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By Brown W.F.

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9) for u in Lp (Ω), v in Lq (Ω), p1 + 1q = 1. The following proposition addresses the same question when weak convergence comes into play. 3 Let 1 ≤ p ≤ +∞ and q such that 1 p + 1 q = 1. (i) If un strongly converges to u in Lp (Ω), and vn weakly converges to v in Lq (Ω), then un vn weakly converges to uv in L1 (Ω). (ii) If un weakly converges (respectively, weakly-⋆ converges if p = +∞) to u in Lp (Ω), and vn weakly converges to v in Lq (Ω), then nothing can be said on the sequence un vn . As for (ii), we emphasize that for the sequence un vn to converge (even weakly, or even in the sense of distributions), more needs to be known about either of the sequence un or vn .

Sermange and R. Temam [208]. ) and all possible settings (weak solutions, strong solutions, mild solutions15 ). Should we do this, the present book would be a pale mirror of existing textbooks of outstanding quality and completeness. However, for the convenience of the reader, for consistency, and also in order to prepare the ground for the analysis of the non-homogeneous case, we find it useful to still devote the present chapter and the following to the one-fluid case. The rule of the game we have fixed for this purpose is the following.

This will be useful below. 49) where Pn is the orthogonal projector in H onto Span (w1 , . . 37). We now make use of the fact mentioned above: in addition to being of norm less than one as an operator on H, Pn is also of norm less than one when considered as an operator on V . The latter holds because of the special choice of the functions wj . From this, we deduce that the adjoint operator of Pn , that is Pn itself since Pn is obviously self-adjoint, is an operator of norm less than one when considered on V ′ .

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