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New PDF release: An Introduction to the Mathematical Theory of the

By Giovanni P. Galdi (auth.)

ISBN-10: 1475738668

ISBN-13: 9781475738667

ISBN-10: 1475738684

ISBN-13: 9781475738681

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Read or Download An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Volume I: Linearised Steady Problems PDF

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

Example text

II}). 3 to the space wm,q(O), m;:::: 2, is formally analogous, provided we introduce a suitable generalisation of the space W 1- 1fq,q(80). To this end, assume 0 of class cm- 1 •1 and let {Bk} and {(k}, k = 1,2, ... 1. Assuming that (k) - Xn- ( ( (k) (k) ) kX1 , ... • ,xn-1•(kX1 , ... 141,2) is finite. It can be shown that the definition of << Uk >>m-1/q,qE 0 48 II. Basic Function Spaces and Related Inequalities wm-l/q,q(80) does not depend on the particular choice of the local representation {Bk}, { (k} of the boundary.

2). Hardy, Littlewood and Polys 1934, Theorem 16, p. 26. 2. 19 . 7) allow us to deduce more general relations, which are contained in the following lemma. 2. Let r E [q, nq/(n- q)J, if q E [1, n), and rE(q,oo), ifq;:::n. 5) where c1 = max(q, r(n- 1)/n), >. = n(r- q)frq. Proof. We shall distinguish the two cases: (i) q :5 r :5 qnf(n- 1), {ii) r ;:::: qnf(n- 1). 5). 2) with lur

1. For a > 0 let C = {:z: E Dln: 0 < :Z:i

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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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