论文标题
立即爆炸了非等分压缩的Navier的真空无效边界问题的熵结合的经典解决方案
Immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations
论文作者
论文摘要
本文考虑了对非异形可压缩Navier-Stokes方程的真空无效边界问题的熵结合的经典解决方案的立即爆炸。 The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $ \barμ> 0，〜2 \barμ+n \barλ\ geq0，〜\barκ\ geq 0 $，以及绝热指数$γ> 1 $）。 随着跨真空边界的初始密度的规定衰减速率，我们证明：（1）对于三维球体对称流，具有非散装的体积粘度和零热电导率，如果在任何小时间内，熵构成的经典溶液都不存在，前提是初始速度在边界附近扩展； （2）对于三维球体对称流，具有非散热热电导率，跨自由边界的经典溶液温度的正常导数不会退化，因此，如果初始密度的衰减速率不超过$ 1/（γ-1），则熵立即爆炸。 （3）对于零热电导率的一维流量，不存在的结果相似，但需要对衰减速率进行更多限制。 Together with our previous results on local or global entropy-bounded classical solutions (Liu and Yuan, SIAM J. Math. Anal. (2) 51, 2019; Liu and Yuan, Math. Models Methods Appl. Sci. (9) 12, 2019), this paper shows the necessity of proper degenerate conditions on the density and temperature across the boundary for the well-posedness of the entropy-bounded classical solutions to the vacuum boundary problem of the viscous 气体。
This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., $μ=\barμ θ^α, ~ λ=\barλ θ^α,~ κ=\barκ θ^α$, for constants $0\leq α\leq 1/(γ-1)$, $\barμ>0,~2\barμ+n\barλ\geq0,~\barκ\geq 0$, and adiabatic exponent $γ>1$). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1) for three-dimensional spherically symmetric flows with non-vanishing bulk viscosity and zero heat conductivity, entropy-bounded classical solutions do not exist for any small time, provided the initial velocity is expanding near the boundary; (2) for three-dimensional spherically symmetric flows with non-vanishing heat conductivity, the normal derivative of the temperature of the classical solution across the free boundary does not degenerate, and therefore the entropy immediately blowups if the decaying rate of the initial density is not of $1/(γ-1)$ power of the distance function to the boundary; (3) for one-dimensional flow with zero heat conductivity, the non-existence result is similar but need more restrictions on the decaying rate. Together with our previous results on local or global entropy-bounded classical solutions (Liu and Yuan, SIAM J. Math. Anal. (2) 51, 2019; Liu and Yuan, Math. Models Methods Appl. Sci. (9) 12, 2019), this paper shows the necessity of proper degenerate conditions on the density and temperature across the boundary for the well-posedness of the entropy-bounded classical solutions to the vacuum boundary problem of the viscous gas.
