报告摘要：
| The entropy is one of the fundamental states of a fluid and, in the viscous case, the equation that it satisfies is both degenerate and singular in the region close to the vacuum. In spite of its importance in the gas dynamics, the mathematical analyses on the behavior of the entropy near the vacuum region, were rarely carried out; in particular, in the presence of vacuum, either at the far field or on the physical boundaries, it was unknown if the entropy remains its boundedness. It will be shown in this talk that the ideal gases retain their uniform boundedness of the entropy, locally or globally in time, for both the Cauchy problem and the initial-boundary value problems, if the vacuum occurs only at the far field or on the physical boundary, as long as the initial density behaves well at the far field or near the boundary. For the Cauchy problem, the density is required to decay slowly enough at the far field, while for the initial-boundary value problem, the $(gamma-1)$-th power of density is required to be equivalent to the distance to the boundary, near the boundary.
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