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1.1 Young measure $B$N7hDjLdBj$K4X$7$F(B

 

Young measure $B$,(B measure $B$K$J$k$3$H$r(B Tartar's functional equation $B$H8F$P$l$k(B commutative relation

$B$+$iF3$$$F$$$k$b$N(B:

L.Tartar [36],
R.J.DiPerna [14],[15],
X.Ding, G.-Q.Chen and P.Luo [1],[11],[12],
G.-Q.Chen and P.T.Kan [5],
P.L.Lions, B.Perthame, E.Tadmore [20],
B.Rubino [26],
D.Serre [29]

[36] (L.Tartar) $B$O(B Tartar $B$K$h$k(B compensated compactness $B$N(B $B86E5$G!"?M9)G4@-$K$h$k6a;w2r$KBP$9$k(B scalar conservation law $B$X$N1~MQ(B $B$b$3$3$K=q$+$l$F$$$k!#(BYoung measure $B$NB8:_DjM}(B (Young $B$,(B 1 $BZL@$7$?;vn $BZL@$b$3$3$K$"$k!#(B

[14] (R.J.DiPerna) $B$O(B compensated compactness $B$r(B p-system $B$N$h$&$J(B $B$NJ]B8B'J}Dx<07O$KE,MQ$7$?O@J8$G!"?M9)G4@-$K$h$k(B $B6a;w2r$H:9J,6a;w2r$K$D$$$F!"$=$N0lMMM-3&@-$,J]>Z$5$l$l$P$=$l$O $B$^$?!"(B[15] (DiPerna) $B$O(B Euler $B:BI87O$NEy%(%s%H%m%T! $B$,(B 7/5, 9/7, 11/9,...$B$N>l9g$K!"(B $BG$0U$N=i4|CM$KBP$7$Fl9g$O2r$K??6u0h$,@8$:$k$3$H$,$"$jF@$k$?$a!"$=$3$G(B vanish $B$9$k$h$&(B $B$J(B entropy (weak entropy) $B$r [1],[11],[12] (X.Ding, G.-Q.Chen and P.Luo) $B$O(B [15] $B$N7k2L$r(B $B$K$^$G3HD%$7$?$b$N(B $B$G!"(B[15] $B$HF1MM$NI>2A$r>\$7$$7W;;$GF@$k$3$H$K$h$j>ZL@$7(B $B$F$$$k!#(B[11]() $B$OZL@$N$&$A!"(BYoung measure $B$r7hDj(B $B$9$kItJ,0J30$N$3$H$,CzG+$K=q$+$l$F$*$j!"(B[15] $B$rFI$`$h$j(B $BJ,$+$j$d$9$$!#(B [11]() $B$O(B $B$KBP$7$F$N(B entropy $B$NI>2A!"(B [1] $B$O0lHL$N(B $B$KBP$9$kI>2A$N7W;;$,9T$o$l$F$$$k!#(B

$B$?$@$7!"(B[1] $B$N7W;;$K$O=$@5$,I,MW$J2U=j$,$"$j!"(B[12] $B$K$O=$@5$,2DG=$H$@$1$7$+=q$+$l$F$$$J$$$,(B [1] $B$NCf9q8lHG$N(B $BO@J8$K$=$N=$@5ItJ,$,=q$+$l$F$"$k!#(B

[5] (G.-Q.Chen and P.T.Kan) $B$O8IN)E@$G$N$_AP6J@-$,l9g$N(B $B$NJ]B8B'7O$N!"$=$N8IN)E@$N6aK5$G$N $B$NO@J8$G!"=i4|CM$N(B $BM-3&@-$N@)8B$r=|$$$?!"(BRiemann $BLdBj$H:9J,6a;w$K$h$k7k2L$,=P$kM=Dj!#$?$@$7!"(BRiemann $BLdBj$,$+$J$j(B $BJ#;($J$N$G!"(B $B$K$h$C$F(B $B$,<+L@$@$H$O8B$i$J$$$h$&$G$"$k!#(B

[20] (P.L.Lions, B.Perthame, E.Tadmore) $B$O(B [15], [11], [1], [12] $B$HF1$8(B Euler $B:BI87O$NJ}Dx(B $B<0$KBP$9$k$b$N$G!"(B $B$K$D$$$F$N7k2L$G$"$k!#$?$@$7!"(B Tartar's functional equation $B$r2r$$$F$$$k$H$3$m$O!"4v$D$+J,$+$j$K(B $B$/$$2U=j$,$"$k!#(B $B6a;w2r$O(B kinetic formulation $B$H$$$&J}K!$G:n$C$F$$$k$h$&$G$"$k!#(B Perthame $B$d(B Tadmor $B$O$=$&$$$C$?I8Bj$NO@J8$r$$$/$D$+=q$$$F$$$k!#(B

[26] (B.Rubino) [5] $B$r>/$73HD%$7$F$$$k7k2L(B $B$i$7$$!#%$%?%j%"$K$OB>$K(B P. Marcati, R. Natalini, J. Milani $B$i(B $B$,(B compensated compactness $B$NO@J8$rH/I=$7$F$$$k!#(B

[29] (D.Serre) $B$O(B [14] $B$N!"(B $B$NJ]B8(B $BB'J}Dx<07O$KBP$9$k(B Tartar's functional equation $B$N2rK!$N2~NI$G$"$j!"(B entropy $B$NK~$?$9(B 2 $B3,@~7AAP6J7?J}Dx<0$H!"J]B8B'7O$NFC@-J}8~$H$N(B $B4X78$r>\$7$/D4$Y$k$3$H$K$h$C$F(B Young measure $B$r7hDj$7$F$$$k!#(B $B$^$?!"J}Dx<0$,A4$F$NFC@-J}8~$K(B degenerate $B$7$F$$$k>l9g!"$^$?$O0l$D$N(B $BFC@-J}8~$K(B degenerate $B$7$F$$$k>l9g$K$D$$$F$N(B Young measure $B$N7hDj$N(B $B8B3&!"$9$J$o$A(B compensated compactness $B$K$h$k2rK!$N8B3&$b<($7$F$$$k!#(B

$B$3$N(B degenerate $B$7$F$$$kFC@-J}8~$r;}$DJ]B8B'7O$N>l9g$K$D$$$F$O(B [3] $B$J$I$G9M;!$5$l$F$$$k!#(B