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.


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