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Regularity results for the 2D critical Oldroyd-B model in the corotational case

Part of: Incompressible viscous fluids Equations of mathematical physics and other areas of application Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  12 March 2019

Zhuan Ye
Zhuan Ye*
Affiliation:
Department of Mathematics and Statistics, Jiangsu Normal University, 101 Shanghai Road, Xuzhou221116, Jiangsu, PR China ([email protected])
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Abstract

This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

Keywords

Oldroyd-B modelglobal regularityclassical solutions

MSC classification

Primary: 76A10: Viscoelastic fluids
Secondary: 76D03: Existence, uniqueness, and regularity theory 76A05: Non-Newtonian fluids 35Q35: PDEs in connection with fluid mechanics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1871 - 1913
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Copyright
Copyright © Royal Society of Edinburgh 2019

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