Well-posedness of an initial-boundary value problem for a vascular blood flow model with a prescribed moving boundary

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Abstract

While fully coupled fluid-structure interaction models are often used to capture cardiovascular hemodynamics, establishing the well-posedness of the fluid equations on a moving domain constitutes a crucial mathematical foundation. Therefore, this paper investigates an initial- boundary value problem for a blood flow model where the vessel wall displacement, denoted η(t,x), is explicitly prescribed as an a priori given function, rather than being treated as a structural unknown. The inherent moving boundary introduces significant mathematical challenges, compounded by operator singularities in cylindrical coordinates and complex nonhomogeneous boundary conditions. To overcome the singularity at the polar axis, we truncate the domain by introducing a sufficiently small positive constant. We then apply the arbitrary Lagrangian-Eulerian mapping to transform the moving boundary problem into a fixed reference domain, enabling the use of standard mathematical analysis methods. The superposition principle is subsequently used to decouple the problem. Specifically, we use BogovskiĬ’s theorem to construct auxiliary functions that resolve the nonhomogeneous boundary conditions, and apply the Galerkin method alongside energy estimates and compactness arguments to solve the resulting homogeneous parabolic problem. Finally, by applying inverse mapping, we establish the existence and uniqueness of weak solutions in the original time-dependent fluid domain. This rigorous mathematical analysis provides a theoretical complement to the numerical simulations of vascular hemodynamics.

Keywords

hemodynamics / fluid-structure interaction / moving boundary / initial-boundary value problem / arbitrary Lagrangian-Eulerian mapping / weak solution

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