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arxiv: 2604.05679 · v1 · submitted 2026-04-07 · 🧮 math.AP · math-ph· math.MP· physics.flu-dyn

Asymptotic models for viscoelastic one-dimensional blood flow

Diego Alonso-Or\'an , Rafael Granero-Belinch\'on , Carlos Yanes P\'erez This is my paper

Pith reviewed 2026-05-10 19:14 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPphysics.flu-dyn
keywords blood flowviscoelastic arteriesasymptotic reductionone-dimensional modelwell-posednessSobolev spacesBBM equationperiodic solutions
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The pith

A reduced unidirectional model for blood flow in viscoelastic arteries has local strong solutions in Sobolev spaces for general parameters and periodic data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a simplified one-dimensional model that approximates blood flow through arteries whose walls respond viscoelastically. It then proves that this model possesses unique strong solutions that exist at least locally in time when the initial data are periodic and have zero mean. For the special case in which the wall is purely elastic and the equation reduces to a BBM-type dispersive model, the same solutions exist globally and decay exponentially when the initial data are sufficiently small. These results supply a mathematically controlled reduced system that can be simulated efficiently while retaining key viscoelastic effects.

Core claim

Starting from the full three-dimensional viscoelastic fluid-structure interaction problem, the authors perform an asymptotic reduction to obtain a unidirectional one-dimensional system. They establish local well-posedness of strong solutions in Sobolev spaces for arbitrary positive parameters and mean-zero periodic initial data. In the purely elastic BBM regime the same solutions exist globally in time and decay exponentially to equilibrium whenever the initial datum is small enough in a suitable norm.

What carries the argument

The unidirectional asymptotic model obtained by scaling the viscoelastic wall law and reducing the three-dimensional system to one space dimension.

If this is right

  • The local existence result justifies short-time numerical simulations of the reduced model across a wide range of viscoelastic parameters.
  • In the purely elastic regime the exponential decay of small solutions implies that the model is asymptotically stable around the zero-flow state.
  • The continuation criterion mentioned in the numerical study links finite-time blow-up to the growth of Sobolev norms, allowing practical monitoring of solution regularity.
  • The framework supplies a mathematically rigorous starting point for studying the effect of varying wall viscosity on pulse propagation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the small-parameter scalings remain valid for real arteries, the reduced model could serve as an efficient surrogate for studying wave reflection at vessel junctions without solving the full three-dimensional problem.
  • The global existence and decay in the elastic case suggest that viscoelastic damping may be the dominant mechanism preventing blow-up in more general regimes.
  • The periodic setting used here could be extended to networks of vessels by matching solutions at junctions, provided the local existence theory carries over to piecewise-smooth data.

Load-bearing premise

The reduction to the one-dimensional model requires specific small-parameter scalings and constitutive relations for the artery wall that are assumed to hold but are not verified against physiological data.

What would settle it

A direct numerical comparison between the full three-dimensional viscoelastic fluid-structure system and the reduced one-dimensional model on a periodic domain that shows large discrepancies in wave speed or dissipation rate for realistic parameter values.

Figures

Figures reproduced from arXiv: 2604.05679 by Carlos Yanes P\'erez, Diego Alonso-Or\'an, Rafael Granero-Belinch\'on.

Figure 1
Figure 1. Figure 1: shows snapshots of fpx, tq for the parameter values ν “ 1, ε “ 1, κ “ 1, β “ 1, A “ 0.10. (5.2) In this case, the computation reaches the final time t “ 10 [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of f for the parameter values given in (5.3). The vertical scale reaches the order of 1010 . The corresponding diagnostics are shown in [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the final profiles for ν “ 0, 0.1, 0.5, 1, 1.5, 2, 3, with A “ 0.1 and all other parameters fixed to 1. The associated diagnostics are displayed in [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Diagnostics for the simulations in [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the final profiles for A “ 0.5, 1, 5, 10, 20, with ν “ 1 and all other parameters fixed to 1. The vertical scale reaches the order of 1012 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Diagnostics for the simulations in [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Numerical results for ν “ 0 and β “ 2. The corresponding diagnostics are shown in [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: presents the corresponding results for β “ 0 [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: presents the results for β “ ´1 [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Numerical results for ν “ 0, β “ ´1, extended up to t “ 20. The corresponding diagnostics for the longer simulation are shown in [PITH_FULL_IMAGE:figures/full_fig_p033_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Diagnostics corresponding to [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
read the original abstract

We derive a unidirectional asymptotic model for one-dimensional blood flow in viscoelastic arteries. We prove local well-posedness of strong solutions in Sobolev spaces for general parameters and mean-zero periodic data. In the purely elastic BBM regime we further establish global existence and exponential decay for sufficiently small initial data. We also present a numerical study of the reduced model, including comparisons across different viscoelastic and amplitude regimes, and discuss the observed dynamics in connection with the continuation criterion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives a unidirectional asymptotic model for one-dimensional blood flow in viscoelastic arteries from the 3D axisymmetric Navier-Stokes system with a viscoelastic wall law. It proves local well-posedness of strong solutions in Sobolev spaces H^s (s>3/2) for general parameters and mean-zero periodic data on the torus. In the purely elastic BBM regime, global existence and exponential decay are established for sufficiently small initial data via a Lyapunov functional. The paper includes a numerical study exploring dynamics across viscoelastic and amplitude regimes, relating observations to the continuation criterion.

Significance. If the results hold, the work strengthens the analytical basis for reduced 1D models in hemodynamics. The explicit small-parameter scalings and viscoelastic constitutive law in §2 justify the reduction, while the local well-posedness for general parameters and global existence/decay in the elastic case provide useful guarantees for long-term behavior and numerical stability. These are standard but carefully executed energy estimates that support further analysis of arterial wave propagation.

minor comments (3)
  1. §2: The scalings and wall law are clearly stated and address potential concerns about unspecified assumptions; a brief sentence comparing the viscoelastic law to standard Kelvin-Voigt or other arterial models would aid readers from the hemodynamics community.
  2. §4: The Lyapunov functional for the global existence proof is effective, but stating the explicit dependence of the decay rate on the smallness parameter and viscosity coefficients would make the result easier to compare with related BBM-type equations.
  3. Numerical section: The regime comparisons are informative, but the manuscript would benefit from a short statement on the spatial/temporal discretization and any observed convergence behavior to support the discussion of the continuation criterion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on the asymptotic unidirectional model for viscoelastic arterial blood flow. We appreciate the recommendation for minor revision and the recognition of the local well-posedness results in Sobolev spaces as well as the global existence and decay in the elastic BBM regime. Since no specific major comments were provided in the report, we have no point-by-point rebuttals to address at this stage.

Circularity Check

0 steps flagged

No significant circularity: derivation and proofs are self-contained

full rationale

The paper explicitly states small-parameter scalings and the viscoelastic constitutive law in §2 to justify the unidirectional reduction from the 3D axisymmetric system. Local well-posedness follows from standard energy estimates in H^s (s>3/2) on the torus with mean-zero data; the elastic BBM case uses a Lyapunov functional for global existence and decay of small data. No load-bearing step reduces to a fitted input, self-citation chain, or definitionally equivalent ansatz. The numerical study is presented as exploration of dynamics, not as validation of a constructed prediction. The central claims (model derivation + existence theorems) remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract does not list explicit free parameters or invented entities. The model derivation necessarily rests on standard assumptions of asymptotic reduction (small aspect ratio, slow variation along the vessel) and a viscoelastic constitutive law for the wall; these are treated as domain assumptions rather than new postulates. No new particles or forces are introduced.

axioms (2)
  • domain assumption Standard assumptions of asymptotic reduction for slender tubes (small aspect ratio and slow axial variation) are invoked to obtain the unidirectional model.
    Typical for 1D blood-flow derivations; required to reduce 3D Navier-Stokes to 1D.
  • domain assumption A viscoelastic constitutive relation for the arterial wall is assumed without specifying the exact form or parameter values.
    Central to the viscoelastic extension; the abstract does not detail the law.

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