IMMEDIATE BLOWUP OF ENTROPY-BOUNDED CLASSICAL SOLUTIONS TO THE VACUUM FREE BOUNDARY PROBLEM OF NONISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of nonisentropic compressible Navier--Stokes equations. The vis-cosities and the heat conductivity could be constants or, more physically, the degenerate, temperature -dependent functions which vanish on the vacuum boundary (i.e., \mu = \mu \=\theta\alpha, \lambda = constants 0 \leq \alpha \leq 1/(- -1), \mu =\ > 0, 2\mu =\+ n\lambda \geq 0, \=appa\k \geq 0, and adiabatic exponent -> 1). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that (1) for three-dimensional spherically symmetric flows with nonvanishing bulk viscosity and zero heat con-ductivity , 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 nonvanishing heat conductivity , the normal derivative of the temperature of the classical solution across the free boundary does not degenerate , and therefore the entropy immediately blows up if the decaying rate of the initial density is not of 1/(- -1) power of the distance function to the boundary; (3) for one-dimensional flow with zero heat conductivity , the nonexistence result is similar but needs more restrictions on the decaying rate. Together with our previous results on local or global entropy-bounded classical solutions [SIAM J. Math. Anal., 51 (2019), pp. 748--789; Math. Models Methods Appl. Sci., 29 (2019), pp. 2271-2320], 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. \lambda \theta\alpha, \kappa = =\kappa \\theta\alpha for