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28a) and 't:, = C([o, TO];V,) n H'([O, T0];L2(R)). 28a) guarantees that u E C([0,To];V:). 4. The case of compressible fluids requires a modification in the equations of motion to include a term of the form grad(divu). 11) may be perturbed by the term (div v, div w), which is insensitive to u. 9b) is no longer valid and the continuity equation must be adjoined. 5 We shall begin with a general uniqueness theorem for equations of the form aTH(u) at + u +Tg(u)+Th(u)=0 almost everywhere in R, 0

In particular, by the Cauchy-Schwarz in- ' Suppressed da. 17) Lno[ruj+ aildo hno r u d a . 16) that uj -, u - a. 14), since the standard norm dominates this norm up to a constant multiplier. 14). 3. 13). 14). We have, then, ( t ,U) = Jn VR,t - Vu dx + in wr(R,tp da. 5. 18). R,t)"2. 1, we may conclude that the mappings R, are continuous and have closed range, and that [F], is topologically equivalent to F, with 1. Formulations for Parabolic Models 32 the standard duality norm. In particular, RolL2(R)is self-adjoint, positivedefinite, and compact.

Ockendon and W. R. ), pp. 267-284. Oxford Univ. Press (Clarendon), London and New York, 1975.

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