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

By Giovanni P. Galdi (auth.)

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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

2, when q < n, the exponent q• = nq/(n- q) is excluded. Actually one proves by means of counterexamples that the strong convergence is, in general, ruled out in this case. For, in the ball B 1 consider the sequence of functions Um(x) = { mCn-q)fq(1 - mjxl) 0 if jxj < 1/m if lxl ~ 1/m m= 1,2, ... with q < n. One has with Ct and C 2 independent of m. e. in B1 m-+oo it follows that no subsequence can converge strongly in L9•(B1 ). 2 are proposed to the reader in the following exercises. 6. ), m ~ 1, q E (1,oo) be uniformly bounded in the norm of wm,q.

Show the following Gauss identity: f ~v . u = ln 1an ~u. n- f ln n be bounded and u . 1. 4. 6). 4 Furth er Inequalities and Compactness Criter ia in wm,q We begin to prove some inequalities relating the Lq-norm of a function with that of its first derivatives (Poinca re 1894, §III and Friedrichs 1933). n: -d/2 < Xn < d/2}. 1. Assume f2 C Ld, for some d > 0. 1) Proof. It is enough to show the theorem for u E Clf(O). 1) for q = oo. If q E [1, oo), employing the Holder inequality yields 50 II. 1).

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. 4), we obtain the following result whose proof is left to the reader as an exercise. 1. Let 0 be locally lipschitzian.

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