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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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References

1Bahouri, H., Chemin, J.-Y. and Danchin, R.. Fourier Analysis and Nonlinear Partial Differential Equations, In Grundlehren der Mathematischen Wissenschaften, vol. 343 (Heidelberg: Springer, 2011.CrossRefGoogle Scholar
2Beale, J. T., Kato, T. and Majda, A.. Remarks on breakdown of smooth solutions for the 3D Euler equations. Comm. Math. Phys. 94 (1984), 61–66.CrossRefGoogle Scholar
3Bejaoui, O. and Majdoub, M.. Global weak solutions for some Oldroyd models. J. Differ. Equ. 254 (2013), 660–685.CrossRefGoogle Scholar
4Brezis, H. and Gallouet, T.. Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4 (1980), 677–681.CrossRefGoogle Scholar
5Brezis, H. and Wainger, S.. A note on limiting cases of Sobolev embedding and convolution inequalities. Comm. Partial Differ. Equ. 5 (1980), 773–789.CrossRefGoogle Scholar
6Chemin, J.-Y. and Masmoudi, N.. About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33 (2001), 84–112.CrossRefGoogle Scholar
7Chen, Q. and Miao, C.. Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces. NonlinearAnal. 68 (2008), 1928–1939.Google Scholar
8Chen, Y. and Zhang, P.. The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm. Partial Differ. Equ. 31 (2006), 1793–1810.CrossRefGoogle Scholar
9Constantin, P.. Euler equations, Navier–Stokes equations and turbulence, In Mathematical foundation of turbulent viscous flows, Lecture Notes in Math., vol. 1871,pp. 1–43 (Berlin: Springer, 2006).CrossRefGoogle Scholar
10Constantin, P. and Kliegl, M.. Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress. Arch. Ration. Mech. Anal 206 (2012), 725–740.CrossRefGoogle Scholar
11Constantin, P. and Sun, W.. Remarks on Oldroyd-B and related complex fluid models. Commun. Math. Sci. 10 (2012), 33–73.CrossRefGoogle Scholar
12Constantin, P. and Vicol, V.. Nonlinear maximum principles for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22 (2012), 1289–1321.CrossRefGoogle Scholar
13Elgindi, T. and Rousset, F.. Global regularity for some Oldroyd-B type models. Comm. Pure Appl. Math. 68 (2015), 2005–2021.CrossRefGoogle Scholar
14Fang, D. and Zi, R.. Global solutions to the Oldroyd-B model with a class of large initial data. SIAM J. Math. Anal. 48 (2016), 1054–1084.CrossRefGoogle Scholar
15Fang, D., Hieber, M. and Zi, R.. Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters. Math. Ann. 357 (2013), 687–709.CrossRefGoogle Scholar
16Fernández-Cara, E., Guillén, F. and Ortega, R.. Mathematical modeling and analysis of viscoelastic fluids of the Oldroyd kind, Handbook of numerical analysis, vol. 8,pp. 543–661 (NorthHolland, Amsterdam: Elsevier, 2002).Google Scholar
17Guillopé, C. and Saut, J.-C.. Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal. 15 (1990), 849–869.CrossRefGoogle Scholar
18Guillopé, C. and Saut, J.-C.. Global existence and one-dimensional nonlinear stability of shearingmotions of viscoelastic fluids of Oldroyd type. RAIROModél. Math. Anal. Numér. 24 (1990), 369–401.CrossRefGoogle Scholar
19Hieber, M., Naito, Y. and Shibata, Y.. Global existence results for Oldroyd-B fluids in exterior domains. J. Differ. Equ. 252 (2012), 2617–2629.CrossRefGoogle Scholar
20Hmidi, T., Keraani, S. and Rousset, F.. Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation. J. Differ. Equ. 249 (2010), 2147–2174.CrossRefGoogle Scholar
21Hmidi, T., Keraani, S. and Rousset, F.. Global well-posedness for Euler-Boussinesq system with critical dissipation. Comm. Partial Differ. Equ. 36 (2011), 420–445.CrossRefGoogle Scholar
22Hu, D. and Lelièvre, T.. New entropy estimates for Oldroyd-B and related models. Commun. Math. Sci. 5 (2007), 909–916.CrossRefGoogle Scholar
23Hu, X. and Wang, D.. Local strong solution to the compressible viscoelastic flow with large data. J. Differ. Equ. 249 (2010), 1179–1198.CrossRefGoogle Scholar
24Hu, X. and Wang, D.. Global existence for the multi-dimensional compressible viscoelastic flows. J. Differ. Equ. 250 (2011), 1200–1231.CrossRefGoogle Scholar
25Karageorghis, A. and Fairweather, G.. The method of fundamental solutions for the numerical solution of the biharmonic equation. J. Comput. Phys. 69 (1987), 434–459.CrossRefGoogle Scholar
26Kato, T. and Ponce, G.. Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), 891–907.CrossRefGoogle Scholar
27Kenig, C., Ponce, G. and Vega, L.. Well-posedness of the initial value problem for the Korteweg-de-Vries equation. J. Amer. Math. Soc. 4 (1991), 323–347.CrossRefGoogle Scholar
28Lei, Z.. On 2D viscoelasticity with small strain. Arch. Ration. Mech. Anal. 198 (2010), 13–37.CrossRefGoogle Scholar
29Lei, Z. and Zhou, Y.. Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit. SIAM J. Math. Anal. 37 (2005), 797–814.CrossRefGoogle Scholar
30Lei, Z., Liu, C. and Zhou, Y.. Global solutions for incompressible viscoelastic fluids. Arch. Ration. Mech. Anal 188 (2008), 371–398.CrossRefGoogle Scholar
31Lei, Z., Masmoudi, N. and Zhou, Y.. Remarks on the blowup criteria for Oldroyd models. J. Differ. Equ. 248 (2010), 328–341.CrossRefGoogle Scholar
32Lemarié-Rieusset, P. G.. Recent developments in the Navier-Stokes problem, Chapman Hall/CRC Research Notes in Mathematics,vol. 431 (Boca Raton, FL: Chapman Hall/CRC, 2002).CrossRefGoogle Scholar
33Lin, F., Liu, C. and Zhang, P.. On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58 (2005), 1437–1471.CrossRefGoogle Scholar
34Lions, P. and Masmoudi, N.. Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann. Math. Ser. B 21 (2000), 131–146.CrossRefGoogle Scholar
35Masmoudi, N.. Global existence of weak solutions to macroscopic models of polymeric flows. J. Math. Pures Appl. 96 (2011), 502–520.CrossRefGoogle Scholar
36Oldroyd, J. G.. Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. Roy. Soc. London Ser. A 245 (1958), 278–297.Google Scholar
37Ye, Z.. On the global regularity of the 2D Oldroyd-B-type model, Annali di Matematica (2018). https://doi.org/10.1007/s10231-018-0784-2.CrossRefGoogle Scholar
38Ye, Z. and Xu, X.. Global regularity for the 2D Oldroyd-B model in the corotational case. Math. Methods Appl. Sci. 39 (2016), 3866–3879.CrossRefGoogle Scholar
39Zi, R.. Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces. Filomat 30 (2016), 3627–3639.CrossRefGoogle Scholar
40Zi, R., Fang, D. and Zhang, T.. Global solution to the incompressible Oldroyd-B model in the critical L p framework: the case of the non-small coupling parameter. Arch. Rational Mech. Anal. 213 (2014), 651–687.CrossRefGoogle Scholar