Weak Solutions of the Navier–Stokes Equations for Compressible Flows in a Half-Space with No-Slip Boundary Conditions

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We consider the Navier–Stokes equations for the motion of compressible, viscous flows in a half-space \({\mathbb{R}^n_+,}\)n = 2, 3, with the no-slip boundary conditions. We prove the existence of a global weak solution when the initial data are close to a static equilibrium. The density of the weak solution is uniformly bounded and does not contain a vacuum, the velocity is Holder continuous in (x, t) and the material acceleration is weakly differentiable. The weak solutions of this type were introduced by D. Hoff in Arch Ration Mech Anal 114(1):15–46, (1991), Commun Pure and Appl Math 55(11):1365–1407, (2002) for the initial-boundary value problem in \({\Omega = \mathbb{R}^n}\) and for the problem in \({\Omega = \mathbb{R}^n_+}\) with the Navier boundary conditions.