Immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations

This paper considers the immediate blowup of classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier--Stokes equations, where 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., $\mu=\bar{\mu} \theta^{\alpha}, ~ \lambda=\bar{\lambda} \theta^{\alpha},~ \kappa=\bar{\kappa} \theta^{\alpha}$, for constants $0\leq \alpha\leq 1/(\gamma-1)$, $\bar{\mu}>0,~2\bar{\mu}+n\bar{\lambda}\geq0,~\bar{\kappa}\geq 0$, and adiabatic exponent $\gamma>1$). In our previous study (Liu and Yuan, Math. Models Methods Appl. Sci. (9) 12, 2019), with three-dimensional spherical symmetry and constant shear viscosity, vanishing bulk viscosity and heat conductivity, we established a class of global-in-time large solutions, with bounded entropy and entropy derivatives, under the condition the decaying rate of the initial density to the vacuum boundary is of any positive power of the distance function to the boundary. In this paper we prove that such classical solutions do not exist for any small time for non-vanishing bulk viscosity, provided the initial velocity is expanding near the boundary. When the heat conductivity does not vanish, it is automatically satisfied that the normal derivative of the temperature of the classical solution across the free boundary does not degenerate; meanwhile, the entropy of the classical solution immediately blowups if the decaying rate of the initial density is not of $1/(\gamma-1)$ power of the distance function to the boundary. The proofs are obtained by the analyzing the boundary behaviors of the velocity, entropy and temperature, and investigating the maximum principles for parabolic equations with degenerate coefficients