The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System

and Applied Analysis 3 For any Banach space B, the space Lp(0, T;B) consists of all strongly measurable functions u : [0, T] → B equipped with the norm ‖u‖ L p (0,T;B) := (∫ T 0 ‖u (t)‖PBdt) 1/p < ∞ (12) for 1 ≤ p < ∞, and ‖u‖ L ∞ (0,T;B) = ess sup t∈[0,T] ‖u(t)‖B < ∞. (13) And the space C([0, T];B) denotes the set of continuous functions u : [0, T] → B with ‖u‖ C([0,T];B) = max t∈[0,T] ‖u(t)‖B < ∞. (14) In this paper, the letter C is a generic constant and its value may change at each appearance. Moreover, every C is independent of the parameters ] and κ. 2. Proof of Theorem 1 In this section, we present the proof of Theorem 1. To this goal, we need the following calculus inequality, the proof of which can be found in [18, 19]. Lemma 3. Assume that s > 0 and p ∈ (1, +∞). If f, g ∈ S(R), the Schwartz class, then 󵄩󵄩󵄩󵄩J s (fg) − f (J s g) 󵄩󵄩󵄩󵄩Lp ≤ C ( 󵄩󵄩󵄩󵄩∇f 󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Hs−1,p2 + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Hs,p3 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Lp4 ) , (15) 󵄩󵄩󵄩󵄩J s (fg) 󵄩󵄩󵄩󵄩Lp ≤ C ( 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Lp1 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Hs,p2 + 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩Hs,p3 󵄩󵄩󵄩󵄩g 󵄩󵄩󵄩󵄩Lp4 ) (16) with p 2 , p 3 ∈ (1, +∞) such that