RENDICONTI DEL SEMINARIO MATEMATICO

42 J. Fan - S. Jiang3. Proof of Theorem 1In this section we pass to the limit for (ρ ǫ , u ǫ ,v ǫ ,w ǫ ) as ǫ → 0 in (1)–(6). First,it is easy to see by the uniform a priori estimates established in the last section andLemma C.1 in [38] that one can extract a subsequence of (ρ ǫ , u ǫ ,v ǫ ,w ǫ ), still denotedby (ρ ǫ , u ǫ ,v ǫ ,w ǫ ) for simplicity, such that as ǫ → 0,(22)(23)(24)(25)(26)ρ ǫ ⇀ ρ weak- ∗ in L ∞ (Q T ), ρ ≤ ρ(x, t) ≤ ρ, a.e.,ρ ǫ → ρ in C([0, T], L γ () − w) for any γ > 1,u ǫ ⇀ u weak- ∗ in L ∞ (0, T; L 2 ())and weakly in L 2 (0, T; H0 1 ()) ∩ L6 (Q T ),(v ǫ ,w ǫ ) ⇀ (v,w) weak- ∗ in L ∞ (Q T ),(ǫ∂ x u ǫ ,ǫ∂ x v ǫ ,ǫ∂ x w ǫ ) → (0, 0, 0) strongly in L 2 (Q T ),and from (23) and the Sobolev compact imbedding theorem, one gets(27) ρ ǫ → ρ in C([0, T], H −1 ()).Using (22)–(24), (21) and Lemma 5.1 in [39], we find that(28)ρ ǫ u ǫ ⇀ ρu weak- ∗ in L ∞ (0, T; L 2 ()) andweakly in L 2 (0, T; L p ()) for all p > 1,and by Lemma C.1 in [38],ρ ǫ u ǫ ⇀ ρu in C([0, T], L 2 () − w),from which and the Sobolev compact imbedding theorem, it follows thatρ ǫ u ǫ ⇀ ρu in C([0, T], H −1 ()).Hence, the above weak convergence together with (24) results in(29) ρ ǫ u 2 ǫ ⇀ ρu2 weakly in L 2 (Q T ).On the other hand, noticing that by virtue of Lemma 1 and the Sobolev imbeddingtheorem, ∫ T0 ‖u ǫ(t)‖ 2 L ∞ dt ≤ C. Therefore,∫ Twhich together with (27) implies(30)∣∫ T00‖u 2 ǫ (t)‖ H 1dt ≤ C uniformly in ǫ,∫∣ ∣∣∣(ρ ǫ − ρ)u 2 ǫ φdxdt → 0 as ǫ → 0, φ ∈ C0 ∞ (Q T).