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 $ \eta(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.
Citation: Jiahui Li, Peicheng Zhu. Well-posedness of an initial-boundary value problem for a vascular blood flow model with a prescribed moving boundary[J]. Electronic Research Archive, 2026, 34(8): 5626-5659. doi: 10.3934/era.2026251
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 $ \eta(t, x) $, is explicitly prescribed as an a priori given function, rather than being treated as a structural unknown. The inherent moving boundary introduces significant mathematical challenges, compounded by operator singularities in cylindrical coordinates and complex nonhomogeneous boundary conditions. To overcome the singularity at the polar axis, we truncate the domain by introducing a sufficiently small positive constant. We then apply the arbitrary Lagrangian-Eulerian mapping to transform the moving boundary problem into a fixed reference domain, enabling the use of standard mathematical analysis methods. The superposition principle is subsequently used to decouple the problem. Specifically, we use Bogovskiĭ's theorem to construct auxiliary functions that resolve the nonhomogeneous boundary conditions, and apply the Galerkin method alongside energy estimates and compactness arguments to solve the resulting homogeneous parabolic problem. Finally, by applying inverse mapping, we establish the existence and uniqueness of weak solutions in the original time-dependent fluid domain. This rigorous mathematical analysis provides a theoretical complement to the numerical simulations of vascular hemodynamics.
| [1] | Global Burden of Cardiovascular Diseases and Risks 2023 Collaborators, Global, regional, and national burden of cardiovascular diseases and risk factors in 204 countries and territories, 1990–2023, JACC, 86 (2025), 2167–2243. https://doi.org/10.1016/j.jacc.2025.08.015 |
| [2] | O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Revised 2$^{nd}$ edition, Gordon and Breach, 1969. |
| [3] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, American Mathematical Society, 2024. https://doi.org/10.1090/chel/343 |
| [4] | Y. C. Fung, Biomechanics: Circulation, 2$^{nd}$ edition, Springer-Verlag, 1997. https://doi.org/10.1007/978-1-4757-2696-1 |
| [5] | L. Formaggia, A. Quarteroni, A. Veneziani, Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer Science & Business Media, 2010. https://doi.org/10.1007/978-88-470-1152-6 |
| [6] |
H. Beirão da Veiga, On the existence of strong solutions to a coupled fluid–structure evolution problem, J. Math. Fluid Mech., 6 (2004), 21–52. https://doi.org/10.1007/s00021-003-0082-5 doi: 10.1007/s00021-003-0082-5
|
| [7] |
C. H. A. Cheng, S. Shkoller, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., 42 (2010), 1094–1155. https://doi.org/10.1137/080741628 doi: 10.1137/080741628
|
| [8] |
S. Schwarzacher, P. Su, Existence of strong solutions for a perfect elastic beam interacting with Navier-Stokes equations, Commun. Math. Phys., 406 (2025), 206. https://doi.org/10.1007/s00220-025-05386-3 doi: 10.1007/s00220-025-05386-3
|
| [9] |
C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008), 716–737. https://doi.org/10.1137/070699196 doi: 10.1137/070699196
|
| [10] |
B. Muha, S. Čanić, Existence of a weak solution to a nonlinear fluid–structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal., 207 (2013), 919–968. https://doi.org/10.1007/s00205-012-0585-5 doi: 10.1007/s00205-012-0585-5
|
| [11] |
S. Čanić, M. Galić, B. Muha, Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction, Trans. Am. Math. Soc., 373 (2020), 6621–6681. https://doi.org/10.1090/tran/8125 doi: 10.1090/tran/8125
|
| [12] |
S. Trifunović, Y. G. Wang, Existence of a weak solution to the fluid–structure interaction problem in 3D, J. Differ. Equations, 268 (2020), 1495–1531. https://doi.org/10.1016/j.jde.2019.09.002 doi: 10.1016/j.jde.2019.09.002
|
| [13] |
M. Kampschulte, B. Muha, S. Trifunović, Global weak solutions to a 3D/3D fluid–structure interaction problem including possible contacts, J. Differ. Equations, 385 (2024), 280–324. https://doi.org/10.1016/j.jde.2023.12.014 doi: 10.1016/j.jde.2023.12.014
|
| [14] |
J. Kuan, S. Čanić, Well-posedness of solutions to stochastic fluid–structure interaction, J. Math. Fluid Mech., 26 (2024), 4. https://doi.org/10.1007/s00021-023-00839-y doi: 10.1007/s00021-023-00839-y
|
| [15] |
A. H. Arnous, M. S. Hashemi, K. S. Nisar, M. Shakeel, J. Ahmad, I. Ahmad, et al., Investigating solitary wave solutions with enhanced algebraic method for new extended Sakovich equations in fluid dynamics, Results Phys., 57 (2024), 107369. https://doi.org/10.1016/j.rinp.2024.107369 doi: 10.1016/j.rinp.2024.107369
|
| [16] |
Y. M. Chu, N. A. Shah, P. Agarwal, J. D. Chung, Analysis of fractional multi-dimensional Navier–Stokes equation, Adv. Differ. Equations, 2021 (2021), 91. https://doi.org/10.1186/s13662-021-03250-x doi: 10.1186/s13662-021-03250-x
|
| [17] |
N. A. Shah, H. A. Alyousef, S. A. El-Tantawy, R. Shah, J. D. Chung, Analytical investigation of fractional-order Korteweg–De-Vries-type equations under Atangana–Baleanu–Caputo operator: Modeling nonlinear waves in a plasma and fluid, Symmetry, 14 (2022), 739. https://doi.org/10.3390/sym14040739 doi: 10.3390/sym14040739
|
| [18] | J. Mazumdar, Biofluid Mechanics, 2$^{nd}$ edition, World Scientific, 2016. https://doi.org/10.1142/9754 |
| [19] | J. P. Raymond, Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions, Ann. Inst. Henri Poincare Anal. Non Linéaire, 24 (2007), 921–951. https://doi.org/10.1016/J.ANIHPC.2006.06.008 |
| [20] | M. E. Bogovskiĭ, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR, 248 (1979), 1037–1040. |
| [21] | M. E. Bogovskiĭ, Solution of some vector analysis problems connected with operators $\mathrm{div}$ and $\mathrm{grad}$, Trudy Seminara S. L. Soboleva, (1980), 5–40. |
| [22] | M. E. Bogovskiĭ, Decomposition of $L_p(\Omega; \mathbb{R}^n)$ into the direct sum of subspaces of solenoidal and potential vector fields, Dokl. Akad. Nauk SSSR, 286 (1986), 781–786. |
| [23] |
V. N. Maslennikova, M. E. Bogovskiĭ, Denseness of finite solenoidal vector fields, Sib. Math. J., 19 (1978), 771–783. https://doi.org/10.1007/BF00973607 doi: 10.1007/BF00973607
|
| [24] | G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Science & Business Media, 2011. https://doi.org/10.1007/978-0-387-09620-9 |
| [25] | A. Kufner, Weighted Sobolev Spaces, John Wiley & Sons, 1985. |
| [26] | K. Yosida, Functional Analysis, Springer Science & Business Media, 1968. https://doi.org/10.1007/978-3-662-11791-0 |
| [27] | E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, 1955. |
| [28] | J. K. Hale, Ordinary Differential Equations, 2$^{nd}$ edition, Krieger Publishing Company, 1980. |
| [29] | L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Mathematical Society, 2010. |
| [30] | R. A. Adams, J. J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Elsevier, 2003. https://doi.org/10.1007/978-0-12-044143-3 |
| [31] | G. Leoni, A First Course in Sobolev Spaces, 2$^{nd}$ edition, American Mathematical Society, 2017. https://doi.org/10.1090/gsm/181 |
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Jiahui Li, Peicheng Zhu. Well-posedness of an initial-boundary value problem for a vascular blood flow model with a prescribed moving boundary[J]. Electronic Research Archive, 2026, 34(8): 5626-5659. doi: 10.3934/era.2026251
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