Singular Perturbations and Boundary Layers by Gung-Min Gie & Makram Hamouda & Chang-Yeol Jung & Roger M. Temam

Author:Gung-Min Gie & Makram Hamouda & Chang-Yeol Jung & Roger M. Temam
Language: eng
Format: epub
ISBN: 9783030006389
Publisher: Springer International Publishing

We now estimate the u ∗j in the L 2-norm.

Lemma 5.12.

The following norm estimates of u ∗j and its derivatives hold: for j, i, m ≥ 0,

(5.131)

where, setting λ 1 = λ 1(j, i, m, p, q) = 1 − 3j − 2i − m + 2p + q as in ( 5.122 ):

(5.132)

Proof.

We recall that . From (5.121) and (5.117) and Lemma 5.10, thanks to the symmetry, we just prove that, for i, m ≥ 0,

(5.133)

Using (5.108) () we see that the last sum is of the order of BD 1(j, i, m) in all cases and thus the lemma follows.

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