On global motion of a compressible barotropic viscous fluid with boundary slip condition

Global-in-time existence of solutions for equations of viscous compressible barotropic fluid in a bounded domain Ω ⊂ ${\displaystyle {ℝ}^3}$ with the boundary slip condition is proved. The solution is close to an equilibrium solution. The proof is based on the energy method. Moreover, in the ${\displaystyle L_2}$-approach the result is sharp (the regularity of the solution cannot be decreased) because the velocity belongs to ${\displaystyle H^{2+\alpha ,1+\frac{\alpha }{2}}(\Omega \times {ℝ}_{+})}$ and the density belongs to ${\displaystyle H^{1+\alpha ,\frac{1}{2}+\frac{\alpha }{2}}(\Omega \times {ℝ}_{+})}$, α ∈ (1/2,1).