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6 $BJd40B,EYK!(B

$BDjM}(B 8 (div-curl $BJdBj(B)   $\Omega(\subset {\mbox{\sl R}}^N)$ $B$rM-3&3+=89g!"(B $v_n, w_n:\Omega\rightarrow {\mbox{\sl R}}^N$ $B$rM-3&4X?tNs$H$7!"(B $v_n\rightarrow v$, $w_n\rightarrow w$ $L^\infty(\Omega;{\mbox{\sl R}}^N) \mbox{weak$\ast$}$ $B$H$9$k!#(B

$B$3$N$H$-!"(B

$$\mathop{\rm div}v_n =\sum_{k=1}^N\frac{\partial}{\partial x... ...i}(w_n)_j -\frac{\partial}{\partial x_j}(w_n)_i\right)_{i<j}$$

$B$,!"(B $\mbox{$H^{-1}_{\rm loc}(\Omega)$}$ $B$N$"$k%3%s%Q%/%H=89g$K4^$^$l$k$J$i$P!"(B

$$v_{n_j}\cdot w_{n_j}=\sum_{k=1}^N(v_{n_j})_k(w_{n_j})_k \r... ...arrow v\cdot w\hspace{1zw}L^\infty(\Omega) \mbox{weak$\ast$}$$

$B$H$J$kItJ,Ns(B $\{n_j\}_j$ $B$,B8:_$9$k!#(B

$B>\:Y$O(B [13] $B;2>H$N$3$H!#$J$*!"(B $\mbox{$H^{-1}_{\rm loc}(\Omega)$}$ $B$O(B $BG$0U$N(B $\phi\in C^\infty_0(\Omega)$ $B$KBP$7$F(B $\phi T\in H^{-1}(\Omega)$ $B$H$J$k(B $T\in{\cal D}'(\Omega)$ $BA4BN$+$i$J$k(B Fréchet $B6u4V!#(B

$BDj5A(B 9   $C^2$ $B5i4X?t(B $\eta(u),q(u):{\mbox{\sl R}}\rightarrow{\mbox{\sl R}}$ $B$,(B

$$q'(u)=\eta'(u)f'(u)$$ (8)

$B$rK~$?$9$H$-!"(B $(\eta(u),q(u))$ $B$rJ}Dx<0(B $u_t+f(u)_x=0$ $B$N(B $B%(%s%H%m%T!$B$H8F$S!"(B $\eta (u)$ $B$r(B$B%(%s%H%m%T!<(B$B!"(B$q(u)$ $B$r(B$B%(%s%H%m%T!$B$H$$$&!#(B

$B$b$7!"(B$u(x,t)$ $B$,(B $u_t+f(u)_x=0$ $B$N3j$i$+$J2r$G$"$k$H$-$O(B

$$\eta(u)_t+q(u)_x = \eta'(u)u_t+q'(u)u_x = \eta'(u)(u_t+f'(u)u_x) =\eta'(u)(u_t+f(u)_x)=0$$

$B$H$J$k$N$G!"%(%s%H%m%T!

$BCm(B 10

$B$J$*!"C1FH$G$J$/O"N)$NJ]B8B'J}Dx<0(B (1) $B$N>l9g$O(B $B%(%s%H%m%T! $(\eta(U),q(U))$ $B$O@~7A$NHyJ,J}Dx<07O(B

$$\nabla q(U)=\nabla\eta(U) F'(U) \hspace{1zw}(\nabla=(\partial_1,\partial_2,\ldots,\partial_N))$$ (9)

$B$GDj5A$5$l!"$3$l$OJ}Dx<0$,(B $N$ $BK\!"L$CN4X?t(B 2 $B8D$N2a>j$JJ}Dx<07O(B $B$G$"$k$N$G%(%s%H%m%T!

$BG4@-6a;w2r(B $u^\varepsilon$ $B$KBP$7$F$O(B

$$\begin{eqnarray*} \lefteqn{\eta(u^\varepsilon )_t+q(u^\varepsilon )_x}\\ & = ... ... _x)_x -\varepsilon \eta''(u^\varepsilon )(u^\varepsilon _x)^2 \end{eqnarray*}$$

$B$H$J$k!#$3$3$+$i(B $\{\eta(u^\varepsilon )_t+q(u^\varepsilon )_x\}_\varepsilon$ $B$N%3%s%Q%/%H@-$r<($9!#(B

$BL?Bj(B 11   ${\mbox{\sl R}}\times(0,\infty)$ $B$NG$0U$NM-3&3+=89g(B $\Omega$ $B$KBP$7$F(B $\{\eta(u^\varepsilon )_t+q(u^\varepsilon )_x\}_\varepsilon$ $B$O(B $\mbox{$H^{-1}_{\rm loc}(\Omega)$}$ $B$GAjBP%3%s%Q%/%H!#(B

$B$3$NL?Bj$N>ZL@$K$O

$BDjM}(B 12   $\Omega(\subset {\mbox{\sl R}}^N)$ $B$,M-3&3+=89g(B,

$$A\subset {\cal M}(\Omega)(\equiv \{\mbox{$\Omega$ $B>e$NId9fIU$-(B Radon $BB,EY(B}\}=C_0(\Omega)^\ast)$$

$B$H$9$k$H$-(B

$$\sup_{\mu\in A,C_0(\Omega)\ni\phi\neq0}\frac{\vert\langle \... ...i\rangle \vert} {\Vert\phi\Vert _{C_0}}\hspace{0.5zw}<\infty$$

$B$J$i$P(B $A$ $B$O%=%\%l%U6u4V(B $W^{-1,p}(\Omega)$ $B$KKd$a9~$^$l!"$=$3$G(B $BAjBP%3%s%Q%/%H(B ($B$?$@$7(B $p$ $B$OG$0U$N(B $1<p<N/(N-1)$)$B!#(B

$BDjM}(B 13 (Murat $B$NJdBj(B)   ${\mbox{\sl R}}^N$ $B$NM-3&$J3+=89g(B $\Omega$ $B$*$h$S(B $1<q \leq 2<r<\infty$ $B$J$kG$0U$N(B $q,r$ $B$KBP$7(B

$$\begin{array}{l} \displaystyle (W^{-1,q}(\Omega)\mbox{ $B$N(B.. ...{-1}_{\rm loc}(\Omega)$}\mbox{ $B$N%3%s%Q%/%H=89g(B}) \end{array}$$

$B>\:Y$O(B [13,21] $B;2>H$N$3$H!#(B

($BL?Bj(B 11 $B$N>ZL@(B)

$I_1=\varepsilon (\eta'(u^\varepsilon )u^\varepsilon _x)_x$, $I_2=-\varepsilon \eta''(u^\varepsilon )(u^\varepsilon _x)^2$ $B$H$9$k!#(B

$u^\varepsilon$ $B$O0lMMM-3&$J$N$G(B $\eta''(u^\varepsilon )$ $B$bM-3&$G!"(B $BL?Bj(B 1 $B$h$j(B

$$\Vert\sqrt{\varepsilon }u^\varepsilon _x\Vert _{L^2}\leq C$$

$B$J$N$G!"G$0U$N(B $\phi\in C_0(\Omega)$ $B$KBP$7$F(B

$$\left\vert\int\hspace{-6pt}\int _\Omega I_2\phi dxdt\right\vert\leq C_1\Vert\phi\Vert _{C(\Omega)}$$

$B$H$J$k!#$h$C$F(B $I_2$ $B$O(B $C_0(\Omega)^\ast={\cal M}(\Omega)$ $B$GM-3&!"(B $B$H8+$k$3$H$,$G$-!"DjM}(B 12 $B$K$h$j(B $1<q<2$ $B$G$"$k(B $q$ $B$KBP$7(B $\{I_2\}_{\varepsilon }$ $B$O(B $W^{-1,q}(\Omega)$ $B$N(B $B$"$k%3%s%Q%/%H=89g$K4^$^$l$k!#(B

$B$^$?(B $I_1$ $B$O!"(BSchwarz $B$NITEy<0$K$h$jG$0U$N(B $\phi\in C_0^1(\Omega)$ $B$KBP$7(B

$$\left\vert\int\hspace{-6pt}\int _\Omega I_1\phi dxdt\right\... ... \Vert\sqrt{\varepsilon }u^\varepsilon _x\Vert _{L^2(\Omega)}$$

$B$H$J$j!"(B $\eta'(u^\varepsilon )$ $B$O0lMMM-3&!"(B$\Omega$ $B$OM-3&$G(B $1<q<2$ $B$h$j(B $q'=q/(q-1)>2$ $B$J$N$G(B

$$\Vert\phi_x\Vert _{L^2(\Omega)}\leq C_2(\Omega)\Vert\phi_x\Vert _{L^{q'}(\Omega)}$$

$B$G$"$j!"$h$C$FL?Bj(B 1 $B$h$j(B

$$\left\vert\int\hspace{-6pt}\int _\Omega I_1\phi dxdt\right\... ...\Omega)} \rightarrow 0\hspace{1zw}(\varepsilon \downarrow 0)$$

$B$H$J$k$N$G(B $\{I_1\}_{\varepsilon }$ $B$b(B $W^{-1,q}(\Omega)$ $B$GAjBP%3%s%Q%/%H$H$J$k!#(B $B$h$C$F(B $I_1+I_2=\eta(u^\varepsilon )_t+q(u^\varepsilon )_x$ $B$O(B $W^{-1,q}(\Omega)$ $B$G(B $BAjBP%3%s%Q%/%H$G$"$k$3$H$,$o$+$k!#(B

$B0lJ}!"(B $\eta(u^\varepsilon )$, $q(u^\varepsilon )$ $B$O0lMMM-3&$J$N$G!"(B $r>1$, $\phi\in C_0^1(\Omega)$ $B$KBP$7$F(B

$$\begin{eqnarray*} \lefteqn{\left\vert\int\hspace{-6pt}\int _\Omega(\eta(u^\vare... ...Vert\phi\Vert _{W^{1,r'}_0(\Omega)}\\ && (r'=\frac{r}{r-1}>1) \end{eqnarray*}$$

$B$H$J$k$N$G!"(B $\{\eta(u^\varepsilon )_t+q(u^\varepsilon )_x\}_{\varepsilon }$ $B$O(B $W^{-1,r}(\Omega)$ $B$GM-3&$G$"$k$3$H$K$J$k!#(B

$B$h$C$F(B Murat $B$NJdBj(B ($BDjM}(B 13)$B$K$h$j(B $\{\eta(u^\varepsilon )_t+q(u^\varepsilon )_x\}_{\varepsilon }$ $B$O(B $\mbox{$H^{-1}_{\rm loc}(\Omega)$}$ $B$G(B $BAjBP%3%s%Q%/%H$G$"$k$3$H$K$J$k!#(B

$B:#!"(B2 $B$D$N%(%s%H%m%T!, $(\hat{\eta},\hat{q})$ $B$r9M$($k!#(B $B$3$l$i$OO"B3$@$+$i!"(BYoung $BB,EY$NDjM}(B 5 $B$K$h$j(B $\{u^\varepsilon \}_{\varepsilon }$ $B$N$"$kItJ,Ns(B $\{u^{\varepsilon '}\}_{\varepsilon '}$ $B$KBP$7$F(B

$$\begin{eqnarray*} && \eta(u^{\varepsilon '})\stackrel{\ast}{\rightharpoonup}\ba... ...hat{q}-\hat{\eta}q} =\langle\nu,\eta\hat{q}-\hat{\eta}q\rangle \end{eqnarray*}$$

$B$H$J$k!#(B

$B0lJ}!"L?Bj(B 11 $B$K$h$j(B

$$\mathop{\rm div}_{(x,t)}(q(u^\varepsilon ),\eta(u^\varepsilo... ...lon )) =\hat{\eta}(u^\varepsilon )_t+\hat{q}(u^\varepsilon )_x$$

$B$O$$$:$l$b$"$k(B $\mbox{$H^{-1}_{\rm loc}(\Omega)$}$ $B$N%3%s%Q%/%H=89g$K4^$^$l$k$N$G!"(B div-curl $BJdBj$K$h$j(B $\{u^{\varepsilon '}\}_{\varepsilon '}$ $B$N$"$kItJ,Ns(B $\{u^{\varepsilon ''}\}_{\varepsilon ''}$ $B$KBP$7$F(B

$$\{(q,\eta)\cdot(-\hat{\eta},\hat{q})\}(u^{\varepsilon ''}) =... ...ightharpoonup} \bar{\eta}\bar{\hat{q}}-\bar{\hat{\eta}}\bar{q}$$

$B$H$J$k!#0J>e$K$h$j(B

$$\overline{\eta\hat{q}-\hat{\eta}q} =\bar{\eta}\bar{\hat{q}}-\bar{\hat{\eta}}\bar{q} \hspace{1zw}\mbox{a.e. $(x,t)$}$$ (10)

$B$,F@$i$l$k!#$3$N<0(B (10) $B$O(B

$$\langle\nu,\left\vert\begin{array}{ll}\eta&q \hat{\eta}&\h... ...\eta}\rangle &\langle\nu,\hat{q}\rangle \end{array}\right\vert$$

$B$H$b=q$+$l!"(BTartar $BJ}Dx<0(B $B$H8F$P$l$k!#(B $BJ}Dx<0(B $u_t+f(u)_x=0$ $B$,@~7A$G$"$k>l9g(B ( $f''(u)\equiv 0$) $B$3$N(B Tartar $BJ}Dx<0$O<+L@$J$b$N$H$J$k$,!"(B $BHs@~7A$N>l9g$K$OI,$:$7$b$=$&$G$O$J$/!"(B $B$3$NJ}Dx<0$+$i(B $\nu=\delta_{\bar{u}}$ $B$G$"$k$3$H$rF3$/$3$H$,$G$-$k!#(B $B$3$l$r