Abstract We are concerned with the time-asymptotic behavior toward rarefaction waves for strong non-vacuum solutions to the Cauchy problem of the one-dimensional compressible Navier-Stokes equations with degenerate density-dependent viscosity. The case when the pressure p ( ρ ) = ρ γ and the viscosity coefficient μ ( ρ ) = ρ α for some parameters α , γ ∈ R is considered. For α ≥ 0 , γ ≥ max ⁡ { 1 , α } , if the initial data is assumed to be sufficiently regular, without vacuum and mass concentrations, we show that the Cauchy problem of the one-dimensional compressible Navier-Stokes equations admits a unique global strong non-vacuum solution, which tends to the rarefaction waves as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction waves can be arbitrarily large. The proof is established via a delicate energy method and the key ingredient in our analysis is to derive the uniform-in-time positive lower and upper bounds on the specific volume.

中文翻译：

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具有密度相关粘度的一维可压缩 Navier-Stokes 方程的渐近性

摘要 我们关注对具有退化密度依赖粘度的一维可压缩纳维-斯托克斯方程的柯西问题的强非真空解的稀疏波的时间渐近行为。考虑压力 p ( ρ ) = ρ γ 和粘度系数 μ ( ρ ) = ρ α 对于某些参数 α , γ ∈ R 的情况。对于 α ≥ 0 , γ ≥ max ⁡ { 1 , α } ，如果假设初始数据足够规则，没有真空和质量浓度，我们表明一维可压缩纳维-斯托克斯方程的柯西问题承认独特的全局强非真空解，随着时间的推移趋于无穷大，趋于稀疏波。这里的初始扰动和稀疏波的强度都可以任意大。