Cardiovascular mathematics · 2006-bio-8

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

unresolved

The exact problem asks for one artificial-boundary treatment that simultaneously uses cross-sectional pressure or flow data, represents the omitted circulation, and yields existence, uniqueness, and continuous dependence for the fully coupled Navier–Stokes/deformable-wall problem. No verified result supplies all three components. The literature contains substantial but non-exhaustive progress:

• “On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels,” Luca Formaggia, Jean-Frédéric Gerbeau, Fabio Nobile, and Alfio Quarteroni, Computer Methods in Applied Mechanics and Engineering 191 (2001), 561–582. DOI: 10.1016/S0045-7825(01)00302-4. This work couples a two- or three-dimensional fluid–structure model to a one-dimensional compliant-vessel model, derives energy bounds under an outflow kinetic-energy-flux condition, and demonstrates numerically that the reduced model absorbs outgoing waves. An energy estimate and numerical coupling do not establish existence, uniqueness, or continuous dependence for the coupled moving-domain problem.

• “On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations,” Luca Formaggia, Alexandra Moura, and Fabio Nobile, ESAIM: Mathematical Modelling and Numerical Analysis 41 (2007), 743–769. DOI: 10.1051/m2an:2007039. The paper derives a global energy estimate for flux/mean-total-stress matching and treats several wall models. It explicitly notes that the cited well-posedness results for defective flow-rate or mean-pressure conditions cover rigid walls, not the deformable-wall system. For wall models requiring endpoint conditions, its estimate uses homogeneous structural conditions and does not establish stability when cross-sectional area continuity is imposed.

• “Absorbing boundary conditions for a 3D non-Newtonian fluid–structure interaction model for blood flow in arteries,” João Janela, Alexandra Moura, and Adélia Sequeira, International Journal of Engineering Science 48 (2010), 1332–1349. DOI: 10.1016/j.ijengsci.2010.08.004. This extends the 3D–1D energy estimate to a shear-thinning generalized-Newtonian fluid with a nonlinear hyperelastic wall and compares 3D–1D and linear absorbing outlet conditions numerically, including a carotid geometry. It does not prove well-posedness of the complete coupled initial-boundary-value problem.

• “Fluid structure system with boundary conditions involving the pressure,” Jean-Jérôme Casanova, Journal of Evolution Equations 21 (2021), 107–149. DOI: 10.1007/s00028-020-00581-2; arXiv: 1707.06382. Theorem 4.3 proves local existence and uniqueness of strong solutions for a two-dimensional rectangular-type Navier–Stokes domain coupled to a damped Euler–Bernoulli beam with dynamic-pressure conditions at inflow and outflow; Theorem 4.4 gives a strong solution on a prescribed time interval for small data. This is a genuine well-posedness result, but only for a special two-dimensional wall model and prescribed pressure conditions, not a general three-dimensional artificial-boundary coupling to the omitted circulation or the flow-rate-data case.

• “Existence of solutions and continuous and semi-discrete stability estimates for 3D/0D coupled systems modelling airflows and blood flows,” Céline Grandmont and Sébastien Martin, ESAIM: Mathematical Modelling and Numerical Analysis 55 (2021), 2365–2419. DOI: 10.1051/m2an/2021055. This proves existence and stability results for Stokes–Windkessel and Navier–Stokes–Windkessel systems, including local-in-time or small-data strong results for the nonlinear system. The three-dimensional lateral walls are explicitly assumed rigid, so the theorem does not cover the coupled arterial-wall unknown in the statement.

• “Well-posedness of an initial-boundary value problem for a vascular blood flow model with a prescribed moving boundary,” Jiahui Li and Peicheng Zhu, Electronic Research Archive 34 (2026), 5626–5659. DOI: 10.3934/era.2026251. The authors prove existence and uniqueness of weak solutions in a moving vascular domain, but the wall displacement is prescribed a priori rather than solved as a structural unknown, and the artificial boundaries are not coupled to a reduced circulation model.

The survey “Geometric multiscale modeling of the cardiovascular system, between theory and practice,” Alfio Quarteroni, Alessandro Veneziani, and Christian Vergara, Computer Methods in Applied Mechanics and Engineering 302 (2016), 193–252. DOI: 10.1016/j.cma.2016.01.007 confirms that flux/pressure data, local–systemic coupling, structural-law mismatch, spurious reflection, and patient-specific parameter identification remain separate mathematical and computational issues. Thus the established constructions and special-case well-posedness theorems do not collectively exhaust the deformable-artery problem stated here; a rigorous general boundary formulation satisfying all of its requirements remains unresolved.