(PDF ¥Õ¥¡¥¤¥ë: paper10.pdf)

8.3 Tartar $BJ}Dx<0(B

$B$3$l$O6a;w2r$H$OD>@\$O4X78$J$/!"%(%s%H%m%T! $B$N>l9g$K$7$+(B Tartar $BJ}Dx<0$O2r$+$l$F$*$i$:!"(B$N=2$ $B$N>l9g$G$b2r$+$l$F$$$kJ}Dx<07O$O(B $B$[$\0J2<$NDL$j$G$"$k!#(B

• $BCF@-7O(B:
  
  $$\left\{\begin{array}{l} v_t-u_x=0,\\ u_t-\sigma(v)_x=0 \end{array}\right.$$
  
  
  $v\sigma''(v)\geq 0$ $B$N>l9g$O(B DiPerna ([14]), $v\sigma''(v)\leq 0$ $B$N>l9g$O(B J.Shearer ([33]), P.Lin ([34]) $B$i$N7k2L$,$"$k$,!"(B $v\sigma''(v)\leq 0$ $B$N>l9g$O(B $\sigma(v)$ $B$K$+$J$j@)Ls$,I,MW$G!"Nc$($P(B $\sigma'\rightarrow 0$ $(\vert v\vert\rightarrow\infty)$ $B$N>l9g$K$OL$2r7h!#(B
• $B5$BN$N%P%m%H%m%T%C%/%b%G%k(B:
  
  $$\left\{\begin{array}{l} \rho_t + (\rho u)_x=0,\\ (\rho u... ...t. \hspace{1zw}(\rho\geq 0, P'(\rho)\geq 0, P''(\rho)\geq0)$$
  
  
  DiPerna ([15]), Ding,Chen,Luo ([26,27],[28]), Lions,Perthame,Tadmor,Souganidis ([19,20]), Chen,LeFloch ([31]), Makino ([30]) $BEy$N7k2L$,$"$k!#(B $B$?$@$705NO4X?t(B $P(\rho)$ $B$K$O6/$$@)Ls$,$D$/!#(B $B$^$?$O!"$3$NJ}Dx<0$NAjBPO@7A$r07$C$?$b$N$b$"$k(B (Makino [44])$B!#(B
• $B$X$=7?$NB`2=@-$r;}$D(B 2$\times$2 $B$NO"N)J}Dx<0(B:
  
  $$f'(U_0) \sim \left[\begin{array}{ll}\lambda_1&0 0&\lambda... ...ambda_1(U) (U\neq U_0), \lambda_2(U_0)=\lambda_1(U_0))$$
  
  
  Chen,Kan ([45])
• $B0lHL$N(B 2$\times$2 $B$NO"N)J}Dx<0$KBP$9$k9M;!(B: Serre [46].

2$\times$2 $B$NO"N)J}Dx<0$N>l9g!"%(%s%H%m%T!9) $B$OL$CN4X?t$N?t$OF1$8$K$J$j!"(B $B=i4|CM$NG$0U@-$NJ,%(%s%H%m%T!2 $B$NJ}Dx<0$G$b(B Tartar $BJ}Dx<0$,2r$+$l$F$$$J$$$b$N$OB?$$!#(B

$B$^$?!"(B$N$ $B$,(B 3 $B0J>e$N>l9g$O%(%s%H%m%T!<$,$[$H$s$I$J$$$N$G!"(B $B:#$N$H$3$m(B Tartar $BJ}Dx<0$r2r$/$K$OIT==J,$G!"(B $BNc$($P%(%M%k%.!