数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1301-1322.doi: 10.1007/s10473-023-0317-y
Yuhui Chen1, Qinghe Yao1, Minling Li2,†, Zheng-an Yao2
收稿日期:2022-02-11
修回日期:2022-06-10
出版日期:2023-06-25
发布日期:2023-06-06
通讯作者:
† Minling Li, E-mail: limling3@mail2.sysu.edu.cn
作者简介:Yuhui Chen, E-mail: chenyh339@mail.sysu.edu.cn; Qinghe Yao, E-mail: yaoqhe@mail.sysu.edu.cn;Zheng-an Yao,E-mail: mcsyao@mail.sysu.edu.cn
基金资助:Yuhui Chen1, Qinghe Yao1, Minling Li2,†, Zheng-an Yao2
Received:2022-02-11
Revised:2022-06-10
Online:2023-06-25
Published:2023-06-06
Contact:
† Minling Li, E-mail: limling3@mail2.sysu.edu.cn
About author:Yuhui Chen, E-mail: chenyh339@mail.sysu.edu.cn; Qinghe Yao, E-mail: yaoqhe@mail.sysu.edu.cn;Zheng-an Yao,E-mail: mcsyao@mail.sysu.edu.cn
Supported by:摘要： In this paper, we consider the initial value problem for the incompressible generalized Phan-Thien-Tanner (GPTT) model. This model pertains to the dynamic properties of polymeric fluids. Under appropriate assumptions on smooth function $f$, we find a particular solution to the GPTT model. In dimension three, we establish the global existence and the optimal time decay rates of strong solutions provided that the initial data is close to the particular solution. The results which are presented here are generalizations of the network viscoelastic models.
Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao. GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN ${\mathbb{R}}^{3}$*[J]. 数学物理学报(英文版), 2023, 43(3): 1301-1322.
Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao. GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN ${\mathbb{R}}^{3}$*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1301-1322.
| [1] Barrett J W, Lu Y, Süuli E.Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model. Commun Math Sci, 2017, 15(5): 1265-1323 [2] Barrett J W, Süuli E.Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains. Math Models Methods Appl Sci, 2011, 21(6): 1211-1289 [3] Barrett J W, Süuli E.Existence and equilibration of global weak solutions to kinetic models for dilute polymers II: Hookean-type bead-spring chains. Math Models Methods Appl Sci, 2012, 22(5): 1150024 [4] Bautista O, Sánchez S, Arcos J C, Méndez F.Lubrication theory for electroosmotic flow in a slit microchannel with the Phan-Thien and Tanner model. J Fluid Mech, 2013, 722: 496-532 [5] Chemin J Y, Masmoudi N.About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J Math Anal, 2001, 33(1): 84-112 [6] Chen Q L, Miao C X.Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces. Nonlinear Anal, 2008, 68(7): 1928-1939 [7] Chen Y H, Li M L, Yao Q H, Yao Z A. Global well-posedness for the three-dimensional generalized Phan- Thien-Tanner model in critical Besov spaces. J Math Fluid Mech, 2021, 23: Art 55 [8] Chen Y H, Li M L, Yao Q H, Yao Z A.The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien-Tanner system with magnetic field. Asymptot Anal, 2022, 129(3/4): 451-484 [9] Chen Y H, Luo W, Yao Z A.Blow up and global existence for the periodic Phan-Thein-Tanner model. J Differential Equations, 2019, 267(11): 6758-6782 [10] Chen Y H, Luo W, Yao Z A.Global existence and optimal time decay rates for the three-dimensional incompressible Phan-Thien-Tanner model. Anal Appl (Singap), 2022. DOI:10.1142/S0219530522500051 [11] Chen Y M, Zhang P.The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm Partial Differential Equations, 2006, 31(12): 1793-1810 [12] Fang D Y, Zi R Z.Global solutions to the Oldroyd-B model with a class of large initial data. SIAM J Math Anal, 2016, 48(2): 1054-1084 [13] Feng Z F, Zhu C J and Zi R Z. Blow-up criterion for the incompressible viscoelastic flows. J Funct Anal, 2017, 272(9): 3742-3762 [14] Fernández-Cara E, Guillén F, Ortega R R.Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind. Ann Scuola Norm Sup Pisa CI Sci, 1998, 26(1): 1-29 [15] Garduňo I E, Tamaddon-Jahromi H R, Walters K, Webster M F. The interpretation of a long-standing rheological flow problem using computational rheology and a PTT constitutive model. J Non-Newton Fluid Mech, 2016, 233: 27-36 [16] Guillopé C, Saut J C.Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal, 1990, 15(9): 849-869 [17] Guillopé C, Saut J C.Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type. RAIRO Modél Math Anal Numér, 1990, 24(3): 369-401 [18] He L B, Xu L.Global well-posedness for viscoelastic fluid system in bounded domains. SIAM J Math Anal, 2010, 42(6): 2610-2625 [19] Hieber M, Wen H Y, Zi R Z.Optimal decay rates for solutions to the incompressible Oldroyd-B model in R3. Nonlinearity, 2019, 32(3): 833-852 [20] Hu X P, Wang D H.Strong solutions to the three-dimensional compressible viscoelastic fluids. J Differential Equations, 2012, 252(6): 4027-4067 [21] Hu X P, Wu G C.Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows. SIAM J Math Anal, 2013, 45(5): 2815-2833 [22] Huang J R, Wang Y H, Wen H Y, Zi R Z.Optimal time-decay estimates for an Oldroyd-B model with zero viscosity. J Differential Equations, 2022, 306: 456-491 [23] Jiang F, Jiang S.Strong solutions of the equations for viscoelastic fluids in some classes of large data. J Differential Equations, 2021, 282: 148-183 [24] Lei Z, Liu C, Zhou Y.Global solutions for incompressible viscoelastic fluids. Arch Ration Mech Anal, 2008, 188(3): 371-398 [25] Lei Z, Masmoudi N, Zhou Y.Remarks on the blowup criteria for Oldroyd models. J Differential Equations, 2010, 248(2): 328-341 [26] Lin F H, Liu C, Zhang P.On hydrodynamics of viscoelastic fluids. Comm Pure Appl Math, 2005, 58(11): 1437-1471 [27] Lin F H, Zhang P.On the initial-boundary value problem of the incompressible viscoelastic fluid system. Comm Pure Appl Math, 2008, 61(4): 539-558 [28] Lions P L, Masmoudi N.Global solutions for some Oldroyd models of non-Newtonian flows. Chinese Ann Math Ser B, 2000, 21(2): 131-146 [29] Lu Y, Pokorný M.Global existence of large data weak solutions for a simplified compressible Oldroyd-B model without stress diffusion. Anal Theory Appl, 2020, 36(3): 348-372 [30] Lu Y, Zhang Z F.Relative entropy, weak-strong uniqueness, and conditional regularity for a compressible Oldroyd-B model. SIAM J Math Anal, 2018, 50(1): 557-590 [31] Masmoudi N.Global existence of weak solutions to macroscopic models of polymeric flows. J Math Pures Appl, 2011, 96(5): 502-520 [32] Mu Y, Zhao G, Wu X, Zhai J.Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models. Appl Math Comput, 2012, 218(17): 8429-8443 [33] Oliveira P J, Pinho F T.Analytical solution for fully developed channel and pipe flow of Phan-Thien-Tanner fluids. J Fluid Mech, 1999, 387: 271-280 [34] Omowunmi S C, Yuan X F.Time-dependent non-linear dynamics of polymer solutions in microfluidic contraction flow a numerical study on the role of elongational viscosity. Rheol Acta, 2013, 52: 337-354 [35] Phan-Thien N.A nonlinear network viscoelastic model. J Rheol, 1978, 22(3): 259-283 [36] Phan-Thien N, Tanner R I.A new constitutive equation derived from network theory. J Non-Newton Fluid Mech, 1977, 2(4): 353-365 [37] Qian J Z, Zhang Z F.Global well-posedness for compressible viscoelastic fluids near equilibrium. Arch Ration Mech Anal, 2010, 198(3): 835-868 [38] Sun Y Z, Zhang Z F.Global well-posedness for the 2D micro-macro models in the bounded domain. Comm Math Phys, 2011, 303(2): 361-383 [39] Wang W J, Wen H Y.The Cauchy problem for an Oldroyd-B model in three dimensions. Math Models Methods Appl Sci, 2020, 30(1): 139-179 [40] Zhang T, Fang D Y.Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical [41] Zhou Z S, Zhu C J, Zi R Z.Global well-posedness and decay rates for the three dimensional compressible Oldroyd-B model. J Differential Equations, 2018, 265(4): 1259-1278 [42] Zhu Y.Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism. J Funct Anal, 2018, 274(7): 2039-2060 [43] Zi R Z, Fang D Y, Zhang T.Global solution to the incompressible Oldroyd-B model in the critical Lp framework: The case of the non-small coupling parameters. Arch Ration Mech Anal, 2014, 213(2): 651-687 |
|