Stability of boundary layers for the inflow compressible Navier-Stokes equations (Q713388)

The 1-D compressible Navier-Stokes equations with small viscosity coefficient \(\varepsilon\) \[ \frac{\partial \varrho}{\partial t}+\frac{\partial (\varrho u)}{\partial x}=0,\quad \frac{\partial (\varrho u)}{\partial t}+ \frac{\partial }{\partial x}(\varrho u^2+p)=\varepsilon\frac{\partial^2 u}{\partial x^2} \tag{1} \] are studied together with the 1-D compressible Euler equations \[ \frac{\partial \varrho}{\partial t}+\frac{\partial (\varrho u)}{\partial x}=0,\quad \frac{\partial (\varrho u)}{\partial t}+u \frac{\partial u}{\partial x}+a\gamma\varrho^{\gamma-2}\frac{\partial \varrho}{\partial x}=0 \tag{2} \] in the domain \(\{(x,t):x>0,t>0\}\). Here \(\varrho\) is the density, \(u\) is the velocity and \(p=a\varrho^\gamma\) is the pressure, \(\gamma>1\). Systems (1) and (2) are supplied by the same initial conditions. A solution to the system (1) satisfies the inflow boundary condition on \(\{x=0\}\). It is supposed that the characteristics of system (2) \[ \lambda_1=u-c,\quad \lambda_2=u+c,\quad \text{where}\;c=\sqrt{a\gamma\varrho^{\gamma-1}} \] satisfy the inequalities \(\lambda_1<0,\quad \lambda_2>0\) on the boundary \(\{x=0\}\) up to some small time. The authors prove the existence of the boundary layers and that away from the boundary the solution of (1) approximates the solution to the Euler system (2) at an optimal rate in \(\varepsilon\).