Compressible Navier-Stokes system in 1-D (Q2747410)

The author proves the global existence of uniformly bounded solutions to the model of the motion of a barotropic viscous compressible fluid in the one-dimensional case with a free boundary given by an initial-boundary value problem for the Navier-Stokes system: \(\varrho (u_t+uu_r)+p_r=\mu u_{rr}-\varrho f, \varrho_t+(\varrho u)_r=0;\) \(u|_{r=0}=0, (\mu u_r-p)|_{S(t)}=-P;\) \(u|_{S(t)}=S'(t);\) \(u(r,0)=u_0(r), \varrho (r,0)=\varrho_0(r)\), where \(\varrho\), \(u\), \(p\), \(f\), \(\mu\) and \(P\) are the density of the fluid, the velocity, the pressure, the external force acting on the fluid, the positive constant viscosity coefficient and the external constant pressure, respectively, and \(S(t)\) is the free boundary. It is assumed that \(p(\varrho)=a\varrho^{\gamma}\) with constants \(a>0\) and \(\gamma >0\), and that \(S(0)=1\). The author shows, moreover, that those solutions tend to the stationary solution as \(t\to\infty\).