RENDICONTI DEL SEMINARIO MATEMATICO

Zero shear viscosity limit in compressible isentropic fluids 41andsupt∈[0,T]∫∣ϕ(x, t)dx∣ ≤ C.Hence, the generalized Poincaré inequality implies(18) ‖ϕ ǫ ‖ L ∞ (Q T ) ≤ C.Utilizing the equations (17), and the estimates (16) and (18), following the samearguments as in [17, Lemma 2.2], we obtain the following lemma, the proof of whichis therefore omitted.LEMMA 2. There are positive constants ρ, ρ independent of ǫ, such that(19) ρ ≤ ρ ǫ (x, t) ≤ ρ ∀ x ∈ ¯, t ≥ 0.As a consequence of Lemmas 1 and 2, one has by the Cauchy-Schwarz inequalitythat(20)∫ T0‖u ǫ ‖ 6 L 6 dt ≤ ‖u ǫ ‖ 2 L ∞ (0,T;L 2 )≤ C≤ C≤ C.∫ T0∫ T0∫ T0‖u ǫ ‖ 4 L ∞dt( ∫ r 2 ) 2dt|u ǫ ∂ x u ǫ |dξr 1‖u ǫ ‖ 2 L 2 ‖∂ x u ǫ ‖ 2 L 2 dtNow, one can apply Lemma 1 and (19) to the parabolic equations (2)–(4) toobtain bounds on the time derivative of (ρ ǫ ,ρ ǫ ⃗v ǫ ):LEMMA 3.(21) ‖∂ t ρ ǫ ‖ L ∞ (0,T;H −1 ) + ‖∂ t(ρ ǫ ⃗v ǫ )‖ L 2 (0,T;H −1 ) ≤ C,where ⃗v ǫ = (u ǫ ,v ǫ ,w ǫ ).The following lemma gives us uniform bounds of (w ǫ ,v ǫ ) in L ∞ -norm, theproof of which is based on using the properties of transport equations and Lemmas 1and 2, and can be found in [17].LEMMA 4.(‖v ǫ ‖ L ∞ (Q T ) ≤ ‖v 0 ‖ L ∞ () exp C≤ C‖v 0 ‖ L ∞ (),‖w ǫ ‖ L ∞ (Q T ) ≤ ‖w 0 ‖ L ∞ ().∫ T0)‖u ǫ ‖ L ∞ ()dt