The stability of one dimensional stationary flows of compressible viscous fluids (Q749397)

The paper proves that every stationary solution of the problem \[ u_ t+uu_ x-\mu \rho^{-1}u_{xx}+\pi (\rho)_ x=f(x), \] \[ \rho_ t+(u\rho)_ x=0,\quad (t,x)\in (0,+\infty)\times (0,1), \] \[ u(t,0)=u(t,1)=0,\quad \int^{1}_{0}\rho (t,x)dx=1 \] is stable. The main theorem involves the existence and uniqueness of the global solution (u(t),\(\rho\) (t)) which satisfies the estimates \(c_ 1\leq \rho (t,x)\leq c_ 2\) for all \((t,x)\in [0,+\infty)\times (0,1)\) and \[ \| u(t)\|^ 2_ 1+\| \rho (t)-\eta \|^ 2_ 1\leq c_ 3(\| u_ 0\|^ 2_ 1+\| \rho_ 0-\eta \|^ 2_ 1)e^{-c_ 4t} \] for all \(t\geq 0\), where \(\eta\) (x) is the stationary solution.