A mathematical analysis reports that a two-dimensional, stress-diffusive Oldroyd-B model admits a unique strong solution for any time horizon, even when the initial polymeric stress is arbitrarily large. The result concerns the zero-Reynolds-number, or creeping-flow, regime and rests on specific conditions for the domain and starting stress.
The work studies a deterministic initial-boundary value problem. It is posed on a smooth, bounded, connected region in two dimensions, with a prescribed initial stress. At the boundary, the velocity is set to zero and the normal derivative of the stress is set to zero, written u = 0 and ∂nτ = 0.
The equations include stress diffusion through the term −εΔτ. In the analysis, this term is treated explicitly as a mathematical regularization of the stress, so the result concerns the diffusive version of the system.
What the theorem requires
The global theorem requires the initial stress τ0 to belong to H1. It also requires the conformation tensor T = τ + αI to be symmetric and nonnegative definite, the positivity condition used for the initial state.
Under those assumptions, T remains nonnegative definite throughout the solution. The paper also gives a trace-based L1 estimate: ||τ||L1 is bounded by ||trT||L1 plus a constant depending on α and the domain. The estimate provides direct control on the polymeric stress within the theorem.
The theorem gives the solution a detailed regularity profile. The velocity is continuous in time with H2 spatial regularity, belongs to L2 over time with H3 spatial regularity, and has a time derivative in L2 over time with H1 spatial regularity. The stress is continuous in time with H1 spatial regularity, belongs to L2 over time with H2 spatial regularity, and has a time derivative in L2 over time. These are the regularity levels stated for the strong solution.
How the analysis gets there
Local existence and uniqueness are obtained using Schauder’s fixed-point theorem. That local result is stated for H1 initial stress on a bounded connected domain with C2,1 boundary, with 0 < α < 1 and ε and γ greater than zero.
The later argument reports uniform a priori estimates and says the proof is completed through a continuity argument. The authors describe it as relying on the elliptic Stokes structure, preservation of conformation non-negativity, and stress diffusion.
In the zero-Reynolds-number setting, the paper reports a clear regularity hierarchy: the velocity field has higher spatial regularity than the polymeric stress tensor, as described through the elliptic Stokes equation.
A result with a narrow reach
The theorem’s reach is limited to the stated two-dimensional, bounded, smooth-domain problem, with stress diffusion and the required H1 and nonnegative-conformation assumptions. It does not establish global well-posedness for the corresponding non-diffusive creeping-flow system or give a three-dimensional global theorem.
The document is a preprint identified as arXiv:2608.25333v1 and dated 26 August 2026. The authors state that no new data were created or analysed, and they declare that they have no conflict of interest.
The supplied analysis leaves open whether the result can be extended to the non-diffusive model, to three dimensions, or to weaker assumptions on the initial stress and conformation tensor. Those questions are outside the theorem reported here.
Paper data and sources
Original title: Global well-posedness of strong solutions to the initial-boundary value problem for a two-dimensional stress-diffusive Oldroyd-B model in the creeping flow regime
Authors: Yinghui Wang, Shihao Zhang, Zhuo Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text