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Stability of planar stationary solutions to the compressible Navier-Stokes equation on the half space. (English) Zbl 1117.35062
The authors consider the large-time behavior of solutions to the compressible Navier-Stokes equation: $$\partial_t\rho+\text{div}(\rho u)=0$$, $$\partial_t(\rho u)+ \text{div}(\rho u\otimes u)+\nabla p(\rho)= u\Delta u+(\mu+\mu')\nabla \text{div}\,u$$, and $$p(\rho)=K\rho^\gamma$$, $$\gamma>1$$, on the half-space $${\mathbb R}_+^n$$ for $$n\geq 2$$, under the initial condition $$(\rho,u)| _{t=0}=(\rho_0,u_0)$$ and the outflow boundary condition $$u| _{x_1=0}=(u_b^1,0,\dots,0)$$, where $$u_b^1<0$$ is a constant, together with the boundary condition $$\rho\rightarrow\rho_+$$, $$u\rightarrow(u_b^1,0,\dots,0)$$ as $$x_1\rightarrow\infty$$, where $$\rho_+>0$$ is a constant. It is shown by means of an a priori estimate that the planar stationary solution is stable with respect to small perturbations in $$H^s({\mathbb R}^n)$$ with $$s\geq[n/2]+1$$ and the perturbations decay in the $$L^\infty$$ norm as $$t\rightarrow\infty$$ provided the perturbations are sufficiently small.

##### MSC:
 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 76E30 Nonlinear effects in hydrodynamic stability 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
a priori estimate; stability; large-time behavior
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##### References:
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