Existence and uniqueness of a time-periodic strong solution to incompressible Navier-Stokes equations in a time-periodic moving domain, describing the blood flow in an artificial heart

Abstract

In this paper, we prove the existence and uniqueness result of a strong time-periodic solution $(u,p)\in L^2\left(H^2\right)\cap H^1\left(L^2\right)\times L^2\left(H^1\right)$ to incompressible Navier-Stokes equations in a time-periodic moving domain in dimension $N=2,3$. The boundary of the moving domain is a $L^{\infty }\left(W^{2-\frac{1}{2^{\prime }},2^{\prime }}\right)\cap H^1\left(H^{\frac{3}{2}}\right)\cap H^2\left(L^2\right)$ local perturbation of the boundary of a $C^{1,1}$ fixed domain, where $2^{\prime }>N$. The proof is based on a trace lifting technique for transforming the moving domain problem into a fixed domain one, a time-dependent strong solution of a divergence problem, and the implicit function theorem. Our study model describes blood movement in a fluid-driven or mechanically powered artificial heart prototype.