Shock waves, increase of entropy and loss of information Page: 27 of 49
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where v = v(x,t); (3.27) holds for a.a.(x,t).
We introduce the abbreviations
(3.28) u = <v,v>, f = <f(v),v>;
then (3.27) can be rewritten as
(3.29) <(v-u)y>,v> = <(f-f)7?,v>.
In the derivation of (3.27) we used second derivatives of rj,
but (3.27) itself depends continuously on 7? in the C topology;
therefore, (3.37) remains true for n piecewise C'. We choose
(3.30) j?(v) = I v-u I;
then from (3.22)
f f(v) - f(u) for u>v
(3.30) ' SP(v) = l
L f(u ) - f(v) for u < v
Setting these choices for n and into (3.29) gives
< I v-u I (f-f(u)),v> = <(f-fj I v-u.fO;
we deduce from this that
(3.31) (f(u)-f) <lv-ul,>l> = 0.
For f strictly convex, it follows from (3.28) and Jensen's inequality
that the first factor (f(ui-f) in (3.31) is positive unless v is
concentrated at the single point u; the same, of course, is true for
the second factor <lv-ul,v>. Thus, it follows from (3.31) that v is
concentrated at the single point u, and so u*c^ tends to u in the
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Lax, P.D. Shock waves, increase of entropy and loss of information, report, October 1, 1984; United States. (https://digital.library.unt.edu/ark:/67531/metadc1087730/m1/27/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.