Existence of weak solutions up to collision for viscous fluid-solid systems with slip. (English) Zbl 1307.35193
This paper is a step forward related with a previous result of the authors [ESAIM, Math. Model. Numer. Anal. 46, No. 5, 1201–1224 (2012; Zbl 1267.76020)] where the collision paradox was solved for a Stokes fluid, by using the Navier boundary conditions (instead of no-slip boundary conditions) on the fluid-solid interface. In the present paper the analysis is extended to a Navier-Stokes fluid. The main purpose is to define and obtain the existence for a weak solution in the case of Navier boundary conditions. The test functions are discontinuous on the fluid-solid interface. Moreover, the notion of weak solution is restricted to configurations up to collision, to avoid possible cuspoidal fluid configurations. The corresponding approximate solutions are considered, whose resolution is carried out through a Galerkin scheme. The main result is the existence of a weak Leray-type solution, up to collision, in the 3-D case.
Reviewer: Gelu Paşa (Bucureşti)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35D30 | Weak solutions to PDEs |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
Keywords:
viscous fluid-solid interaction; collision paradox; Navier boundary conditions; weak solutions; Leray solutionsCitations:
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