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doi: 10.3934/eect.2021041
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Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model

School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

* Corresponding author: Yuzhu Wang

Received  April 2021 Revised  June 2021 Early access August 2021

Fund Project: This work was supported in part by the NNSF of China (Grant No. 11871212) and the Basic Research Project of Key Scientific Research Project Plan of Universities in Henan Province(Grant No. 20ZX002)

In this paper, we investigate the optimal time-decay rates of global classical solutions for the compressible Oldroyd-B model in $ \mathbb{R}^n(n = 2,3) $. Global classical solutions in two space dimensions are still open. We first complete the proof of global classical solutions in two space dimensions. Based on global classical solutions and Fourier spectrum analysis, we obtain the optimal time-decay rates of global classical solutions in two and three space dimensions. More precisely, if the initial data belong to $ L^1 $, the optimal time-decay rate of solutions and time-decay rates of $ l(l = 1,\cdot\cdot\cdot,m) $ order derivatives under additional assumptions are established.

Citation: Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations & Control Theory, doi: 10.3934/eect.2021041
References:
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J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. Edinburgh Sect. A, 245 (1958), 278-297.  doi: 10.1098/rspa.1958.0083.  Google Scholar

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J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration Mech. Anal., 198 (2010), 835-868.  doi: 10.1007/s00205-010-0351-5.  Google Scholar

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[30]

Z. ZhouC. Zhu and R. Zi, Global well-posedness and decay rates for the three dimensional compressible Oldroyd-B model, J. Differential Equations, 265 (2018), 1259-1278.  doi: 10.1016/j.jde.2018.04.003.  Google Scholar

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Y. Zhu, Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism, J. Funct. Anal., 274 (2018), 2039-2060.  doi: 10.1016/j.jfa.2017.09.002.  Google Scholar

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R. ZiD. Fang and T. Zhang, 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.  doi: 10.1007/s00205-014-0732-2.  Google Scholar

show all references

References:
[1]

R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Fluid Mechanics, 1, 2nd edn, Wiley, New York, 1987. Google Scholar

[2]

J. Chemin and N. Masmoud, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112.  doi: 10.1137/S0036141099359317.  Google Scholar

[3]

Q. Chen and C. Miao, Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Anal., 68 (2008), 1928-1939.  doi: 10.1016/j.na.2007.01.042.  Google Scholar

[4]

Q. Chen and X. Hao, Global well-posedness in the critical Besov spaces for the incompressible Oldroyd-B model without damping mechanism, J. Math. Fluid Mech., 21 (2019), 23 pp. doi: 10.1007/s00021-019-0446-1.  Google Scholar

[5]

T. Elgindi and F. Rousset, Global regularity for some Oldroyd-B type models, Comm. Pure Appl. Math., 68 (2015), 2005-2021.  doi: 10.1002/cpa.21563.  Google Scholar

[6]

T. Elgindi and J. Liu, Global well-posedness to the generalized Oldroyd type models in $\mathbb{R}^3$, J. Differential Equations, 259 (2015), 1958-1966.  doi: 10.1016/j.jde.2015.03.026.  Google Scholar

[7]

D. Fang and R. Zi, Global solutions to the Oldroyd-B model with a class of large initial data, SIAM J. Math. Anal., 48 (2016), 1054-1084.  doi: 10.1137/15M1037020.  Google Scholar

[8]

D. Fang and R. Zi, Incompressible limit of Oldroyd-B fluids in the whole space, J. Differential Equations, 256 (2014), 2559-2602.  doi: 10.1016/j.jde.2014.01.017.  Google Scholar

[9]

C. Guillopé and J. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869.  doi: 10.1016/0362-546X(90)90097-Z.  Google Scholar

[10]

C. Guillopé and J. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model Math. Anal. Numér., 24 (1990), 369-401.  doi: 10.1051/m2an/1990240303691.  Google Scholar

[11]

C. GuillopeZ. Salloum and R. Talhouk, Regular flows of weakly compressible viscoelastic fluids and the incompressible limit, Discrete Contin. Dyn. Syst. Ser B, 14 (2010), 1001-1028.  doi: 10.3934/dcdsb.2010.14.1001.  Google Scholar

[12]

X. Hu, Global existence of weak solutions to two dimensional compressible viscoelastic flows, J. Differential Equations, 265 (2018), 3310-3167.  doi: 10.1016/j.jde.2018.05.001.  Google Scholar

[13]

X. Hu and D. Wang, Formation of sigularity for compressible viscoelasticity, Acta Math. Sci. Ser B (Engl Ed.), 32 (2012), 109-128.  doi: 10.1016/S0252-9602(12)60007-1.  Google Scholar

[14]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[15]

Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814.  doi: 10.1137/040618813.  Google Scholar

[16]

Z. Lei, Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann Math. Ser. B., 27 (2006), 565-580.  doi: 10.1007/s11401-005-0041-z.  Google Scholar

[17]

Z. LeiN. Masmoudi and Y. Zhou, Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.  doi: 10.1016/j.jde.2009.07.011.  Google Scholar

[18]

Z. Lei and F. Wang, Uniform bound of the highest energy for the three dimensional incompressible elastodynamics, Arch. Ration. Mech. Anal., 216 (2015), 593-622.  doi: 10.1007/s00205-014-0815-0.  Google Scholar

[19]

Z. LeiT. Sideris and Y. Zhou, Almost global existence for 2-D incompressible isotropic elastodynamics, Trans. Amer. Math. Soc., 367 (2015), 8175-8197.  doi: 10.1090/tran/6294.  Google Scholar

[20]

Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2072-2106.  doi: 10.1002/cpa.21633.  Google Scholar

[21]

Z. Lei, Incompressible elastic waves and viscoelastic fluids, Proceedings of the Sixth International Congress of Chinese Mathematicians, Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, II (2017), 575–586.  Google Scholar

[22]

F. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.  doi: 10.1002/cpa.20074.  Google Scholar

[23]

P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B., 21 (2000), 131-146.  doi: 10.1142/S0252959900000170.  Google Scholar

[24]

Y. Lu and Z. Zhang, Relative entropy, weak-strong uniqueness, and conditional regularity for a compressible oldroyd-B model, SIAM J. Math. Anal., 50 (2018), 557-590.  doi: 10.1137/17M1128654.  Google Scholar

[25]

J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. Edinburgh Sect. A, 245 (1958), 278-297.  doi: 10.1098/rspa.1958.0083.  Google Scholar

[26]

J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration Mech. Anal., 198 (2010), 835-868.  doi: 10.1007/s00205-010-0351-5.  Google Scholar

[27]

W. Wang and Y. Zhao, On the Rayleigh-Taylor instability in compressible viscoelastic fluids, J. Math. Anal. Appl., 463 (2018), 198-221.  doi: 10.1016/j.jmaa.2018.03.018.  Google Scholar

[28]

X. Zhai, Global solutions to the n-dimensional incompressible Oldroyd-B model without damping mechanism, J. Math. Phys., 62 (2021), 021503. doi: 10.1063/5.0010742.  Google Scholar

[29]

X. Zhai, Optimal decay for the n-dimensional incompressible Oldroyd-B model without damping mechanism, preprint, arXiv: 1905.02604v1. Google Scholar

[30]

Z. ZhouC. Zhu and R. Zi, Global well-posedness and decay rates for the three dimensional compressible Oldroyd-B model, J. Differential Equations, 265 (2018), 1259-1278.  doi: 10.1016/j.jde.2018.04.003.  Google Scholar

[31]

Y. Zhu, Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism, J. Funct. Anal., 274 (2018), 2039-2060.  doi: 10.1016/j.jfa.2017.09.002.  Google Scholar

[32]

Y. Zhu, Global existence of classical solutions for the 3D generalized compressible Oldroyd-B model, Math. Meth. Appl. Sci., 43 (2020), 6517-6528.  doi: 10.1002/mma.6393.  Google Scholar

[33]

R. ZiD. Fang and T. Zhang, 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.  doi: 10.1007/s00205-014-0732-2.  Google Scholar

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