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arxiv: 2605.13598 · v1 · pith:CFKXT2OLnew · submitted 2026-05-13 · 🧮 math.AP

Sharp decay characterization for the incompressible Oldroyd-B model in critical L^p spaces

Zhi Chen , Mingwen Fei , Lvqiao Liu , Jiahong Wu This is my paper

Pith reviewed 2026-05-14 17:51 UTC · model grok-4.3

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keywords incompressiblecriticaldecaymodeloldroyd-btensorcharacterizationdamping
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An L2-type low-frequency condition on initial data is almost necessary and sufficient for optimal upper and lower bounds on temporal decay of solutions to the incompressible Oldroyd-B model without viscosity or damping in critical Besov spaces.

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

The incompressible Oldroyd-B model describes flows of viscoelastic fluids such as polymer solutions. This work examines the case with no fluid viscosity and no damping on the stress tensor, a physically relevant regime that is mathematically delicate because standard smoothing mechanisms are missing. The authors show that if the low-frequency components of the initial velocity and stress tensor satisfy an L2-type integrability condition, then the solutions decay at optimal rates in critical Besov spaces, and this condition is nearly necessary for such decay to hold. To reach this conclusion they introduce a decomposition of the stress tensor into incompressible and compressible parts together with an effective tensor that compensates for regularity loss in the high-frequency velocity field. This combination yields both upper and lower bounds, giving a complete sharp characterization of the decay. The result is presented as the first of its kind for this model in the absence of viscosity and damping, using Fourier analysis and Besov-space estimates to control the linear and nonlinear interactions.

Core claim

an L^2-type condition on the low-frequencies part of the initial data (u_0, τ_0) is almost both necessary and sufficient for obtaining optimal upper and lower bounds on the temporal decay of solutions in critical Besov spaces

Load-bearing premise

The new decomposition of the stress tensor into incompressible and compressible parts together with the effective tensor successfully controls the loss of regularity in high-frequency velocity without introducing uncontrolled errors or extra assumptions on the data.

read the original abstract

This paper establishes a sharp characterization of temporal decay rates for the incompressible Oldroyd-B model in a critical $L^p$ framework, covering the physically relevant and mathematically delicate case where both the fluid viscosity and the stress tensor damping are absent. We prove that an $L^2$-type condition on the low-frequencies part of the initial data $(u_0, \tau_0)$ is almost both necessary and sufficient for obtaining optimal upper and lower bounds on the temporal decay of solutions in critical Besov spaces. A key contribution is a new decomposition of the stress tensor into its incompressible and compressible parts, combined with the introduction of an effective tensor to handle the loss of regularity in the high-frequencies velocity field. This is the first result to reveal such precise two-sided asymptotics for the incompressible Oldroyd-B model without viscosity or damping.

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

1 major / 1 minor

Summary. The manuscript establishes a sharp characterization of temporal decay rates for the incompressible Oldroyd-B model in critical L^p spaces when both fluid viscosity and stress-tensor damping are absent. It proves that an L^2-type condition on the low-frequency components of the initial data (u_0, τ_0) is almost necessary and sufficient for optimal upper and lower bounds on the decay of solutions in critical Besov spaces. The argument rests on a new decomposition of the stress tensor into incompressible and compressible parts together with an effective tensor that is introduced to control the loss of regularity in the high-frequency velocity field.

Significance. If the central estimates hold, the result supplies the first precise two-sided asymptotics for this model in the critical regime without viscosity or damping. The necessity-sufficiency statement in critical Besov spaces would constitute a sharp decay characterization that strengthens the literature on viscoelastic fluids.

major comments (1)
  1. [Decomposition and effective tensor (key technical contribution)] The lower-bound necessity statement (abstract and main theorem) requires that commutator terms generated by the decomposition of τ into incompressible/compressible parts plus the effective tensor are absorbed into the low-frequency L^2 control without destroying sharpness. The manuscript must exhibit the precise estimates showing these terms do not produce an extra decay loss that would invalidate the necessity claim.
minor comments (1)
  1. [Abstract and Theorem 1.1] Clarify the precise meaning of 'almost' in the necessity-sufficiency statement; state the exceptional set of data explicitly in the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The manuscript aims to provide the first sharp two-sided decay characterization for the incompressible Oldroyd-B model without viscosity or damping. We address the single major comment below and will revise the manuscript to strengthen the presentation of the key estimates.

read point-by-point responses

  1. Referee: The lower-bound necessity statement (abstract and main theorem) requires that commutator terms generated by the decomposition of τ into incompressible/compressible parts plus the effective tensor are absorbed into the low-frequency L^2 control without destroying sharpness. The manuscript must exhibit the precise estimates showing these terms do not produce an extra decay loss that would invalidate the necessity claim.

    Authors: We agree that explicit verification of the commutator absorption is essential to preserve sharpness in the necessity direction. The decomposition of the stress tensor and the effective tensor are introduced in Section 2. In the proof of the lower bound (Theorem 1.2, Section 4), the commutator terms are controlled via the estimates (4.12)–(4.15), which bound them by the low-frequency L^2 norm of the initial data without introducing an extra decay factor. To make this absorption fully transparent, we will insert a new remark immediately after the statement of the effective-tensor estimates (following Lemma 3.3) that isolates the commutator contributions and confirms they remain absorbed into the L^2 low-frequency control, thereby preserving the optimal rate. This addition does not alter the main theorems but directly addresses the referee’s request for explicit display of the estimates. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Only the abstract is available; the paper introduces a new decomposition and effective tensor whose precise assumptions are not detailed here. Standard Besov-space and Fourier-analysis tools are presumed.

invented entities (1)
  • effective tensor no independent evidence
    purpose: to handle the loss of regularity in the high-frequencies velocity field
    Newly introduced to manage technical difficulties in high-frequency estimates.

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