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Exterior stationary problems for the equations of motion of compressible viscous and heat-conductive fluids. (English) Zbl 0679.76076
Differential equations, Proc. EQUADIFF Conf., Xanthi/Greece 1987, Lect. Notes Pure Appl. Math. 118, 473-479 (1989).
[For the entire collection see Zbl 0675.00011.]
The stationary motion of viscous compressible fluid which fills an unbounded do main \(\Omega\) in \(R^3\) is described by the following system of equations for the unknown functions, the density \(\rho\), the fluid velocity \(u = (u_1,u_2,u_3)\) and the absolute temperature \(\theta\) : \[ \begin{aligned} &\nabla\cdot(\rho u) = 0,\qquad \nabla\cdot (\rho u\otimes u + pI) = \nabla\cdot \Pi +\rho f, \\ &\nabla\cdot(\rho u(e + | u|^2/2) + pu = \nabla\cdot(\kappa\nabla\theta + \Pi\cdot u) + \rho uf, \end{aligned} \tag{1} \] where \(x=(x_1,x_2,x_3) \in \Omega\), \(\nabla = (\partial/\partial x_1, \partial/\partial x_2, \partial/\partial x_3)\), \(p\) is the pressure, \(I\) is the \(3\times 3\) unit matrix, \(\Pi\) is the viscous tensor whose components are given by \(\Pi_{ij} = \mu (\partial u_ i/\partial x_ j + \partial u_ j/\partial x_ i) + \mu'(\nabla \cdot u)\delta_{ij}\), \(\mu\) and \(\mu'\) are the viscosity coefficients, \(f = (f_1,f_2,f_3)\) is the external force, \(e\) is the internal energy per unit mass, \(\kappa\) is the coefficient of heat- conduction.
As basic assumptions, we assume the followings: (A1) \(\Omega\) is the half space \(R^3_+ = \{x\in R^3, x_3>0\}\), or the exterior domain of a compact simply connected set with smooth boundary \(\partial\Omega\), (A2) \(p\) and \(e\) are smooth functions of \((\rho,\theta)>0\) satisfying \(e_\theta, p_\rho, p_\theta >0\) for \((\rho,\theta)>0\) where \(e_\theta = \partial e/\partial\theta\) and so on, (A3) \(\kappa\), \(\mu\), \(\mu'\) are constants satisfying \(\kappa >0\), \(\mu >0\) and \(2\mu +3\mu'\geq 0\).
We consider the boundary value problem for the system (1) with the boundary conditions (2) \(u |_{\partial\Omega} = 0\), \(\theta |_{\partial\Omega} = \theta_ w\), \(u |_\infty = 0\), \(\theta |_\infty =\bar\theta\), \(\rho |_\infty =\bar\rho\).
We further make the assumptions of \(f\) and \(\theta_ w\), and make some definitions of solution spaces: (A4) \(f\in H^{2+k}(\Omega) \cap L^ 1(\Omega)\), \(| x| f\in L^ 2(\Omega) \cap L^ 1(\Omega)\), \(| x|^2 f\in L^ 2(\Omega)\), \(\theta_ w - \bar\theta \in H^{7/2+k}(\partial\Omega)\), for some integer \(k\geq0\) where \(H^ k\) is a usual Sobolev space with the norm \(\|\cdot\|_ k\) and we simply write \(\|\cdot\|\) when \(k=0\); \(\mathcal H^ k\) (\(k\geq1\)) is the completion of \(C_0^\infty(\bar\Omega)\) w.r.t. \(\| D\cdot\|_{k-1}\), where \(D\) represents the collection of all first order derivatives.
Then we have Theorem 1. Suppose (A1)\(\sim\)(A4). Then there exist positive constants \(\varepsilon_0\) and \(C_0\) such that if \(\Phi_ k \equiv \| f\|_{2k+k} + \| | x| f\|_{L^1} + \| | x|^2 f\|_{L^\infty} + \| \theta_ w - \bar\theta \|_{k+7/2} < \varepsilon_0\), then the stationary problem (1), (2) has a unique solution \((\rho,u,\theta)\) satisfying \(\rho - \bar\rho \in \mathcal H^{3+k}\), \((u,\theta-\bar\theta) \in \mathcal H^{4+k}\), \(| x|(\rho-\rho',u,\theta-\bar\theta,Du,D\theta) \in L^\infty\), \((p-\bar p, Du, D\theta) = o(| x|^{-1})\) for \(| x| >>1\), and \(\|| \rho-\bar\rho, u, \theta-\bar\theta \||_{3+k} \leq C_0 \Phi_1\), where \(\|| \rho, u, \theta \||_{3+k} = \| D\rho \|_{2+k} + \| Du, D\theta \|_{3+k} + \|| x| (\rho,u,Du,D\theta)\|_{L^\infty}\).

MSC:
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations