Abstract
In this paper, we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner model. By a refined energy method, we prove the global existence under the assumption that the H3 norm of the initial data is small, but that the higher order derivatives can be large. If the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces, we obtain the time decay rates of the solution and its higher order spatial derivatives. Moreover, we also obtain the usual Lp − L2 (1 ≤ p ≤ 2) type of the decay rate without requiring that the Lp norm of initial data is small.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bautista O, Sánchez S, Arcos J C, Méndez F. Lubrication theory for electro-osmotic flow in a slit microchannel with the Phan-Thien and Tanner model. J Fluid Mech, 2013, 722: 496–532
Bird R B, Armstrong R C, Hassager O. Dynamics of Polymeric Liquids. Volume 1. New York: Wiley, 1977
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: 6758–6782
Chen Y H, Luo W, Zhai X P. Global well-posedness for the Phan-Thein-Tanner model in critical Besov spaceswithout damping. J Math Phys, 2019, 60: 061503
Duan R J, Ukai S, Yang T, Zhao H J. Optimal convergence rate for compressible Navier-Stokes equations with potential force. Math Models Methods Appl Sci, 2007, 17: 737–758
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: 1054–1084
Fang D Y, Zi R Z. Strong solutions of 3D compressible Oldroyd-B fluids. Math Methods Appl Sci, 2013, 36: 1423–1439
Fang D Y, Zi R Z. Incompressible limit of Oldroyd-B fluids in the whole space. J Differential Equations, 2014, 256: 2559–2602
Garduño 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
Guillopé C, Saut J C. Existence results for the flow of viscoelastic fluids with a differential constitutive law. Nonlinear Anal, 1990, 15: 849–869
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, 190, 24: 369–401
Guo Y, Wang Y J. Decay of dissipative equations and negative Sobolev spaces. Comm Partial Differential Equations, 20112, 37: 2165–2208
Hu X P, Wang D. Local strong solution to the compressible viscoelastic flow with large data. J Differential Equations, 2010, 249: 1179–1198
Hu X P, Wu G C. Global existence and optimal decay rates for three-dimensionalcompressible viscoelastic flows. SIAM J Math Anal, 2013, 45: 2815–2833
Ju N. Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space. Comm Math Phys, 2004, 251: 365–376
Lei Z. Global existence of classical solutions for some Oldroyd-B model via the incompressible limit. Chin Ann Math, 2006, 27B: 565–580
Lei Z, Liu C, Zhou Y. Global solutions for incompressible viscoelastic fluids. Arch Ration Mech Anal, 2008, 188: 371–398
Li Y, Wei R Y, Yao Z A. Optimal decay rates for the compressible viscoelastic flows. J Math Phys, 2016, 57: 111506
Lin F H, Liu C, Zhang P. On hydrodynamics of viscoelastic fluids. Comm Pure Appl Math, 2005, 58: 1437–1471
Matsumura A. An Energy Method for the Equations of Motion of Compressible Viscous and Heat-conductive Fluids. University of Wisconsin-Madison MRC Technical Summary Report, 1986, 2194: 1–16
Matsumura A, Nishida T. The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc Japan Acad Ser A, 1979, 55: 337–342
Matsumura A, Nishida T. The initial value problems for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67–104
Molinet L, Talhouk R. On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law. NoDEA Nonlinear Differential Equations Appl, 2004, 11: 349–359
Nirenberg L. On elliptic partial differential equations. Ann Scuola Norm Sup Pisa, 1959, 13: 115–162
Oldroyd J G. Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc Roy Soc London Ser A, 1958, 245: 278–297
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
Ponce G. Global existence of small solutions to a class of nonlinear evolution equations. Nonlinear Anal, 1985, 9: 339–418
Qian J Z, Zhang Z F. Global well-posedness for compressible viscoelastic fluids near equilibrium. Arch Ration Mech Anal, 2010, 198: 835–868
Schonbek M E, Wiegner M. On the decay of higher-order norms of the solutions of Navier-Stokes equations. Proc Roy Soc Edinburgh Sect A, 1996, 126: 677–685
Sohinger V, Strain R M. The Boltzmann equation, Besov spaces, and optimal time decay rates in ℝ nx . Advances in Mathematics, 2014, 261: 274–332
Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1970
Strain R M, Guo Y. Almost exponential decay near Maxwellian. Comm Partial Differential Equations, 2006, 31: 417–429
Tan Z, Wang Y J. On hyperbolic-dissipative systems of composite type. J Differential Equations, 2016, 260: 1091–1125
Tan Z, Wang Y J, Wang Y. Decay estimates of solutions to the compressible Euler-Maxwell system in ℝ3. J Differential Equations, 2014, 257: 2846–2873
Tan Z, Wu W P, Zhou J F. Global existence and decay estimate of solutions to magneto-micropolar fluid equations. J Differential Equations, 2019, 266: 4137–4169
Wang Y J. Decay of the Navier-Stokes-Poisson equations. J Differential Equations, 2012, 253: 273–297
Wei R Y, Li Y, Yao Z A. Decay of the compressible viscoelastic flows. Commun Pure Appl Anal, 2016, 15: 1603–1624
Zhang T, Fang D. Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical Lp framework. SIAM J Math Anal, 2012, 44: 2266–2288
Zi R Z. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete Contin Dyn Syst Ser A, 2017, 37: 6437–6470
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: 1259–1278
Zhu Y. Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism. J Funct Anal, 2018, 274: 2039–2060
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is partially supported by the National Natural Science Foundation of China (11926354, 11971496), Natural Science Foundation of Guangdong Province (2019A1515011320, 2021A1515010292, 2214050001249), Innovative team project of ordinary universities of Guangdong Province (2020KCXTD024), Characteristic innovation projects of ordinary colleges and universities in Guangdong Province (2020KTSCX134), and the Education Research Platform Project of Guangdong Province (2018179).
Rights and permissions
About this article
Cite this article
Wei, R., Li, Y. & Yao, Za. The Global Existence and a Decay Estimate of Solutions to the Phan-Thein-Tanner Model. Acta Math Sci 42, 1058–1080 (2022). https://doi.org/10.1007/s10473-022-0314-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-022-0314-6