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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