Abstract We prove that there exists a weak solution to a system governing an unsteady flow of a viscoelastic fluid in three dimensions, for arbitrarily large time interval and data. The fluid is described by the incompressible Navier-Stokes equations for the velocity v, coupled with a diffusive variant of a combination of the Oldroyd-B and the Giesekus models for a tensor 𝔹. By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of physics) and for the energy dissipation, we are able to prove that 𝔹 enjoys the same regularity as v in the classical three-dimensional Navier-Stokes equations. This enables us to handle any kind of objective derivative of 𝔹, thus obtaining existence results for the class of diffusive Johnson-Segalman models as well. Moreover, using a suitable approximation scheme, we are able to show that 𝔹 remains positive definite if the initial datum was a positive definite matrix (in a pointwise sense). We also show how the model we are considering can be derived from basic balance equations and thermodynamical principles in a natural way.

中文翻译：

---

具有应力扩散的速率型粘弹性流体三维非定常流动的大数据存在性理论

摘要 我们证明，对于任意大的时间间隔和数据，在三个维度上控制粘弹性流体非定常流动的系统存在弱解。流体由速度 v 的不可压缩 Navier-Stokes 方程以及张量 𝔹 的 Oldroyd-B 和 Giesekus 模型组合的扩散变体来描述。By a proper choice of the constitutive relations for the Helmholtz free energy (which, however, is non-standard in the current literature, despite the fact that this choice is well motivated from the point of view of physics) and for the energy dissipation,我们能够证明 𝔹 与经典三维纳维-斯托克斯方程中的 v 具有相同的正则性。这使我们能够处理 𝔹 的任何类型的客观导数，从而也获得了扩散 Johnson-Segalman 模型类的存在结果。此外，使用合适的近似方案，我们能够证明如果初始数据是正定矩阵（在逐点意义上），𝔹 仍然是正定的。我们还展示了我们正在考虑的模型如何以自然的方式从基本平衡方程和热力学原理推导出来。