## Rocky Mountain Journal of Mathematics

### Global existence and decay rate of strong solution to incompressible Oldroyd type model equations

#### Abstract

This paper investigates the global existence and the decay rate in time of a solution to the Cauchy problem for an incompressible Oldroyd model with a deformation tensor damping term. There are three major results. The first is the global existence of the solution for small initial data. Second, we derive the sharp time decay of the solution in $L^{2}$-norm. Finally, the sharp time decay of the solution of higher order Sobolev norms is obtained.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1703-1720.

Dates
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/euclid.rmjm/1539936042

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1703

Mathematical Reviews number (MathSciNet)
MR3866565

Zentralblatt MATH identifier
06958798

#### Citation

Yuan, Baoquan; Liu, Yun. Global existence and decay rate of strong solution to incompressible Oldroyd type model equations. Rocky Mountain J. Math. 48 (2018), no. 5, 1703--1720. doi:10.1216/RMJ-2018-48-5-1703. https://projecteuclid.org/euclid.rmjm/1539936042

#### References

• J.Y. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal. 33 (2001), 84–112.
• Y.M. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in $2$ and $3$ space dimensions, Comm. Part. Diff. Eqs. 31 (2006), 1793–1810.
• C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal. 267 (2014), 1034–1056.
• M.E. Gurtin, An introduction to continuum mechanics, Math. Sci. Eng. 158 (1981).
• X.P. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discr. Contin. Dynam. Syst. 35 (2015), 3437–3461.
• J.X. Jia, J. Peng and Z.D. Mei, Well-posedness and time-decay for compressible viscolelastic fluids in critical Besov space, J. Math. Appl. 418 (2014), 638–675.
• Q.S. Jiu and H. Yu, Decay of solutions to the three-dimensional generalized Navier-Stokes equations, Asympt. Anal. 94 (2015), 105–124.
• T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.
• C. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-Vries equation, J. Amer. Math. Soc. 4 (1991), 323–347.
• R.G. Larson, The structure and rheology of complex fluids, in Topics in chemical engineering, Oxford University Press, New York, 1998.
• Z. Lei, C. Liu and Y. Zhou, Global existence for a $2$D incompressible viscoelastic model with small strain, Comm. Math. Sci. 5 (2007), 595–616.
• ––––, Global solutions for incompressible viscoelastic fluids, Arch. Rat. Mech. Anal. 188 (2008), 371–398.
• Z. Lei and Y. Zhou, Global existence of classical solutions for $2$D Oldroyd model via the incompressible limit, SIAM J. Math. Anal. 37 (2005), 797–814.
• F.H. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math. 58 (2005), 1437–1471.
• F.H. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math. 61 (2008), 539–558.
• P.L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. 21 (2000), 131–146.
• A.J. Majda and A.L. Bertozzi, Vorticity and incompressible flow, Cambridge University Press, Cambridge, 2002.
• A. Matsumura and T. Nishita, The initial value problem for the equations of motion of viscous and heat conductive gases, Kyoto J. Math. 20 (1980), 67–104.
• J.Z. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlin. Anal. 72 (2010), 3222–3234.
• M.E. Schonbek, $L^{2}$ decay for weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 88 (1985), 209–222.
• ––––, Large time behaviour of solutions of the Navier-Stokes equations, Comm. Partial Diff. Eqs. 11 (1986), 733–763.
• R.Y. Wei and Y. Li, Decay of the compressible viscoelastic flows, Comm. Pure Appl. Anal. 5 (2016), 1603–1624.
• B.Q. Yuan and R. Li, The blow-up criteria of smooth solutions to the generalized and ideal incompressible viscoelastic flow, Math. Meth. Appl. Sci. 38 (2015), 4132–4139.
• B.Q. Yuan and Y. Liu, Local existence for the incompressible Oldroyd model equations, Chinese Quart. J. Math. 32 (2017), 331–334.
• T. Zhang and D.Y. Fang, Global existence of strong solution for equations related to the incompressible viscoelastic fluids in the critical $L^{p}$ framework, SIAM J. Math. Anal. 44 (2012), 2266–2288.