Non-existence of strong solutions to the two-dimensional radially symmetric compressible Navier–Stokes equations

Under the assumption that the smooth initial data are of small total energy, Li–Xin [Ann. PDE 5 (2019), p. 37] estabished the global existence of strong solutions in homogeneous Sobolev space (without the information of ‖ u ‖ L 2 \|\mathbf {u\|_{L^2}} ) to the Cauchy problem of two-dimensional isentropic compressible Navier–Stokes equations with vacuum as far field density. In particular, the initial density can even have compact support. On the other hand, Luo [Math. Methods Appl. Sci. 37 (2014), pp. 1333–1352] showed that the two-dimensional radially symmetric isentropic compressible Navier–Stokes equations has no non-trivial global strong solutions in the inhomogeneous Sobolev space if the initial density is compactly supported. In this paper, we are concerned with the well-posedness of strong solutions to the Cauchy problem of two-dimensional radially symmetric isentropic compressible Navier–Stokes equations, and prove that the strong solution does not exist in the inhomogeneous Sobolev space for any short time provided that the smooth initial data are of small total energy.