An Introduction to the Mathematical Theory of the - download pdf or read online

By Giovanni P. Galdi (auth.)

ISBN-10: 1475738668

ISBN-13: 9781475738667

ISBN-10: 1475738684

ISBN-13: 9781475738681

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

Example text

Since Xo E an is arbitrary, we may form an open covering (} of an constituted by domains of the type G. •. , Gm} satisfying all conditions in the lemma, which is thus completely proved. Other relevant properties related to star-shaped domains are described in the following exercises. 1. 1, the outer unit normal n on 80 exists almost everywhere. Then, setting F(x) := n · x, show that essinfF(x) > 0. 2. Assume fl bounded and locally lipschitzian. Prove that where each G; is star-shaped with respect to every point of a ball B; with B; C G;.

1) defines a norm in Lq, with respect to which Lq becomes a Banach space. 2) is a norm and that L 00 endowed with this norm is a Banach space (Miranda 1978, §47). For q = 2, Lq is a Hilbert space under the scalar product (u,v) = k uv, Whenever confusion of domains might occur, we shall use the notation II ·llq,o, ll·lloo,n, and {·,·) 0 . We want now to collect some inequalities in Lq spaces that will be frequently used throughout. 1). The number q' is called the Holder conjugate of q. 3) shows t,hat the bilinear form (u,v) is meaningful whenever u E Lq(O) and v E Lq (0).

34 II. Basic Function Spaces and Related Inequalities shown for the first time by Ladyzhenskaya (1958, 1959a, eq. (6)). 6). l), for some r E [1, oo). 6. 3 and set um(x) = cp(x)exp(-mlxl), mE IN. Obviously, {um} C C8"(1Rn). Show that for n = 3 the following inequality holds with c a positive number. Since R(m)-+ oo as m-+ oo, a constant"( E (O,oo) such that does not exist. 2 can be further strengthened, as shown by the following lemma. 3. Let q > n. 12) with c2 =max q-l)(q-l)/q} {1, (-q-n . Proof.

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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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