By Giovanni P. Galdi (auth.)

ISBN-10: 1475738668

ISBN-13: 9781475738667

ISBN-10: 1475738684

ISBN-13: 9781475738681

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Additional resources for An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems

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Let n c Ld, for some d > 0. Show that for all u E W~· 2 (0) n W 2 •2 (0). 4. _ )n f i q

Picard (Picone 1946, §160). 1 In fact, assume as before 0 c Ld for some d > 0 and consider the function u(x) oo + d/2)/d], u E Co (0). U(x) = sin[7T(Xn Since U(x) is bounded in Ld and vanishes at -d/2, d/2, integrating by parts we find ~u(x) cot [1T(Xn ; 0 $ ld/2 { 8u -d/ 2 8xn d d/2)] }2 dxn = ld/2 -d/2 (:uXn )2 dxn . -2 [7T(Xn + -7T21d/2 u 2 { sm d d/2)] -cot2 [7T(Xn + d d/2)]} dx n· lP -d/2 1 This proof was brought to my attention by Professor L. Pepe. 4. Further Inequalities and Compac tness Criteria in wm,q 51 Hence which implies Therefore, one deduces and, if 0 is bounded, J1.

If £9 is reflexive, this result has a sort of converse, which is referred to as the weak compactness property. 2. Let {um} C £9(0), 1 < q < oo, and assume that there is a number M > 0 such that llumll Then there exist {um'} ~ ~ M, for all mE lN. 1). 1. Most of the results stated in this section concerning the topological properties of the spaces £9 are in fact valid in general Banach spaces. 111). 2 is a special case of a result that states that every bounded sequence in a reflexive Banach space has a weakly convergent subsequence (Miranda 1978, §§28, 30).

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An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems by Giovanni P. Galdi (auth.)


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