Ordinary differential equations, transport theory and Sobolev spaces 545

Furthermore , ф.еСЦО, Г], ЬЩ^)) n L*(0, Г, L°"([R^)) (for all^G [1, oo)) and we have

T T

] ^ u.ij/Jtdx- ^ и^ф,{0)ах = ^ j u.xdtdx (100)

0 [RN ^N 0 jJ^N

Next , the arguments given above show that ф^ is bounded in L'^(0, Г, L''(IR^)) for all 1 ^ ^ ^ 00 Then, passing weakly to the limit m (99) and using the uniqueness results of section II, we deduce that ф^ converges weakly in Z>^(0, Г, L''((R^)) (l<^^oo) to the solution (AgC([0, Г], L^(R^))n L^(0, Г, L*(!R^)) {\f\ Sq< oo)

- ^ - а1\ { Ьф ) = х in(0, Г)хК^ iAUr = OonR^ (101)

ct

To prove strong convergence, we first observe that the same proof as the one used to show (98) yields

- || \ф,\их- l(q~\)d^^b\il/A4xS0, (102)

while we already know (see section II) that ф satisfies

- - J \ф\Чх- J ((?-l)divb|i/^|^Jx = 0 (103)

ct ^s ^s

This , exactly as we did in the proof of the stability result Theorem II4, implies that Ф, converges to ф in C([0, Г], L'^iU^)) for all 1 g g < oo

Now , if Wg converges weakly in L'*(0, Г, L^{U^)) to some u, we may pass to the limit in (100) and we deduce

^ ^ uфdtdx-^ ^ u''ф{0)dtdx = ^ j uxdtdx (104)

0 irn 0 j^n Ou"

And by the results of section II 5, we know that и is the unique renormalized solution of (11) with с = 0, for the initial condition u^

The proof of the strong convergence follows then from (98) and the fact that и satisfies

- - j \u\'dx-h j (diwb)\u\Pdx = 0 ,

ct ^Ы ^N

by the same arguments as those used in the proof of Theorem II4 Л

IV 4 Remarks

In this section, we just want to indicate some variants or extensions of the results presented above