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The zero dissipation limit of the compressible heat-conducting NavierStokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient and the viscosity coefficient satisfy = O (), = O (), c> 0, as 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the NavierStokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of . The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].

In this paper, we study a class of analytical solutions to the 3-D compressible Navier-Stokes equations with density-dependent viscosity coefficients, where the shear viscosity h ( ) = 0 , and the bulk viscosity g ( ) = ( > 0 ). By constructing a class of radial symmetric and self-similar analytical solutions in R N ( N 2 ) with both the continuous density condition and the stress free condition across the free boundaries separating the fluid from vacuum, we derive that the free boundary expands outward in the radial direction at an algebraic rate in time and we also have shown that such solutions exhibit interesting new information such as the formation of a vacuum at the center of the symmetry as time tends to infinity and explicit regularities, and we have large time decay estimates of the velocity