An Introduction to the Mathematical Theory of the by Giovanni P. Galdi (auth.)

By Giovanni P. Galdi (auth.)

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

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1. The Lebesgue Spaces L" 21 Other useful properties are those connecting the weak convergence of a sequence with its boundedness in norm. 11). 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.

4) with A = n(r- q)fq(r - 1) and C = C(n, r,q, 0). Denote now by 'Y the linear map which associates to every function f E C«f(O) its value at the bounda ry -y(f) = /lan and let u E W 1·9(0). k} c C«f(O) converging to u in W 1•9(0). 4) this sequence will also converge in Lr(80), for suitable r, to a function u E Lr(80). ) into Lr(80) that ascribes, in a well-defined sense, to every function from W 1•9(0) a function on the bounda ry which, for smooth functions u, reduces to the usual trace ulan· This result can be fairly extended to more general domains 0 and spaces wm,q with m > 1.

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. From the identity {lx-yi 8u(x + re) u(x)-u(y)=-Jo 8r dr, y- X e=ly-xl' 4 The problem of the behavior at large spatial distances of function gradients in Lq(JR") will be fully analysed in Section 7.

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