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\(L^p\)-strong solution to fluid-rigid body interaction system with Navier slip boundary condition. (English) Zbl 1479.35645

Summary: We study a fluid-structure interaction problem describing movement of a rigid body inside a bounded domain filled by a viscous fluid. The fluid is modelled by the generalized incompressible Naiver-Stokes equations which include cases of Newtonian and non-Newtonian fluids. The fluid and the rigid body are coupled via the Navier slip boundary conditions and balance of forces at the fluid-rigid body interface. Our analysis also includes the case of the nonlinear slip condition. The main results assert the existence of strong solutions, in an \(L^p -L^q\) setting, globally in time, for small data in the Newtonian case, while existence of strong solutions in \(L^p\)-spaces, locally in time, is obtained for non-Newtonian case. The proof for the Newtonian fluid essentially uses the maximal regularity property of the associated linear system which is obtained by proving the \(\mathcal{R}\)-sectoriality of the corresponding operator. The existence and regularity result for the general non-Newtonian fluid-solid system then relies upon the previous case. Moreover, we also prove the exponential stability of the system in the Newtonian case.

MSC:

35Q35 PDEs in connection with fluid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76A05 Non-Newtonian fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D35 Strong solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
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