Weak solutions to the barotropic Navier–Stokes system with slip boundary conditions in time dependent domains

Abstract

We consider the compressible (barotropic) Navier-Stokes system on time dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.

On a Nonlinear Model for Tumor Growth in a Cellular Medium

“…A penalty approach to slip conditions for stationary incompressible flow was proposed by Stokes and Carey [24] (see also [11,16]). In the present setting, the variational (weak) formulation of the Forchheimer's equation is supplemented by a singular forcing term 14) penalizing the normal component of the velocity on the boundary of the tumor domain.…”

“…A penalty approach to slip conditions for stationary incompressible fluids was proposed by Stokes and Carey [24]. Compressible fluid flows in time dependent domains, supplemented with the no-slip boundary conditions, were examined in [15] by means of Brinkman's penalization method and in [16] treating a slip boundary condition. A penalty approach to the analysis of a tumor growth model was presented in [11] treating the case of a mixed-type tumor growth model.…”

On the low Mach number limit of compressible flows in exterior moving domains

“…Originated by the pioneering result of Lions [14] on the existence of large data weak solutions for the compressible Navier-Stokes system, Desjardins et al [4], [5] Lions and Masmoudi [15] (see also the surveys Danchin [3], Masmoudi [16], Schochet [18] and the references cited therein) employed the framework of weak solutions to singular incompressible limits for problems confined to fixed spatial domains. Similar problems on a bounded time dependent domain, with prescribed boundary motion, have been studied only recently in [8].…”

“…The following existence result of weak solutions to the compressible Navier-Stokes system in moving domains was proved in [8]. Strictly speaking, the result of [8] covers the case of a bounded physical space, however, the extension to the exterior problem is straightforward, see Sýkora [19].…”

“…Strictly speaking, the result of [8] covers the case of a bounded physical space, however, the extension to the exterior problem is straightforward, see Sýkora [19]. Now we define weak solutions to the target system, the incompressible Navier-Stokes equations.…”