In this paper we study a free boundary problem for the viscous, compressible, heat conducting, onedimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ) = ρ α , where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with α ∈ (0, 1 2 ] for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.

Both the global well-posedness for large data and the vanishing shear viscosity limit with a boundary layer to the compressible Navier-Stokes system with cylindrical symmetry are studied under a general condition on the heat conductivity coefficient that, in particular, includes the constant coefficient. The thickness of the boundary layer is proved to be almost optimal. Moreover, the optimal L 1 convergence rate in terms of shear viscosity is obtained for the angular and axial velocity components.

A free-boundary problem is studied for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity that decreases (to zero) with decreasing density, i.e., µ = Aρ θ , where A and θ are positive constants. The existence and uniqueness of the global weak solutions are obtained with θ ∈ (0, 1], which improves the previous results and no vacuum is developed in the solutions in a finite time provided the initial data does not contain vacuum.

No abstract

A global existence result is established for a free boundary problem of planar magnetohydrodynamic fluid flows with radiation and large initial data. Particularly, it is novelty to embrace the constant transport coefficient. As a by-product, the free boundary is shown to expand outward at an algebraic rate from above in time.