RENDICONTI DEL SEMINARIO MATEMATICO

Zero shear viscosity limit in compressible isentropic fluids 39(13)∫ T0∫} ∫{xρwφ t + xρuwφ x − ǫxw x φ x dxdt + xρ 0 w 0 φ(x, 0)dx = 0,for any ϕ,φ ∈ C 1 ( ¯Q T ), φ ∈ C([0, T], H0 1 ) and ϕ(·, T) = φ(·, T) = 0.ii) We call (ρ, u,v,w)(x, t) a global weak solution of (1)–(6) with ǫ = 0, if for anyT > 0, ρ(x, t) ≥ 0 on [0, T] × , andρ,v,w ∈ L ∞ (Q T ), u ∈ L ∞ (0, T; L 2 ) ∩ L 2 (0, T; H 1 0 ),and ρ, u,v,w satisfy the equations (10)–(13) with ǫ = 0.Thus, the main result of this paper reads:THEOREM 1. Assume that the initial data satisfy (9). Then there exists a globalweak solution (ρ ǫ , u ǫ ,v ǫ ,w ǫ ) of the problem (1)–(6). Moreover, there is a sequenceǫ n ↓ 0, such that as ǫ n → 0,(ρ ǫn ,v ǫn ,w ǫn ) → (ρ,v,w) strongly in L p (Q T ), u ǫn → u strongly in L s (Q T ),∂ x u ǫn → u x strongly in L 2 (Q T )for any p ∈ [1,∞) and s ∈ [1, 6). In addition, the limit (ρ, u,v,w) is a global weaksolution of (1)–(6) with ǫ = 0.REMARK 1. (i) If inf ρ 0 = 0 and (ρ 0 , u 0 ) satisfies a natural compatibility condition,then we can prove that the problem (1)–(6) has a unique global smooth solution.For the proof, see [7] when γ ≥ 2 and [13] when 1 < γ ≤ 2.(ii) A similar result has been obtained recently for the magnetohydrodynamicequations by Fan [12].The next section gives the uniform estimates which will be used in the finalsection to complete the proof of Theorem 1.As the end of this section, we introduce the notation used throughout this paper.L p (I, B) respectively ‖ · ‖ L p (I,B) denotes the space of all strongly measurable, pthpowerintegrable (essentially bounded if p = ∞) functions from I to B respectively itsnorm, I ⊂ R an interval, B a Banach space. C(I, B − w) is the space of all functionswhich are in L ∞ (I, B) and continuous in t with values in B endowed with the weaktopology. We will use the abbreviation:L q (0, T; W m,p ) ≡ L q (0, T; W m,p ()),‖ · ‖ L q (0,T;W m,p ) ≡ ‖ · ‖ L q (0,T;W m,p ()),‖ · ‖ L p ≡ ‖ · ‖ L p ().on ǫ.The same letter C will denote various positive constants which do not depend2. Uniform a priori estimatesWe denote the weak solution of (1)–(6) by (ρ ǫ , u ǫ ,v ǫ ,w ǫ ) throughout the rest of thispaper. This section is devoted to the derivation of a priori estimates of (ρ ǫ , u ǫ ,v ǫ ,w ǫ )