## Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow.(English)Zbl 0946.35058

The authors study the large time behaviour of weak solutions of Navier-Stokes equations for a compressible isotropic flow in a three dimensional domain $$\Omega$$. The system is forced by a term like $$\rho(t)\nabla F$$, where the potential $$F$$ is a bounded Lipschitz continuous function in $$\overline\Omega$$ such that $$\{x |F(x)>k\}$$ is connected, and with a control of the growth at infinity. The pressure has the form $p(\rho)=a\rho^\gamma$ (or at least has the same behaviour) with $$\gamma>1$$ if $$\Omega$$ is bounded and $$\gamma>{3\over 2}$$ if $$\Omega$$ is unbounded.
The main result is that, given any weak solution $$(\rho,u)$$, where $$\rho$$ is the density scalar field and $$u$$ is the velocity vector field, there exists a stationary state $$\rho_s$$, that is a solution $\nabla p(\rho_s)=\rho_s\nabla F$ such that $$\rho(t)\to\rho_s$$ and $$u\to 0$$ as $$t\to\infty$$, and the convergences hold in suitable spaces.
The stationary state $$\rho_s$$ is uniquely determined by the asymptotic energy $$E_\infty$$, which is the limit, for $$t\to\infty$$, of $E(t)={1\over 2}\int_\Omega \rho(t)|u(t)|^2+{a\over{\gamma -1}}\rho(t)^\gamma-\rho(t) F.$
In fact, even if the mass $m=\int_\Omega \rho(t)$ is a conserved quantity along the dynamics, in the limit there can be a loss of mass at infinity (obviously, if $$\Omega$$ is bounded, this cannot happen). The limit stationary state is characterized as the maximal stationary state of mass less than $$m$$. This state realizes $$E_\infty$$ as its energy.

### MSC:

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