## Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density.(English)Zbl 0937.35134

The author investigates the Navier-Stokes equations in $$\mathbb{R}^d$$, $$d\geq 1$$, equations for a compressible, heat-conducting, isotropic Newtonian fluid. The main result states that a smooth solution cannot exist globally in time if the diameter of the support of the density grows sublinearly in time and if the entropy remains bounded from below. Then it is shown that this assumption is fulfilled if the density has compact support initially. The elegant proof rests on the fact that the total pressure $$\int p(x, t) dx$$ decays faster if the density has compact support. The assumption on the density is decisive, and the results are in contrast to the classical theory of small solutions that exist for all time if the initial state is close to a constant, where in particular the density is close to a constant $$\rho_0>0$$.

### MSC:

 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35B40 Asymptotic behavior of solutions to PDEs
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