Strong solutions to a nonlinear fluid structure interaction system (Q833294)

A fluid structure interaction model describes the elastic motion of a solid inside a fluid. Mathematically, the model is comprised of a fluid modeled by the Navier-Stokes equation interacting with an elastic body at the boundary of the solid, with stress and velocity matching boundary conditions. The elastic body motion is described by the wave equation. If \(u_{0}\) and \(w_{0}, w_{1}\) denote the initial datum for the fluid velocity and the wave equation, respectively, then the main result of the paper states as follows: There exists a unique local-in-time smooth solution when \(u_{0}\) belongs to the space \(H^{1}\) while \(w_{0}\) and \(w_{1}\) belong to the spaces \(H^{2}(\Omega_{e})\) and \(H^{1}(\Omega_{e})\), respectively. The regularity of the solutions obtained is \(u \in L^{2}([0,T]; H^{2}(\Omega_{f})) \cap L^{\infty}([0,T]; H^{1}(\Omega_{e}))\) while \((w, w_{t}) \in L^{\infty}([0,T]; H^{7/4-\varepsilon}(\Omega_{e}) \times H^{3/4-\varepsilon}(\Omega_{e}))\), for any \(\varepsilon > 0\).