RENDICONTI DEL SEMINARIO MATEMATICO

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 1 (2007)Subalpine Rhapsody in DynamicsJ. Fan - S. Jiang †ZERO SHEAR VISCOSITY LIMIT FOR THE NAVIER-STOKESEQUATIONS OF COMPRESSIBLE ISENTROPIC FLUIDSWITH CYLINDRIC SYMMETRY ∗Abstract. We study the problem of the limit process as the shear viscosity goes to zero forglobal weak solutions to the Navier-Stokes equations of compressible isentropic fluids withcylindric symmetry between two circular cylinders. We prove that the limit of the globalweak solutions is a weak solution of the corresponding system with zero shear viscosity.1. IntroductionWe shall study the convergence of solutions of the the Navier-Stokes equations fora compressible isentropic fluid with cylindric symmetry, as the shear viscosity goesto zero. In this paper we restrict ourselves to isentropic flows between two circularcoaxial cylinders and assume that the motion of the flows depends only on the radialvariable and the time variable. The corresponding symmetric form of the compressibleisentropic Navier-Stokes equations, which express the conservation of mass and thebalance of momentum, can be written as [35](1) ρ t + (ρu) x + ρux = 0,(ρu) t + (ρu 2 ) x + ρu2(2)(3)(4)x(ρv) t + (ρuv) x + 2ρuvx(ρw) t + (ρuw) x + ρuwx(+ P x = (λ + 2ǫ) u x + u )x− ρv2x(= ǫ v x + v ),x x(= ǫ w xx + w xxHere ρ is the density, u, v and w are the radial, angular and axial components of thevelocity vector ⃗v, respectively, x is the radial variable;).P ≡ P(ρ) = aρ γ , γ > 1denotes the pressure, and λ, ǫ, a and γ are positive constants which stand for the bulk(expansion) viscosity coefficient, shear viscosity coefficient, gas constant and specificheat ratio, respectively.x,∗ Supported by the NSFC (Grant No. 10301014, 10225105) and the National Basic Research Program(Grant No. 2005CB321700) of China.† Corresponding author.35