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Compressible flow in a half-space with Navier boundary conditions. (English) Zbl 1095.35025

The following initial-boundary value problem is considered in the half-space \(Q=\{(x,t): x\in \mathbb R^3\), \(x_3>0\), \(t>0\}\) \[ \begin{aligned} &\frac{\partial\rho }{\partial t}+\text{div}\,(\rho v)=0 \quad \text{in} \;Q,\\ &\frac{\partial }{\partial t}\left(\rho v_i\right)+\text{div}\, (\rho v_iv)+\frac{\partial }{\partial x_i}p(\rho)-\mu\Delta v_i -\lambda\text{div}\,\left(\frac{\partial v}{\partial x_i}\right) =\rho f_i, \quad i=1,2,3\quad \text{in}\;Q,\\ &v=K(x)\left(\frac{\partial v_1}{\partial x_3}, \frac{\partial v_2}{\partial x_3},0\right), \quad x_3=0,\quad t>0,\\ &(\rho,v)(x,0)=(\rho_0,v_0)(x),\quad x\in \mathbb R^3_+. \end{aligned} \] Here the velocity \(v(x,t)=(v_1,v_2,v_3)\) and the density \(\rho(x,t)\) are unknown functions, \(p=p(\rho)\) is the pressure, \(f\) is a given external force, \(\lambda\) and \(\mu\) are viscosity constants, \(K\) is a smooth positive function.
The global in time existence of weak solutions is proved if the data of the problem are small in energy norms. Estimates of solutions are obtained.

MSC:

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