# zbMATH — the first resource for mathematics

Resolvent estimates for the linearized compressible Navier-Stokes equation in an infinite layer. (English) Zbl 1180.35413
The author deals with the following resolvent problem: (1) $$(\lambda+L) u=f$$ in an infinite layer $$\Omega=\mathbb{R}^{n-1}x(0,a)$$, $$n\geq 2$$, where $$\lambda \in\mathbb{C}$$ is a parameter, $$f=f(x)$$ is a given function with values in $$\mathbb{R}^{n+1}$$, $$u=^T(\varphi,m)$$ is the unknown function with $$\varphi= \varphi (x)\in\mathbb{R}$$ and $$m=^T(m^1(x),\dots,m^n(x))\in\mathbb{R}^n$$, and $$L$$ is an operator defined by $L=\left(\begin{matrix} 0 & \gamma\text{div} \\ \gamma\nabla & -\nu\Delta I_n-\widetilde\nu\nabla\text{div}\end{matrix}\right)$ with positive constants $$\nu$$ and $$\gamma$$ and a nonnegative constant $$\widetilde \nu$$. Here $$x=^T(x',x_n)\in\Omega$$ with $$x'\in\mathbb{R}^{n-1}$$, $$x_n\in (0,a)$$; the superscript $$T(x',x_n)$$ stands for the transposition; $$I_n$$ is the $$(n\times n)$$ identity matrix; and $$\text{div},\nabla$$ and $$\Delta$$ are the usual divergence, gradient and Laplacian with respect to $$x$$. The author considers (1) under the boundary condition $$m|_{\partial \Omega}=0(2)$$. The author establishes the $$L^p$$ estimates for the solution of (1)–(2) for $$1\leq p\leq\infty$$. The estimates show that $$-L$$ generates an analytic semigroup in $$W^{1,p}\times L^p$$ for $$1<p< \infty$$. Based on the estimates the author obtains short-time estimates for the semigroup in $$L^p$$ norms for all $$1\leq p\leq\infty$$. Moreover, the author establishes the estimates for the high frequency part of the resolvent, which lead to the exponential decay of the corresponding part of the semigroup.

##### MSC:
 35Q30 Navier-Stokes equations 76N15 Gas dynamics (general theory)
Full Text: