By Giovanni P. Galdi (auth.)

ISBN-10: 1475738668

ISBN-13: 9781475738667

ISBN-10: 1475738684

ISBN-13: 9781475738681

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

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

**Example text**

6) to the anisotropic case. Let 1 $ q; < oo, i = 1, ... , n. lull .. {n' i:l qi-l > 1, s = (E? 8) If qi = q, for all i = 1, ... , n, (2. 6). On the other hand, if for some i (=1, say), ql < q q2 = ... 7) we deduce = s=r + nq( ql - q) < (q-qt)+ qt(n-q) r. 5} are now considered. 9) 3 It is needless to say that the possibility of lowering the exponent s depends on the particular problem. 34 II. Basic Function Spaces and Related Inequalities shown for the first time by Ladyzhenskaya (1958, 1959a, eq.

5) is known as the Cauch y inequa lity. 3) are the . 7) valid for all u E £8(0) n Lr(O) with 1 $ s $ q $ r $ oo, and q- 1 = Os- 1 + (1 - O)r- 1 , 0 E [0, 1]. 1. Assume 0 is bounded. 2. 5). Hint: Minimise the function t 9 /q-t+ l/q'. 3. 7). We want now to list some of the basic properties of the spaces Lq. For 1 $ q < oo, Lq is separable, C0 (0) being, in partic ular, a dense subse t (Mira nda 1978, §51; note that this is not true if q = oo, since C(O) is a closed subspace of £ 00 (0)). In fact, one can prove something more, namely, that every function in Lq, 1 $ q < oo, can be approximated by functi ons from Ccf(O).

We shall give, for simplicity, the proof in the caBe n = 3. 5} for u E C 1 (0). r• :~(e,x2,x3)£1e ~ hl 3 :u(Yli'7•X3)dq '7 Integrating over they-variables and raising to the qth power, we deduce ju(x1,x2,x3) -ulq ~ ICI-q [a foa IVu(e,x2,x3)lde 3 +a2 foa loa jVu(Yli'7•X3)Idyld'7 +a Ia IVuldC] q. 4. v 53 which completes the proof. 3. 2). 1. Suppose 0 is a cube of side a and subdivide it into N equal cubes Ci, each having sides of length a/N 1fn. -q)/q Xi(x), with Xi characteristic function of the cube Ci, from the previous inequality one has the following result due to Friedrichs {1933).

### 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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