Global existence and stability for the 2D Oldroyd-B model with mixed partial dissipation
HTML articles powered by AMS MathViewer
- by Wen Feng, Weinan Wang and Jiahong Wu PDF
- Proc. Amer. Math. Soc. 150 (2022), 5321-5334 Request permission
Abstract:
This paper focuses on a two-dimensional incompressible Oldroyd-B model with mixed partial dissipation. The goal here is to establish the small data global existence and stability in the Sobolev space $H^2(\mathbb R^2)$. The velocity equation itself, without coupling with the equation of the non-Newtonian stress tensor, is an anisotropic 2D Navier-Stokes whose solutions are not known to be stable in Sobolev spaces due to potential rapid growth in time. By unearthing the hidden wave structure of the system and exploring the smoothing and stabilizing effect of the non-Newtonian stress tensor on the fluid, we are able to solve the desired global existence and stability problem.References
- Olfa Bejaoui and Mohamed Majdoub, Global weak solutions for some Oldroyd models, J. Differential Equations 254 (2013), no. 2, 660–685. MR 2990047, DOI 10.1016/j.jde.2012.09.010
- R.B. Bird, C.F. Curtiss, R.C. Armstrong, and O. Hassager, Dynamics of Polymetric Liquids, vol. 1, Fluid Mechanics, 2nd edn., Wiley, New York, (1987).
- Chongsheng Cao and Jiahong Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math. 226 (2011), no. 2, 1803–1822. MR 2737801, DOI 10.1016/j.aim.2010.08.017
- Jean-Yves Chemin and Nader Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (2001), no. 1, 84–112. MR 1857990, DOI 10.1137/S0036141099359317
- Qionglei Chen and Xiaonan Hao, Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism, J. Math. Fluid Mech. 21 (2019), no. 3, Paper No. 42, 23. MR 3978494, DOI 10.1007/s00021-019-0446-1
- Qionglei Chen and Changxing Miao, Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal. 68 (2008), no. 7, 1928–1939. MR 2388753, DOI 10.1016/j.na.2007.01.042
- Peter Constantin, Lagrangian-Eulerian methods for uniqueness in hydrodynamic systems, Adv. Math. 278 (2015), 67–102. MR 3341785, DOI 10.1016/j.aim.2015.03.010
- Peter Constantin, Analysis of hydrodynamic models, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 90, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. MR 3660694, DOI 10.1137/1.9781611974805.ch1
- Peter Constantin and Markus Kliegl, Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Ration. Mech. Anal. 206 (2012), no. 3, 725–740. MR 2989441, DOI 10.1007/s00205-012-0537-0
- Peter Constantin and Weiran Sun, Remarks on Oldroyd-B and related complex fluid models, Commun. Math. Sci. 10 (2012), no. 1, 33–73. MR 2901300, DOI 10.4310/CMS.2012.v10.n1.a3
- Peter Constantin, Jiahong Wu, Jiefeng Zhao, and Yi Zhu, High Reynolds number and high Weissenberg number Oldroyd-B model with dissipation, J. Evol. Equ. 21 (2021), no. 3, 2787–2806. MR 4350254, DOI 10.1007/s00028-020-00616-8
- Tarek M. Elgindi and Jianli Liu, Global wellposedness to the generalized Oldroyd type models in $\Bbb {R}^3$, J. Differential Equations 259 (2015), no. 5, 1958–1966. MR 3349425, DOI 10.1016/j.jde.2015.03.026
- Tarek M. Elgindi and Frederic Rousset, Global regularity for some Oldroyd-B type models, Comm. Pure Appl. Math. 68 (2015), no. 11, 2005–2021. MR 3403757, DOI 10.1002/cpa.21563
- Daoyuang Fang, Matthias Hieber, and Ruizhao Zi, Global existence results for Oldroyd-B fluids in exterior domains: the case of non-small coupling parameters, Math. Ann. 357 (2013), no. 2, 687–709. MR 3096521, DOI 10.1007/s00208-013-0914-5
- Daoyuan Fang and Ruizhao Zi, Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal. 48 (2016), no. 2, 1054–1084. MR 3473592, DOI 10.1137/15M1037020
- Wen Feng, Farzana Hafeez, and Jiahong Wu, Influence of a background magnetic field on a 2D magnetohydrodynamic flow, Nonlinearity 34 (2021), no. 4, 2527–2562. MR 4246463, DOI 10.1088/1361-6544/abb928
- Enrique Fernández Cara, Francisco Guillén, and Rubens R. Ortega, Existence et unicité de solution forte locale en temps pour des fluides non newtoniens de type Oldroyd (version $L^s$–$L^r$), C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 411–416 (French, with English and French summaries). MR 1289322
- C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal. 15 (1990), no. 9, 849–869. MR 1077577, DOI 10.1016/0362-546X(90)90097-Z
- C. Guillopé and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modél. Math. Anal. Numér. 24 (1990), no. 3, 369–401 (English, with French summary). MR 1055305, DOI 10.1051/m2an/1990240303691
- Anupam Gupta and Dario Vincenzi, Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence, J. Fluid Mech. 870 (2019), 405–418. MR 3948679, DOI 10.1017/jfm.2019.224
- Matthias Hieber, Huanyao Wen, and Ruizhao Zi, Optimal decay rates for solutions to the incompressible Oldroyd-B model in $\Bbb R^3$, Nonlinearity 32 (2019), no. 3, 833–852. MR 3909068, DOI 10.1088/1361-6544/aaeec7
- D. Hu and T. Lelièvre, New entropy estimates for Oldroyd-B and related models, Commun. Math. Sci. 5 (2007), no. 4, 909–916. MR 2375053
- Alexander Kiselev and Vladimir Šverák, Small scale creation for solutions of the incompressible two-dimensional Euler equation, Ann. of Math. (2) 180 (2014), no. 3, 1205–1220. MR 3245016, DOI 10.4007/annals.2014.180.3.9
- J. La, On diffusive 2D Focker-Planck-Navier-Stokes systems, arXiv:1804.05168, ARMA (2019). https://doi.org/10.1007/s00205-019-01450-0.
- Joonhyun La, Global well-posedness of strong solutions of Doi model with large viscous stress, J. Nonlinear Sci. 29 (2019), no. 5, 1891–1917. MR 4007623, DOI 10.1007/s00332-019-09533-8
- Junghaeng Lee, Wook Ryol Hwang, and Kwang Soo Cho, Effect of stress diffusion on the Oldroyd-B fluid flow past a confined cylinder, J. Non-Newton. Fluid Mech. 297 (2021), Paper No. 104650, 11. MR 4316727, DOI 10.1016/j.jnnfm.2021.104650
- Fang-Hua Lin, Chun Liu, and Ping Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58 (2005), no. 11, 1437–1471. MR 2165379, DOI 10.1002/cpa.20074
- Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
- P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B 21 (2000), no. 2, 131–146. MR 1763488, DOI 10.1142/S0252959900000170
- Zhen Lei, Nader Masmoudi, and Yi Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations 248 (2010), no. 2, 328–341. MR 2558169, DOI 10.1016/j.jde.2009.07.011
- J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London Ser. A 245 (1958), 278–297. MR 94085, DOI 10.1098/rspa.1958.0083
- J. Pedlosky, Geophysical fluid dynamics, Springer-Verlag, New York, 1987.
- R. Sureshkumar and A.N. Beris, Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows , J. Non-Newtonian Fluid Mech. 60 (1995), 53–80.
- Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
- B. Thomases, An analysis of the effect of stress diffusion on the dynamics of creeping viscoelastic flow, J. Non-Newtonian Fluid Mech. 166 (2011), 1221–1228.
- Renhui Wan, Some new global results to the incompressible Oldroyd-B model, Z. Angew. Math. Phys. 70 (2019), no. 1, Paper No. 28, 29. MR 3899180, DOI 10.1007/s00033-019-1074-6
- Peixin Wang, Jiahong Wu, Xiaojing Xu, and Yueyuan Zhong, Sharp decay estimates for Oldroyd-B model with only fractional stress tensor diffusion, J. Funct. Anal. 282 (2022), no. 4, Paper No. 109332, 55. MR 4348795, DOI 10.1016/j.jfa.2021.109332
- Jiahong Wu and Jiefeng Zhao, Global regularity for the generalized incompressible Oldroyd-B model with only stress tensor dissipation in critical Besov spaces, J. Differential Equations 316 (2022), 641–686. MR 4377168, DOI 10.1016/j.jde.2022.01.059
- J. Wu, J. Zhao, Global regularity for the generalized incompressible Oldroyd-B model with only velocity dissipation and no stress tensor damping, preprint.
- Xiaoqian Xu, Fast growth of the vorticity gradient in symmetric smooth domains for 2D incompressible ideal flow, J. Math. Anal. Appl. 439 (2016), no. 2, 594–607. MR 3475939, DOI 10.1016/j.jmaa.2016.02.066
- Zhuan Ye, On the global regularity of the 2D Oldroyd-B-type model, Ann. Mat. Pura Appl. (4) 198 (2019), no. 2, 465–489. MR 3927165, DOI 10.1007/s10231-018-0784-2
- Zhuan Ye and Xiaojing Xu, Global regularity for the 2D Oldroyd-B model in the corotational case, Math. Methods Appl. Sci. 39 (2016), no. 13, 3866–3879. MR 3529389, DOI 10.1002/mma.3834
- Xiaoping Zhai, Global solutions to the $n$-dimensional incompressible Oldroyd-B model without damping mechanism, J. Math. Phys. 62 (2021), no. 2, Paper No. 021503, 17. MR 4210721, DOI 10.1063/5.0010742
- Yi Zhu, Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism, J. Funct. Anal. 274 (2018), no. 7, 2039–2060. MR 3762094, DOI 10.1016/j.jfa.2017.09.002
- Ruizhao Zi, Daoyuan Fang, and Ting Zhang, Global solution to the incompressible Oldroyd-B model in the critical $L^p$ framework: the case of the non-small coupling parameter, Arch. Ration. Mech. Anal. 213 (2014), no. 2, 651–687. MR 3211863, DOI 10.1007/s00205-014-0732-2
Additional Information
- Wen Feng
- Affiliation: Department of Mathematics, 5795 Lewiston Rd, Niagara University, New York 14109
- Email: wfeng@niagara.edu
- Weinan Wang
- Affiliation: Department of Mathematics, University of Arizona, Tucson, Arizona 85721
- MR Author ID: 1314789
- Email: weinanwang@math.arizona.edu
- Jiahong Wu
- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- MR Author ID: 367820
- ORCID: 0000-0001-9496-9709
- Email: jiahong.wu@okstate.edu
- Received by editor(s): January 7, 2021
- Received by editor(s) in revised form: February 12, 2022
- Published electronically: June 16, 2022
- Additional Notes: The second author was partially supported by an AMS-Simons Travel Grant. The third author was partially supported by the National Science Foundation of the United States (DMS 2104682) and the AT&T Foundation at Oklahoma State University.
- Communicated by: Ariel Barton
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 5321-5334
- MSC (2020): Primary 35Q30, 35Q35, 35Q92
- DOI: https://doi.org/10.1090/proc/16039