Computational Techniques for Fluid Dynamics: Volume 2: by Clive A. J. Fletcher

By Clive A. J. Fletcher

The aim of this textbook is to supply senior undergraduate and postgraduate engineers, scientists and utilized mathematicians with the particular suggestions, and the framework to increase abilities in utilizing the suggestions, that experience confirmed powerful within the numerous brances of computational fluid dynamics.

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Extra resources for Computational Techniques for Fluid Dynamics: Volume 2: Specific Techniques for Different Flow Categories (Scientific Computation)

Example text

13) In [6], G. Anzellotti proves the lower semicontinuity of the functional GIjI in case of a bounded domain and cp = 1. Adapting the results in [6] or [4] we can also prove the lower semicontinuity of G IjI in our case. Indeed we have the following Lemma. 4. Assume that g(x,~, t) is lower-semicontinuous on]RN x]RN X [0, +00[, (t t)foreachfixed x E ]RN, and g(x,~) :::: all~lI- bforall x and~. Then,for convex in any sequence Un E BV(]RN, cp dx) such that Un -+ u in L}oc(]RN) one has 48 F. ANDREU, V.

1) cI>cp : L2(~N) -+ [0, +00] defined by cp(u):= { f(x, Du) lP, if u E ( Jrt N BV(~N, lP dx) n L2(~N) if u E L2(~N) \ BV(~N, lP dx). 13). 4, we have that cI>cp is lower-semicontinuous. Therefore, the subdifferential acI>cp of cI>cp, is a maximal monotone operator in L2(~N) (see [9]). 14) E L2(~N) follows immediately from the nonlinear semigroup theory (see [9]). Now, to get the full strength of the abstract result derived from semigroup theory we need to characterize a cp. To get this characterization, we introduce the following operator Bcp in L2(~N).

3, 2003 The Cauchy problem for linear growth functionals Setting Vrr(x) = n 2: n r , we have Integrating from 0, since a > N, we can find rE E N such N;;" (iN IV

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