##
**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.

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.

Reviewer: Marco Romito (Firenze)

### MSC:

35Q30 | Navier-Stokes equations |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

35B40 | Asymptotic behavior of solutions to PDEs |