A uniqueness result for 3D incompressible fluid-rigid body interaction problem. (English) Zbl 1460.35296
Summary: We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy the Prodi-Serrin \(\mathrm{L}^r-\mathrm{L}^s\) condition are unique in the class of Leray-Hopf weak solutions.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 35D30 | Weak solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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