RENDICONTI DEL SEMINARIO MATEMATICO

Zero shear viscosity limit in compressible isentropic fluids 49which implies v 2 ≤ v 2 a.e. in Q T , since φ t can be nonpositive and arbitrary. Consequently,(64) v ǫ → v strongly in L 2 (Q T ).Hence, it follows from (63) and the form of (61) in Lagrangian coordinates that〈ǫxvx 2 ,φ〉 = 0.Again multiplying (2) by 2xu ǫ φ with φ ∈ C ∞ (Q T ), φ ≥ 0 and φ(·, T) = 0,then integrating over (0, T) × , integrating by parts and employing the boundarycondition for u, one gets after a straightforward calculation that(65)∫ T0∫= 2(λ + 2ǫ)∫xρ ǫ u 2 ǫ (φ t + u ǫ φ x )dxdt +∫ T0∫+ u2 }ǫx φ + u2 ǫ φ x dxdt − 2xρ 0 u 2 0φ(x, 0)dx{x(∂ x u ǫ ) 2 φ + 2u ǫ ∂ x u ǫ φ + xu ǫ ∂ x u ǫ φ x∫ T0∫{ρ ǫ u ǫ v 2 ǫ φ + P(ρ ǫ)(xu ǫ φ) x}dxdt.Letting ǫ → 0 in (65) and making use of (24), (31), (45) and (64), we arrive at(66)∫ T0∫∫xρu 2 (φ t + uφ x )dxdt +∫ T ∫= 2〈(λ + 2ǫ)xu 2 x ,φ〉 + 2λ+ u2x φ + u2 φ x}dxdt − 20 ∫ T ∫0xρ 0 u 2 0φ(x, 0)dx{2uu x φ + xuu x φ x{ρuv 2 φ + P(ρ)(xuφ) x}dxdt,where (λ + 2ǫ)xu 2 x , the weak limit of (λ + 2ǫ)x(∂ xu ǫ ) 2 , is a nonnegative Radon measure.Now, multiplying (47) by 2xuφ in L 2 (Q T ) with the same φ as in (65), andrecalling v 2 = v 2 , we find, in the same manner as in (65), that(67)∫ T0∫∫xρu 2 (φ t + uφ x )dxdt +∫ T ∫= 2〈λxu 2 x ,φ〉 + 2λ0+ u2x φ + u2 φ x}dxdt − 2Combining (66) with (67), we concludexρ 0 u 2 0φ(x, 0)dx{2uu x φ + xuu x φ x∫ T0∫{ρuv 2 φ + P(ρ)(xuφ) x}dxdt.〈λxu 2 x ,φ〉 = 〈(λ + ǫ)xu2 x ,φ〉 ≥ 〈λxu2 x ,φ〉, ∀φ ∈ C∞ (Q T ), φ ≥ 0, φ(·, T) = 0,