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Oldroyd-B Model with High Weissenberg Number and Fractional Velocity Dissipation

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Abstract

This paper focuses on a high Weissenberg number Oldroyd-B model of complex fluids with fractional frequency velocity dissipation. Mathematically the fluid velocity u satisfies the Navier–Stokes equations with fractional dissipation \((-\Delta )^\alpha u\) while the equation of the non-Newtonian tensor \(\tau \) involves no diffusion or damping mechanism. The aim here is to solve the small-data global well-posedness and stability problem with the least amount of dissipation and minimal regularity requirement. We are able to establish the desired well-posedness and stability result in a hybrid homogeneous Besov setting for any fractional power in the range \(1/2\le \alpha \le 1\). To deal with the difficulties due to the weak velocity dissipation and the lack of diffusion or damping in the \(\tau \)-equation, we exploit the coupling and interaction of this Oldroyd-B model to reveal the hidden wave structure and make extensive use of the associated smoothing and stabilizing effect.

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Acknowledgements

Jiefeng Zhao was partially supported by the National Natural Science Foundation of China (Nos.11901165, 11971446) and the Doctoral Fund of HPU (No.B2016-61). The work of Jiahong Wu was partially supported by NSF grant DMS 2104682 and DMS 2309748. This work was completed during Zhao’s visit to the Department of Mathematics at Oklahoma State University from 2018 to 2019 and he thanks the department for its hospitality.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, 454003, People’s Republic of China

    Jiefeng Zhao

  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jiahong Wu

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  1. Jiefeng Zhao
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  2. Jiahong Wu
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Zhao, J., Wu, J. Oldroyd-B Model with High Weissenberg Number and Fractional Velocity Dissipation. J Geom Anal 33, 296 (2023). https://doi.org/10.1007/s12220-023-01361-3

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