For the three-dimensional vacuum free boundary problem with physical singularity where the sound speed is $$C^{1/2}$$ C 1 / 2 -Hölder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding porous media equations simplified via Darcy’s law. It is proved that if the initial perturbation is of the size of $$\varepsilon $$ ε , then the existing time for smooth solutions is at least of the order of $$\exp (\varepsilon ^{-2/3})$$ exp ( ε - 2 / 3 ) . The key issue for the analysis is the slow sub-linear growth of vacuum boundaries of the order of $$t^{1/(3\gamma -1)}$$ t 1 / ( 3 γ - 1 ) , where $$\gamma >1$$ γ > 1 is the adiabatic exponent for the gas. This is in sharp contrast to the currently available global-in-time existence theory of expanding solutions to the vacuum free boundary problems with physical singularity of compressible Euler equations for which the expanding rate of vacuum boundaries is linear. The results obtained in this paper are closely related to the open question in multiple dimensions framed by T.-P. Liu’s construction of particular solutions in 1996.

中文翻译：

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巴伦拉特解周围物理真空中带阻尼的三维等熵无粘性流的几乎全局解

对于声速为 $$C^{1/2}$$ C 1 / 2 -Hölder 连续跨越带阻尼的可压缩欧拉方程的真空边界的具有物理奇点的三维真空自由边界问题，没有任何对称性假设，当初始数据是通过达西定律简化的相应多孔介质方程的 Barenblatt 自相似解的小扰动时，我们证明了平滑解的几乎全局存在。证明如果初始扰动的大小为 $$\varepsilon $$ ε ，那么平滑解的存在时间至少为 $$\exp (\varepsilon ^{-2/3}) $$ exp ( ε - 2 / 3 ) 。分析的关键问题是 $$t^{1/(3\gamma -1)}$$ t 1 / ( 3 γ - 1 ) 量级的真空边界的缓慢次线性增长，其中 $$ \gamma >1$$ γ > 1 是气体的绝热指数。这与当前可用的全局时间存在理论形成鲜明对比，即扩展真空自由边界问题的解的可压缩欧拉方程的物理奇异性，其中真空边界的扩展率是线性的。本文得到的结果与T.-P框定的多维开放问题密切相关。Liu 于 1996 年构建的特定解。本文得到的结果与T.-P框定的多维开放问题密切相关。Liu 于 1996 年构建的特定解。本文得到的结果与T.-P框定的多维开放问题密切相关。Liu 于 1996 年构建的特定解。