(PDF ¥Õ¥¡¥¤¥ë: paper10.pdf)

7 Tartar $BJ}Dx<0$N2rK!(B

$BC1FH$NJ]B8B'J}Dx<0$N>l9g!"G$0U$N(B $C^2$ $B4X?t(B $\eta (u)$ $B$KBP$7(B

$$q(u)=\int^u \eta'(v)f'(v)dv$$

$B$H$9$l$P(B (8) $B$K$h$j(B $(\eta(u),q(u))$ $B$O(B $B%(%s%H%m%T!o$K$?$/$5$s$N%(%s%H%m%T!10) $B$r2r$/$3$H$K$9$k!#(B $B0J2<$G=R$Y$kJ}K!$O!"(BTartar ([13]), Chen-Lu ([24]) $B$i$K$h$k!#(BChen ([22]) $B$b;2>H$N$3$H!#(B

$B$^$:!"(B $(\hat{\eta}(u),\hat{q}(u))=(u,f(u))$ $B$H$9$k(B ($(u,f(u))$ $B$b%(%s%H%m%T!, $\bar{f}$ $B$O(B $(x,t)$ $B$N4X?t$G!"(BYoung $BB,EY(B $\nu=\nu(u)$ $B$G(B $B$N@QJ,$K4X$7$F$ODj?t$H$_$l$k$N$G(B

$$\begin{eqnarray*} \lefteqn{\overline{\eta\hat{q}-\hat{\eta}q} -(\bar{\eta}\bar... ...& = & \langle\nu,\eta(u)(f(u)-\bar{f})-(u-\bar{u})q(u)\rangle \end{eqnarray*}$$

$B$HJQ7A$G$-$k!#$h$C$F!"(B

$$\langle\nu,\eta(u)(f(u)-\bar{f})-(u-\bar{u})q(u)\rangle =0 \hspace{1zw}\mbox{a.e. $(x,t)$}$$ (11)

$B$H$J$k!#(B $B:#!"Nc$($P$3$N(B $\nu$ $B$K4X$9$kHo@QJ,4X?t(B $\eta (u)(f(u)-\bar {f})-(u-\bar {u})q(u)$ $B$,!"(B $B$b$7(B $\eta$, $q$ $B$rE,Ev$KA*$V$3$H$G(B

$$\eta(u)(f(u)-\bar{f})-(u-\bar{u})q(u) \left\{\begin{array}{ll} =0 & (u=\bar{u})\\ <0 & (u\neq\bar{u})\end{array}\right.$$

$B$N$h$&$K$G$-$k$J$i$P!"(B

$B?^(B 3: $\eta (u)(f(u)-\bar {f})-(u-\bar {u})q(u)$ $B$NM}A[E*$J$b$N(B

(11) $B$+$i(B $\mathop{\rm supp}\nu = \{\bar{u}\}$ ($B0lE@(B) $B$G$"$k$3$H$,$o$+$j!"$h$C$F(B $\nu=\delta_{\bar{u}}$ $B$G$"$k$3$H$K$J$k!#(B

$B$=$l$K;w$?Lr3d$r$9$k(B $(\eta,q)$ $B$H$7$F(B $BC1FH$NJ]B8B'J}Dx<0$NM}O@$GNI$/MQ$$$i$l$k(B

$$\eta(u)=\vert u-\bar{u}\vert,\hspace{1zw}q(u)=\{f(u)-f(\bar{u})\}\mathop{\rm sign}(u-\bar{u})$$

$B$r $BCm(B 14

$B$?$@$7!"$3$N(B $(\eta,q)$ $B$O(B $u=\bar{u}$ $B$G$OHyJ,2DG=$G$O$J$/!"(B $B$h$C$F(B $C^2$ $B5i4X?t$G$b$J$$!#$7$+$7!"(B$C^\infty$ $B5i4X?t(B $\bar{\eta}_n$ $B$G(B $B$3$N(B $\eta (u)$ $B$r0lMM$K6a;w$9$k$3$H$O$G$-!"(B

$B?^(B 4: $\eta (u)$ $B$H(B$\eta _n(u)$

$q_n(u)=\int^u\eta_n'(v)f'(v)dv$ $B$H$9$l$P(B $(\eta_n,q_n)$ $B$O(B (11) $B$rK~$?$7!"$=$7$F(B $n\rightarrow\infty$ $B$N$H$-$K!"(BYoung $BB,EY(B $\nu$ $B$N@QJ,$K4X$9$k(B Lebesgue $B<}B+DjM}$K$h$j$=$N6K8B$H$7$F$3$N(B $(\eta,q)$ $B$KBP$7$F(B (11) $B$,@.$jN)$D$3$H$,<($5$l$k!#(B

$B$3$N(B $(\eta,q)$ $B$KBP$7!"(B

$$\begin{eqnarray*} \lefteqn{\eta(u)(f(u)-\bar{f})-(u-\bar{u})q(u)}\\ & = & (f(... ...\{f(u)-f(\bar{u})\} =\vert u-\bar{u}\vert\{f(\bar{u})-\bar{f}\} \end{eqnarray*}$$

$B$H$J$j!"(B $f(\bar{u})-\bar{f}$ $B$O(B $u$ $B$K4X$7$F$ODj?t$J$N$G(B (11) $B$O(B

$$\langle\nu,\vert u-\bar{u}\vert\{f(\bar{u})-\bar{f}\}\rangle... ...angle\nu,\vert u-\bar{u}\vert\rangle =0\hspace{1zw}\mbox{a.e.}$$

$B$H$J$k!#$h$C$F!"(B

$f(\bar{u})=\bar{f}$ $B$^$?$O(B $\langle\nu,\vert u-\bar{u}\vert\rangle =0$ a.e.

$B$9$J$o$A(B

$f(\bar{u})=\bar{f}$ $B$^$?$O(B $\nu=\delta_{\bar{u}}$ a.e.

$B$,8@$($?$3$H$K$J$k!#(B

$B$=$7$F!"(B $\nu=\delta_{\bar{u}}$ $B$J$i$P$b$A$m$s(B $f(\bar{u})=\bar{f}$ $B$G(B $B$b$"$k$N$G!"(B $B$h$C$F$I$A$i$K$7$F$b85$NL\I8$G$"$C$?(B (7) $B$,8@$($?$3$H$K$J$j!"(B $\bar{u}$ $B$, $B$?$@$7!"%(%s%H%m%T!<>r7o!"$^$?$O(B

$$u^{\varepsilon ''}\rightarrow\bar{u}\hspace{1zw}\mbox{a.e.}$$

$B$r<($9$K$O$d$O$j(B $\nu=\delta_{\bar{u}}$ $B$G$"$k$3$H$r<($9I,MW$,$"$k!#(B $B$h$C$F0J2<$G$=$l$r9M;!$9$k!#(B

$B:#EY$O(B

$$\eta(u)=f(u)-f(\bar{u}),\hspace{1zw}q(u)=\int_{\bar{u}}^u(f'(v))^2dv$$

$B$H$7$F(B (11) $B$KBeF~$7$F$_$k!#$3$N>l9g(B

$$H(u)=\eta(u)(f(u)-\bar{f})-(u-\bar{u})q(u)$$

$B$H$*$/$H!"(B $f(\bar{u})=\bar{f}$ $B$J$N$G(B

$$H(u)=(f(u)-f(\bar{u}))^2-(u-\bar{u})\int_{\bar{u}}^u(f'(v))^2dv$$

$B$H$J$k$,!"(BSchwarz $B$NITEy<0$K$h$j(B

$$\vert f(u)-f(\bar{u})\vert=\left\vert\int_{\bar{u}}^uf'(v)dv... ...-\bar{u}\vert}\left\vert\int_{\bar{u}}^u(f'(v))^2dv\right\vert$$

$B$H$J$k$N$G(B $H(u)\leq 0$ $B$G!"$=$NEy9f$,@.N)$9$k$N$O(B $u=\bar{u}$ $B$+(B $B$^$?$O(B $u$ $B$H(B $\bar{u}$ $B$H$N4V$G(B $f'(u)$ $B$,Dj?t$G$"$k>l9g!"$H$J$k!#(B

$B?^(B 5: $BItJ,E*$K@~7A$J(B $f(u)$

$B$3$N>l9g!"(B(11) $B$N(B

$$\langle\nu,H(u)\rangle =0$$

$B$K$h$j(B $\mathop{\rm supp}\nu$ $B>e(B $H(u)$ $B$O(B 0 $B$G$J$/$F$O$J$i$:!"$h$C$F(B $\mathop{\rm supp}\nu$ $B$NFLJq$N>e$G(B $f'(u)$ $B$,Dj?t$G$J$1$l$P$J$i$J$$$3$H$K$J$k!#(B

$B8N$K!"Nc$($P(B $f(u)=u^2/2$ (Burgers $BJ}Dx<0(B) $B$N$h$&$K(B $f''(u)>0$ $B$rK~$?$9(B $BC1FHJ]B8B'J}Dx<0$N>l9g$O(B $\mathop{\rm supp}\nu$ $B$O(B 1 $BE@$K$J$j!"7k6I(B $\nu=\delta_{\bar{u}}$ $B$G$"$k$3$H$,8@$($k!#(B

$BCm(B 15

$BFL$G$O$J$/!" ($B?^(B 5) $B$KBP$7$F$O(B Tartar $BJ}Dx<0(B (10) $B$+$i$O2?$bF@$i$l$J$$!#(B $BNc$($P(B $f(u)$ $B$,6h4V(B $(a,b)$ $B>e$G(B $f'(u)\equiv c_0$ ($BDj?t(B) $B$G$"$k$H$9$k!#(B

$B$3$N$H$-!"(B$a\leq u\leq b$ $B$KBP$7$F(B

$$\begin{eqnarray*} q(u) & = & q(a)+\int_a^u\eta'(v)f'(v)dv=q(a)+c_0\int_a^u\eta'(v)dv =c_0\eta(u)+c_1\\ && (c_1=q(a)-c_0\eta(a)) \end{eqnarray*}$$

$B$H$J$j(B

$$\left\vert\begin{array}{ll}\eta&q \hat{\eta}&\hat{q}\end{... ...0\hat{\eta}+c_2\end{array}\right\vert =c_2\eta-c_1\hat{\eta}$$

$B$H$J$k$N$G!"(B $\mathop{\rm supp}\nu\subset[a,b]$ $B$G$"$k>l9g!"(B

$$\begin{eqnarray*} \lefteqn{\langle\nu,\left\vert\begin{array}{ll}\eta&q \hat{... ...(\mbox{Young $BB,EY$OA4B,EY(B 1 $B$h$j(B $\langle\nu,c_j\rangle =c_j$}) \end{eqnarray*}$$

$B$H$J$k!#$9$J$o$A!"(B $\mathop{\rm supp}\nu\subset[a,b]$ $B$+$D(B $\langle\nu,1\rangle =1$ $B$G$"$k(B $BG$0U$NB,EY$KBP$7(B Tartar $BJ}Dx<0$,@.$jN)$C$F$7$^$&$3$H$K$J$j!"(B $B8@$$49$($k$H(B Tartar $BJ}Dx<0(B (10) $B$+$i$O(B Young $BB,EY$K4X$9$k>pJs$O2?$bF@$i$l$J$$$3$H$K$J$k!#(B

$B$D$^$j!"(BTartar $BJ}Dx<0$O@~7A$NJ}Dx<0$K$O<+L@$J4X78<0$G$"$j(B $B$=$3$+$i$O2?$bF@$i$l$J$$$,!"Hs@~7A@-$,6/$$J}Dx<0$K$O0RNO$rH/4x$9$k!"(B $B$H$$$C$?@- $B$h$C$F!"Jd40B,EYK!$K$h$C$F(B Tartar $BJ}Dx<0$rF3$/