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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)
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