paper10.pdf)

4 $BC1FHJ]B8B'J}Dx<0(B

$B0J2<$G$O!"(BTartar ([13]) $B$K$h$k!"(B $BC1FH$NJ]B8B'J}Dx<0$N=i4|CMLdBj(B

$$\left\{\begin{array}{ll} u_t+f(u)_x=0 & (t>0, x\in{\mbox{... ...})\\ u(x,0)=u_0(x) & (x\in{\mbox{\sl R}}) \end{array}\right.$$ (2)

$B$KBP$9$kJd40B,EYK!$rMQ$$$?ZL@$r!"(B Chen ([22]), Chen-Lu ([24]) $B$K$h$k2~NI$K4p$E$$$F(B $B>R2p$9$k!#(B $B$3$3$G!"(B $u=u(x,t)\in{\mbox{\sl R}}$$B!"(B$f(u)$ $B$O(B $u$ $B$K4X$7$F(B $C^2$ $B4X?t(B $B$G$"$k$H2>Dj$7!"(B $u_0\in L^2\cap L^\infty$ $B$H$9$k!#(B $BNc$($P(B $f(u)=u^2/2$ $B$N$H$-$O$3$NJ}Dx<0$OHsG4@-(B Burgers $BJ}Dx<0(B

$$u_t+\left(\frac{u^2}{2}\right)_x=0\hspace{1zw}(u_t+uu_x=0)$$

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

$B$3$NJ}Dx<0$N6a;w2r$H$7$F!"G4@-6a;wJ}Dx<0(B

$$\left\{\begin{array}{ll} u_t+f(u)_x=\varepsilon u_{xx} & (... ...=u^\varepsilon _0(x) & (x\in{\mbox{\sl R}}) \end{array}\right.$$ (3)

$B$N2r(B $u=u^\varepsilon$ ( $\varepsilon >0$) $B$r $u^\varepsilon _0$ $B$O!"(B$u_0$ $B$r(B $\varepsilon$ $B$K0MB8$7$?(B $B%Q%i%a!<%?$GE,Ev$KJ?3j2=$7$?==J,3j$i$+$J4X?t$G$"$k$H$7!"(B $\varepsilon$ $B$K4X$7$F0lMM$K(B

$$\vert u^\varepsilon _0(x)\vert\leq M,\hspace{1zw}\Vert u^\varepsilon _0\Vert _{L^2}\leq C$$

$B$G!"(B $\varepsilon \rightarrow 0$ $B$N;~$K(B $u^\varepsilon _0\rightarrow u_0$ $B$H$J$k$b$N$H$9$k!#(B

$BL?Bj(B 1   $B$3$N$H$-!"E,Ev$J3j$i$+$5$r;}$C$?(B (3) $B$N(B $B2r(B $u=u^\varepsilon (x,t)$ $B$,B8:_$7!"(B

$$\vert u^\varepsilon (x,t)\vert\leq M,\hspace{1zw} \Vert\sq... ...arepsilon _x(\cdot,\cdot)\Vert _{L^2}\leq \frac{1}{\sqrt{2}}C$$ (4)

$B$rK~$?$9!#(B

$B>ZL@(B

[23] $B$K$h$k!#4JC1$N$?$a!"(B$u^\varepsilon$ $B$r(B $u$ $B$H=q$/$3$H$H$9$k!#(B

$u^\varepsilon _0$ $B$O(B $u(\cdot,t)\in H^2\cap C^2$ $B$H$J$k0L==J,$KJ?3j2=$7$F$*$/!#(B $t_0>0$ $B$H$7!"(B$u(x,t_0)$ $B$r9M$($k$H!"(B $u(\cdot,t_0)\in H^2$ $B$h$j(B $\vert x\vert\rightarrow\infty$ $B$KBP$7$F$O(B $u(x,t_0)\rightarrow 0$ $B$H$J$k$N$G(B $u(x,t_0)$ $B$OFbIt$G:GBgCM$r $B$r(B $x_0$ $B$H$9$k!#(B

$B?^(B 1: $u(t_0,\cdot )$

$$\max_{x\in{\mbox{\scriptsize\sl R}}}u(x,t_0)=u(x_0,t_0)$$

$B$3$N$H$-!"(B$x=x_0$ $B$G6KBg$G$"$k$N$G(B

$$\left.\frac{d}{dx}u(x,t_0)\right\vert _{x=x_0} = 0$$ (5)

$$\left.\frac{d^2}{dx^2}u(x,t_0)\right\vert _{x=x_0} \leq 0$$

$B$G$"$j!"(B(3) $B$h$j(B

$$u_t=\varepsilon u_{xx}-f'(u)u_x$$

$B$H$J$k$N$G!"(B

$$\left.\frac{d}{dt}u(x,t)\right\vert _{x=x_0,t=t_0} = \left. \varepsilon u_{xx}-f'(u)u_x\right\vert _{x=x_0,t=t_0}\leq 0$$

$B$H$J$k!#$3$l$H(B (5) $B$H$O!"(B $u(x,t)$ $B$,(B $(x,t)=(x_0,t_0)$ $B$G$O(B $t$ $BJ}8~$K$OA}2C$7$J$$$3$H$r0UL#$9$k(B $B$N$G!"$h$C$F(B $\max u(\cdot,t)$ $B$O(B $t$ $B$K4X$7$FHsA}2C$H$J$j!"(B

$$\max u(\cdot,t)\leq\max u(\cdot,0)=\max u_0(\cdot) \leq M$$

$B$,8@$($k!#F1MM$K$7$F!"(B

$$\min u(\cdot,t) \geq \min u_0(\cdot) \geq -M$$

$B$b8@$(!":G=i$NITEy<0$,<($5$l$k!#(B

$B$^$?!"J}Dx<0(B $u_t+f'(u)u_x = \varepsilon u_{xx}$ $B$r(B $u$ $BG\$9$k$H(B

$$\left(\frac{u^2}{2}\right)_t + F_0(u)_x = \varepsilon (uu_x)_x -\varepsilon u_x^2 \hspace{1zw}(F_0(u)=\int_0^u f'(v)v dv)$$

$B$H=q$1$k!#$3$l$r(B ${\mbox{\sl R}}\times[0,T]$ $B>e@QJ,$9$k$H(B

$$\int_0^Tdt\int_{\mbox{\scriptsize\sl R}}\left(\frac{u^2}{2}... ...-\varepsilon \int_0^Tdt\int_{\mbox{\scriptsize\sl R}}u_x^2 dx$$

$B$H$J$k$,!"(B $F_0(u)_x = uf'(u)u_x$ $B$O(B $u,u_x\in L^2$, $\vert u\vert\leq M$ $B$h$j(B $f'(u)\in L^\infty$ $B$H$J$k$N$G(B $F_0(u)_x\in L^1$ $B$G$"$j!"(B $F_0(u(\pm\infty,t))=F_0(0)=0$ $B$J$N$G(B

$$\int_{\mbox{\scriptsize\sl R}}F_0(u)_xdx = 0,$$

$(uu_x)_x=u_x^2+uu_{xx}$ $B$b(B $u,u_x,u_{xx}\in L^2$ $B$J$N$G(B $(uu_x)_x\in L^1$ $B$G$"$j!"(B $B$^$?!"(B $u_x,u_{xx}\in L^2$ $B$h$j(B $u_x\in L^\infty$ $B$G$"$k$+$i(B $\vert x\vert\rightarrow\infty$ $B$N$H$-(B $uu_x\rightarrow 0$ $B$H$J$k!#(B $B$h$C$F(B

$$\int_{\mbox{\scriptsize\sl R}}(uu_x)_xdx = 0$$

$B$H$J$k$N$G!"7k6I(B

$$\int_{\mbox{\scriptsize\sl R}}\frac{1}{2}u^2(x,T)dx-\int_{\... ...-\varepsilon \int_0^Tdt\int_{\mbox{\scriptsize\sl R}}u_x^2 dx$$

$B$H$J$j!"$3$l$K$h$j(B

$$\Vert\sqrt{\varepsilon }u_x\Vert _{L^2({\mbox{\scriptsize\s... ...2 \leq \frac{1}{2}\Vert u_0\Vert _{L^2}^2\leq \frac{1}{2}C^2$$

$B$,F@$i$l$k!#(B

$B2) $B$N

$BDj5A(B 2   $u=u(x,t)\in L^\infty({\mbox{\sl R}}\times[0,T))$ $B$,=i4|CMLdBj(B (2) $B$N(B $0\leq t < T$ $B$G$N(B $B $B$G$"$k$H$O(B $BG$0U$N(B $\phi\in C^1_0({\mbox{\sl R}}\times[0,T))$ $B$KBP$7$F(B

$$\int_0^T dt\int_{\mbox{\scriptsize\sl R}}\{\phi_t u + \phi_... ...)\}dx + \int_{\mbox{\scriptsize\sl R}}\phi(x,0)u_0(x)dx = 0$$ (6)

$B$rK~$?$9$3$H!#(B

$BCm(B 3

$B$?$@$7!"J]B8B'J}Dx<0$K$*$$$F$O!"o$O0l0U@-$N$?$a$K%(%s%H%m%T!<>r7o$H8F$P$l$k$b$N$rK~$?$93 $B@a$G=R$Y$?6a;w2r$N6K8B$O!"$$$:$l$b$=$N>r7o$rK~$?$9$3$H$,(B $BCN$i$l$F$$$k!#(B

$T=\infty$ $B$KBP$7$FG4@-6a;w2r(B $u=u^\varepsilon$ $B$r(B (6) $B$N:8JU$K(B $BBeF~$9$k(B ($B$=$l$r(B $I^\varepsilon$ $B$H$9$k(B) $B$H!"(B

$$\begin{eqnarray*} I^\varepsilon & = & \int_0^\infty dt\int_{\mbox{\scriptsi... ...mbox{\scriptsize\sl R}}\phi(x,0)\{u_0(x)-u^\varepsilon _0(x)\}dx \end{eqnarray*}$$

$B$H$J$k$,!"2>Dj$K$h$j(B $\varepsilon \rightarrow 0$ $B$K4X$7$F(B $u_0(x)-u^\varepsilon _0(x)=o(1)$ $B$G!"$^$?L?Bj(B 1 $B$H(B Schwarz $B$NITEy<0$K$h$j(B

$$\left\vert\varepsilon \int_0^\infty dt\int_{\mbox{\scriptsiz... ...eq\frac{C\sqrt{\varepsilon }}{\sqrt{2}}\Vert\phi_x\Vert _{L^2}$$

$B$J$N$G!"7k6I(B

$$I^\varepsilon = o(1)\hspace{1zw}(\varepsilon \rightarrow 0)$$

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

$B0lJ}!"(B $L^\infty(\Omega)$ $B$NHF $B!V(B $\Vert g_n\Vert _{L^\infty(\Omega)}\leq C$ $B$J$i$P(B $B$"$kItJ,Ns(B $\{g_{n_j}\}_j$ $B$H$"$k4X?t(B $g\in L^\infty(\Omega)$ $B$,$"$C$F(B

$$g_{n_j}\rightarrow g \hspace{1zw}L^\infty(\Omega) \mbox{weak$\ast$}$B!W(B$$

$B$H(B $u^\varepsilon$, $f(u^\varepsilon )$ $B$N0lMMM-3&@-$K$h$j!"(B $B$"$kItJ,Ns(B $\{\varepsilon _n\}$$B!"$"$kM-3&$J4X?t(B $\bar{u}$, $\bar{f}$ $B$,$"$C$F(B

$$u^{\varepsilon _n}\rightarrow\bar{u},\hspace{1zw} f(u^{\vare... ... _n})\rightarrow\bar{f}\hspace{1zw}L^\infty \mbox{weak$\ast$}$$

$B$H$J$k$N$G!"(B

$$\int_0^\infty dt\int_{\mbox{\scriptsize\sl R}}\{\phi_t u^{\v... ...{\scriptsize\sl R}}\phi(x,0)u_0(x)dx =I^{\varepsilon _n}=o(1)$$

$B$K$*$$$F(B $n\rightarrow\infty$ $B$H$9$k$H(B

$$\int_0^\infty dt\int_{\mbox{\scriptsize\sl R}}\{\phi_t\bar{u... ...{f})\}dx + \int_{\mbox{\scriptsize\sl R}}\phi(x,0)u_0(x)dx =0$$

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

$B$h$C$F!"$b$7(B

$$\bar{f}(x,t)=f(\bar{u}(x,t))\hspace{1zw}\mbox{a.e.}$$ (7)

$B$,8@$($l$P(B $\bar{u}(x,t)$ $B$,

$BNc(B 4

$v_n(x)=\cos nx$ $B$H$9$k$H!"G$0U$N(B $\phi\in L^1({\mbox{\sl R}})$ $B$KBP$7$F(B Riemann-Lebesgue $B$NDjM}$K$h$j(B

$$\int_{\mbox{\scriptsize\sl R}}\phi(x)\cos nx dx\rightarrow 0 \hspace{1zw}(n\rightarrow\infty)$$

$B$H$J$j!"$3$l$O(B

$$\cos nx\rightarrow 0\hspace{1zw}L^\infty \mbox{weak$\ast$}$$

$B$r0UL#$9$k!#0lJ}!"(B

$$\cos^2 nx = \frac{1+\cos 2nx}{2}$$

$B$h$j!"(B

$$\int_{\mbox{\scriptsize\sl R}}\phi(x)\cos^2 nx dx = \frac... ...x{\scriptsize\sl R}}\phi(x)dx\hspace{1zw}(n\rightarrow\infty)$$

$B$H$J$k$N$G!"(B

$$\cos^2 nx\rightarrow \frac{1}{2}\hspace{1zw}L^\infty \mbox{weak$\ast$}$$

$B$G$"$j!"$9$J$o$A(B

$$w^\ast\!-\!\lim \cos^2 nx \neq (w^\ast\!-\!\lim \cos nx)^2$$

$B$G$"$k$3$H$K$J$k!#(B

$B$3$NNc$+$i$bJ,$+$k$h$&$K!"0lHL$K(B $u^\varepsilon \stackrel{\ast}{\rightharpoonup}\bar{u}$ $B$G$b(B $f(u^\varepsilon )\stackrel{\ast}{\rightharpoonup}f(\bar{u})$ $B$H$O8B$i$J$$!#(B

$B$7$+$7!"$3$N(B $\bar{f}=w^\ast\!-\!\lim f(u^\varepsilon )$ $B$r(B $f(u)$ $B$rMQ$$$F5-=R$9$k(B $B$3$H$r2DG=$H$9$k(B Young $BB,EY$H8F$P$l$k$b$N$,$"$k!#(B