严格双曲型.

3.  均熵流 ($S=\const$): $$\beex \bea \cfrac{\p\rho}{\p t}+\cfrac{\p }{\p x}(\rho u)&=0,\\ \cfrac{\p u}{\p t}+u\cfrac{\p u}{\p x} +\cfrac{c^2}{\rho}\cfrac{\p \rho}{\p x}&=F. \eea \eeex$$