By Jerry Scutts

ISBN-10: 1902579046

ISBN-13: 9781902579047

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**Example text**

In other words, we may assume that pn converges to p weakly in LOO - *, that p2(t) - t and that pi is bounded. Furthermore, the same argument as the above truncation argument shows, by mollification, that we may assume without loss of generality that pi is smooth on [0, oo), say C1. Finally, writing pi (t) = fo 1(t>,) p' (s) ds (pi (s) > 0, V s > 0), we see that we can furthermore simply look at pi (t) = 1(t>a) for an arbitrary positive a. a) + a 1(p>a) < (p - a)+ + a 1(p>a) = (p - a)+ + a 1(p2:a) = p 1(p>a).

Indeed, we know that p E L°°(0,T; Ll(R2)) and pItI2 E L°O(0,T; L1(R2)). Therefore, p u E L°O(0,T; Li(R2)). Then, letting cp E Co (R2), 0 < W:5 1, cp . 1. 22). We begin with the convergence of pnu". In order to prove it, we use once more a mollifier rcc = rc(E) where r. E Co 00 (RN), 'c > 0, fRN rc dx = 1, Supprc C B, and we let gE = g * cE for an arbitrary function g: notice that, in the case of Dirichlet boundary conditions, if g is defined on Si x (0, T), gE is defined on iE x (0, T) when SzE = {x E Si / dist (x, 8S2) > e}.

0 - Compactness results 36 We conclude this section with a simple observation about the bounds which have to be satisfied by f" in order to apply the above arguments. 1. 18) by f" = fl' + f2, fl' (resp. f2) converges weakly to fi (resp. a x (0,T)) (resp. 72) where a > q/(q - 1), b > 1, c > s/(s - 1). 72). 5 In this section, we shall investigate a variant (or extension ) of the preceding results and proofs. e. p is assumed to be a continuous non-decreasing function on [0, oo) vanishing at 0 (this is an irrelevant normalization of p).

### Basic Aviation Modeling by Jerry Scutts

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