Non-existence of Classical Solutions with Finite Energy to the Cauchy Problem of the Compressible Navier-Stokes Equations

The well-posedness of classical solutions with finite energy to the compressible Navier-Stokes equations (CNS) subject to arbitrarily large and smooth initial data is a challenging problem. In the case when the fluid density is away from vacuum (strictly positive), this problem was first solved for the CNS in either one-dimension for general smooth initial data or multi-dimension for smooth initial data near some equilibrium state (that is, small perturbation) (Antontsev etal. in Boundary value problems in mechanics of nonhomogeneous fluids, North-Holland Publishing Co., Amsterdam, 1990; Kazhikhov in Sibirsk Mat Zh 23:60-64, 1982; Kazhikhov etal. in Prikl Mat Meh 41:282-291, 1977; Matsumura and Nishida in Proc Jpn Acad Ser A Math Sci 55:337-342, 1979, J Math Kyoto Univ 20:67-104, 1980, Commun Math Phys 89:445-464, 1983). In the case that the flow density may contain a vacuum (the density can be zero at some space-time point), it seems to be a rather subtle problem to deal with the well-posedness problem for CNS. The local well-posedness of classical solutions containing a vacuum was shown in homogeneous Sobolev space (without the information of velocity in L-2-norm) for general regular initial data with some compatibility conditions being satisfied initially (Cho etal. in J Math Pures Appl (9) 83:243-275, 2004; Cho and Kim in J Differ Equ 228:377-411, 2006, Manuscr Math 120:91-129, 2006; Choe and Kim in J Differ Equ 190:504-523 2003), and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations (Huang etal. in Commun Pure Appl Math 65:549-585, 2012). However, it was shown that any classical solutions to the compressible Navier-Stokes equations in finite energy (inhomogeneous Sobolev) space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported (Xin in Commun Pure Appl Math 51:229-240, 1998). In this paper, we investigate the well-posedess of classical solutions to the Cauchy problem of Navier-Stokes equations, and prove that the classical solution with finite energy does not exist in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum. This implies, in particular, that the homogeneous Sobolev space is as crucial as studying the well-posedness for the Cauchy problem of compressible Navier-Stokes equations in the presence of a vacuum at far fields even locally in time.