Mechanics and Mathematics of Fluids of the Differential Type by D. Cioranescu V. Girault & K.R. Rajagopal

Author:D. Cioranescu, V. Girault & K.R. Rajagopal
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

where

This proof is the only delicate point of this section. Let and consider the following function:

(4.4.11)

Notice that is continuous on and

(4.4.12)

The proof of the monotonicity is done in two steps. We start by showing that is differentiable on the interval ]0, 1[. Afterwards, we apply the Taylor formula to the right-hand side of (4.4.12) for some , and finally we estimate it by making use of hypothesis (4.4.3).

For and set

(4.4.13)

therefore,

We will show that F is differentiable with respect to at any , and moreover, that its modulus is bounded by an integrable function on , independently of . The Lebesgue dominated convergence theorem will then give the result.

To achieve this, consider the following two subsets of :

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