compensated compactness $B$HJ]B8B'J}Dx<0$K$D$$$F(B (PDF ¥Õ¥¡¥¤¥ë: paper10.pdf)

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$$U_t+F(U)_x=0\hspace{1zw}(x\in{\mbox{\sl R}}, t>0)$$ (1)

$B$N7A$NO"N)JPHyJ,J}Dx<0$r;X$9!#$?$@$7!"(B$U$,$F(U)$ $B$O $B

$$U=U(x,t)={}^t(u_1,u_2,\ldots,u_N), \hspace{1zw} F(U)={}^t(f_1(U),f_2(U),\ldots,f_N(U))$$

$B$G$"$j!"(B

$$F'(U)=\left[ \begin{array}{cccc} \partial_1f_1(U) & \parti... ...ce{1zw}\left(\partial_j = \frac{\partial}{\partial u_j}\right)$$

$B$O!"(B$N$ $B8D$NAj0[$J$k

$$\lambda_1(U)<\lambda_2(U)<\cdots<\lambda_N(U)$$

$B$r;}$D$H$9$k!#3F8GM-CM(B $\lambda_j(U)$ $B$KBP$9$k1&8GM-%Y%/%H%k$r(B $R_j(U)$ $B$H=q$/$H$-!"(B$F(U)$ $B$NDj5A$5$l$F$$$kNN0h(B $D$ $BFb$G>o$K(B

$$\nabla\lambda_j(U)\cdot R_j(U)\neq 0 \hspace{1zw}(\nabla = (\partial_1,\partial_2,\ldots,\partial_N))$$

$B$G$"$k$H$-!"(B$j$-$BFC@-J}8~$O(B $D$ $BFb$G(B$B??@-Hs@~7A(B (genuinely nonlinear) $B$G$"$k$H$$$$!"(B$D$ $BFb$G(B

$$\nabla\lambda_j(U)\cdot R_j(U)\equiv 0$$

$B$G$"$k$H$-!"(B$j$-$BFC@-J}8~$O(B $D$ $BFb$G(B$B@~7AB`2=(B (linearly degenerate) $B$G$"$k$H$$$&!#(B

$BJ]B8B'7O$+$i$OOC$,30$l$k$,!"$3$N??@-Hs@~7A!"@~7AB`2=$N>r7o$O!"(B$F'(U)$ $B$,(B $BBP3Q9TNs(B

$$F'(U)=\left[ \begin{array}{cccc} \lambda_1(U) & 0 & \cdots... ...& \vdots\\ 0 & 0 & \cdots & \lambda_N(U) \end{array}\right]$$

$B$N7A!"$D$^$jJ}Dx<0$,(B

$$\left\{ \begin{array}{l} (u_1)_t + \lambda_1(U)(u_1)_x = 0... ...dots\\ (u_N)_t + \lambda_N(U)(u_N)_x = 0 \end{array}\right.$$

$B$N7A$G$"$k>l9g$K$O!"(B$R_j(U)=e_j$ ($=j$ $BHVL\$N4pK\%Y%/%H%k(B) $B$H$J$k$N$G!"(B

$$\begin{array}{lll} \mbox{$B??@-Hs@~7A(B} & \Longleftrightarrow ... ...e{1zw}\mbox{($\lambda_j(U)$ $B$,(B $u_j$ $B$K$h$i$J$$(B)} \end{array}$$

$B$G$"$j!"(B$j$-$BFC@-J}8~$NHs@~7A@-$K4X$9$k>r7o$G$"$k$3$H$,8+$F $B0lHL$N>l9gFC@-6J@~$K4X$7$F$$$($P!"??@-Hs@~7A$O!"FC@-B.EY(B $\lambda_j(U)$ $B$,(B $j$-$BFC@-6J@~$K$=$C$FC1D4$G$"$k$3$H$r0UL#$7!"@~7AB`2=$O!"(B $BFC@-B.EY(B $\lambda_j(U)$ $B$,(B $j$-$BFC@-6J@~$K$=$C$FDj?t$G$"$k$3$H$r0UL#$9$k!#(B

compensated compactness $B$HJ]B8B'J}Dx<0$K$D$$$F(B