paper10.pdf)

3 $B

$BNI$/CN$i$l$F$$$k$h$&$K!"J]B8B'J}Dx<0$O=i4|CM$,3j$i$+$G$b!"(B $BM-8B;~9oFb$K2r$KITO"B3@-$,8=$o$l$k8=>]$,5/$3$jF@$k!#(B $B$3$l$O5$BNNO3X$G$N>W7bGH$KBP1~$7$F$$$F!"$h$C$F$=$l$r4^$`$h$&$J(B $B2r$r9M$($kI,MW$,$"$j!"$=$N$?$a$KDL>o$O $B8=:_$^$G$KCN$i$l$F$$$k!"ZL@J}K!$K$O(B

1. $BC1FHJ]B8B'$KBP$9$kA4JQF0$ND>@\I>2A(B
2. Glimm $B$N:9J,K!(B
3. $BGHLLDI@WK!(B (front-tracking method)
4. $BJd40B,EYK!(B (compensated compactness)

$B$N$h$&$JJ}K!$,CN$i$l$F$$$k!#(B $B$3$l$i$K$h$k@h6nE*$J7k2L$rI=$K$^$H$a$k$H0J2<$N$h$&$K$J$k!#(B

	$B7O(B ($N$)	$B6a;w2r(B	$B<}B+@-(B	$B=i4|CM(B
Oleinik [10] (1957)	1	L-F $B:9J,(B (*1)	Helly (*3)	large data
		$BG4@-6a;w(B (*2)
Glimm [11](1965)	$N$	Glimm $B:9J,(B	Helly	small data (*4)
$B@>ED(B [12] (1968)	2 (*5)	Glimm $B:9J,(B	Helly	large data
Tartar [13] (1979)	1	$BG4@-6a;w(B	$BJd40B,EYK!(B	large data
DiPerna [14],[15] (1983)	2 (*6)	L-F $B:9J,(B	$BJd40B,EYK!(B	large data
		$BG4@-6a;w(B
DiPerna [16] (1976)	2 (*7)	$BGHLLDI@WK!(B	Helly	small data
Bressan [17] (1992)	$N$	$BGHLLDI@WK!(B	Helly	small data

(*1).

Lax-Friedrichs $B:9J,6a;w2r(B: $D_t U + D_x F(U) = 0$ $(\Delta t,\Delta x\rightarrow 0)$

$$\begin{array}{l} \displaystyle \hspace{1zw}D_t U = \frac{U... ...F(U(x+\Delta x,t))-F(U(x-\Delta x,t))}{2\Delta x} \end{array}$$

(*2).

$BG4@-6a;w(B: $U_t + F(U)_x = \varepsilon U_{xx}$ $(\varepsilon \rightarrow 0)$

(*3).

Helly: Helly $B$NA*=PDjM}(B

$B!V(B$[a,b]$ $B>e0lMMM-3&$G0lMMM-3&JQF0$G$"$k4X?tNs(B $B$+$i$O!"$"$kM-3&JQF04X?t$K3FE@<}B+$9$kItJ,Ns$r

(*4).

small data: $B0lHL$K$O=i4|CM$NA4JQF0$,==J,>.$5$$!"$9$J$o$A=i4|CM$,Dj?t$K==J,6a$$!"(B $B$H$$$&>r7o$,I,MW(B

(*5).

2: $BEy295$BN(B (Lagrange $B:BI87O(B) $B$NJ}Dx<0(B

$$v_t - u_x = 0, u_t + (a/v)_x = 0 \hspace{1zw}(\mbox{$u$: $BB.EY!$$

(*6).

2: $BCF@-BN$NJ}Dx<0(B ([14])

$$v_t - u_x = 0, u_t - \sigma(v)_x = 0 \hspace{1zw}(\sigma'(v)>0, v\sigma''(v)\geq 0)$$

$B5Z$S!"Ey%(%s%H%m%T!<5$BN(B (Euler $B:BI87O(B) $B$NJ}Dx<0(B ([15])

$$\begin{array}{l} \rho_t + (\rho u)_x = 0, (\rho u)_t + ... ...$B.H(B}\\ \mbox{ $N=1,2,3,\ldots$, $a>0$: $BDj?t(B}) \end{array}$$

(*7).

2: $B0lHL$N??@-Hs@~7A$J(B 2$\times$2 $B$NAP6J7?J]B8B'7O(B

$B$3$l$i$N7k2L$O2~NI$5$l$F$$$kItJ,$b$"$k$7!"$^$?6a;w2r$H$7$F(B $BF0NO3XE*6a;w$rMQ$$$k(B Lions,Perthame,Tadmor,Sourganidis $B$i$NJ}K!$b$"$k$,!"(B $B>\:Y$O>JN,$9$k(B ([19,20] $B;2>H(B)$B!#(B

$B$$$/$D$+$NJ}K!$NCf$GJd40B,EYK!$N0lHV$NFCD'$O!"(B $BBg$-$$=i4|CM$KBP$7$F2r$NB8:_$r<($;$k$3$H$G$"$m$&!#(B $B:G6a(B Bressan $B$i$K$h$k>.$5$$=i4|CM$KBP$9$kE,@Z@-(B ($BB8:_!"0l0U@-!"O"B30MB8@-(B) $B$N8&5f$,6/NO$K?J$a$i$l$F$$$k$,!"(B $BBg$-$$=i4|CM$KBP$9$k7k2L$OJd40B,EYK!$K$h$k$b$N0J30$O!"(B $B:#$N$H$3$mFCJL$JJ}Dx<0(B (*5) $B$KBP$9$k$b$N$7$+$J$$!#(B

$B$J$*!"(BHelly $B$NDjM}$rMQ$$$k>l9g$H!"Jd40B,EYK!$rMQ$$$k>l9g$K;H$o$l$k(B $B6a;w2r$NI>2A$O0[$J$C$F$$$F!"$=$l$i$O0J2<$NDL$j$G$"$k!#(B

• Helly $B$NA*=PDjM}$N>l9g(B:
  
  $$\left\{\begin{array}{l} T.V._x U^\varepsilon (\cdot,t)\leq... ...\vert (\mbox{$B$^$?$O(B }o(\vert t-s\vert)) \end{array}\right.$$
  
  

• $BJd40B,EYK!(B:
  
  $$\left\{\begin{array}{l} \Vert U^\varepsilon (\cdot,\cdot)\... ...repsilon _x(\cdot,\cdot)\Vert _{L^2}\leq C \end{array}\right.$$