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March 2018, 17(2): 347-374. doi: 10.3934/cpaa.2018020

## Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating

 1 School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China 2 College of Sciences, Henan University of Engineering, Zhengzhou, 451191, China

Received  February 2017 Revised  April 2017 Published  March 2018

Fund Project: YXW is supported by NNSF grant No.11101144

In this paper, we consider the initial value problem for the compressible viscoelastic flows with self-gravitating in $\mathbb{R}^n(n≥ 3)$. Global existence and decay rates of classical solutions are established. The corresponding linear equations becomes two similar equations by using Hodge decomposition and then the solutions operator is derived. The proof is mainly based on the decay properties of the solutions operator and energy method. The decay properties of the solutions operator may be derived from the pointwise estimate of the solution operator to two linear wave equations.

Citation: Yinxia Wang, Hengjun Zhao. Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating. Communications on Pure & Applied Analysis, 2018, 17 (2) : 347-374. doi: 10.3934/cpaa.2018020
##### References:
 [1] Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 ane 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. [2] X. Hu, Wellposedness of self-gravitating Hookean elastodynamics, preprint. [3] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198. doi: 10.1016/j.jde.2010.03.027. [4] X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231. doi: 10.1016/j.jde.2010.10.017. [5] X. Hu and D. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equations, 252 (2012), 4027-4067. doi: 10.1016/j.jde.2011.11.021. [6] X. Hu and D. Wang, The initial-boundary value problem for the compressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 917-934. doi: 10.3934/dcds.2015.35.917. [7] X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833. doi: 10.1137/120892350. [8] X. Hu and F. Lin, Scaling limit for compressible viscoelastic fluids, Frontiers in Differential Geometry, Partial Differential Equations and Mathematical Physics, 243-269, World Sci. Publ., Hackensack, NJ, 2014. [9] X. Hu and F. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Comm. Pure Appl. Math., 69 (2016), 372-404. doi: 10.1002/cpa.21561. [10] X. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461. doi: 10.3934/dcds.2015.35.3437. [11] B. Han, Global strong solution for the density dependent incompressible viscoelastic fluids in the critical $L^p$ framework, Nonlinear Anal., 132 (2016), 337-358. doi: 10.1016/j.na.2015.11.011. [12] Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. [13] Z. Lei, C. Liu and Y. Zhou, Global solutions of incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. [14] Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. [15] Z. Lei, Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 2861-2871. doi: 10.3934/dcds.2014.34.2861. [16] 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. [17] Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math. , doi: 10.1002/cpa.21633. [18] F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. [19] J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Rational Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5. [20] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. [21] Y.-Z. Wang, F. G. Liu and Y. Z. Zhang, Global existence and asymptotic of solutions for a semi-linear wave equation, J. Math. Anal. Appl., 385 (2012), 836-853. doi: 10.1016/j.jmaa.2011.07.010. [22] Y.-Z. Wang and K. Y. Wang, Large time behavior of solutions to the nonlinear pseudo-parabolic equation, J. Math. Anal. Appl., 417 (2014), 272-292. doi: 10.1016/j.jmaa.2014.03.030. [23] Y.-Z. Wang and K. Y. Wang, Asymptotic behavior of classical solutions to the compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 259 (2015), 25,-47. doi: 10.1016/j.jde.2015.01.042. [24] Y.-Z. Wang and K. Y. Wang, Long time behavior of solutions to the compressible MHD system in multi-dimensions, J. Math. Anal. Appl., 429 (2015), 1033-1058. doi: 10.1016/j.jmaa.2015.04.045. [25] S. -M. Zheng, Nonlinear Evolution Equations, CRC Press, New York, 2004. [26] F. Xu, X. Zhang, Y. Wu and L. Liu, The optimal convergence rates for the multi-dimensioanl compressible viscoelastic flows, Z. Angew. Math. Mech., 96 (2016), 1490-1504. doi: 10.1002/zamm.201500095. [27] T. Zhang and D. 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. doi: 10.1137/110851742.

show all references

##### References:
 [1] Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 ane 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960. [2] X. Hu, Wellposedness of self-gravitating Hookean elastodynamics, preprint. [3] X. Hu and D. Wang, Local strong solution to the compressible viscoelastic flow with large data, J. Differential Equations, 249 (2010), 1179-1198. doi: 10.1016/j.jde.2010.03.027. [4] X. Hu and D. Wang, Global existence for the multi-dimensional compressible viscoelastic flows, J. Differential Equations, 250 (2011), 1200-1231. doi: 10.1016/j.jde.2010.10.017. [5] X. Hu and D. Wang, Strong solutions to the three-dimensional compressible viscoelastic fluids, J. Differential Equations, 252 (2012), 4027-4067. doi: 10.1016/j.jde.2011.11.021. [6] X. Hu and D. Wang, The initial-boundary value problem for the compressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 917-934. doi: 10.3934/dcds.2015.35.917. [7] X. Hu and G. Wu, Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows, SIAM J. Math. Anal., 45 (2013), 2815-2833. doi: 10.1137/120892350. [8] X. Hu and F. Lin, Scaling limit for compressible viscoelastic fluids, Frontiers in Differential Geometry, Partial Differential Equations and Mathematical Physics, 243-269, World Sci. Publ., Hackensack, NJ, 2014. [9] X. Hu and F. Lin, Global solutions of two-dimensional incompressible viscoelastic flows with discontinuous initial data, Comm. Pure Appl. Math., 69 (2016), 372-404. doi: 10.1002/cpa.21561. [10] X. Hu and H. Wu, Long-time behavior and weak-strong uniqueness for incompressible viscoelastic flows, Discrete Contin. Dyn. Syst., 35 (2015), 3437-3461. doi: 10.3934/dcds.2015.35.3437. [11] B. Han, Global strong solution for the density dependent incompressible viscoelastic fluids in the critical $L^p$ framework, Nonlinear Anal., 132 (2016), 337-358. doi: 10.1016/j.na.2015.11.011. [12] Z. Lei, C. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. [13] Z. Lei, C. Liu and Y. Zhou, Global solutions of incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. [14] Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. [15] Z. Lei, Rotation-strain decomposition for the incompressible viscoelasticity in two dimensions, Discrete Contin. Dyn. Syst., 34 (2014), 2861-2871. doi: 10.3934/dcds.2014.34.2861. [16] 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. [17] Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math. , doi: 10.1002/cpa.21633. [18] F. Lin, C. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074. [19] J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Rational Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5. [20] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990. [21] Y.-Z. Wang, F. G. Liu and Y. Z. Zhang, Global existence and asymptotic of solutions for a semi-linear wave equation, J. Math. Anal. Appl., 385 (2012), 836-853. doi: 10.1016/j.jmaa.2011.07.010. [22] Y.-Z. Wang and K. Y. Wang, Large time behavior of solutions to the nonlinear pseudo-parabolic equation, J. Math. Anal. Appl., 417 (2014), 272-292. doi: 10.1016/j.jmaa.2014.03.030. [23] Y.-Z. Wang and K. Y. Wang, Asymptotic behavior of classical solutions to the compressible Navier-Stokes-Poisson equations in three and higher dimensions, J. Differential Equations, 259 (2015), 25,-47. doi: 10.1016/j.jde.2015.01.042. [24] Y.-Z. Wang and K. Y. Wang, Long time behavior of solutions to the compressible MHD system in multi-dimensions, J. Math. Anal. Appl., 429 (2015), 1033-1058. doi: 10.1016/j.jmaa.2015.04.045. [25] S. -M. Zheng, Nonlinear Evolution Equations, CRC Press, New York, 2004. [26] F. Xu, X. Zhang, Y. Wu and L. Liu, The optimal convergence rates for the multi-dimensioanl compressible viscoelastic flows, Z. Angew. Math. Mech., 96 (2016), 1490-1504. doi: 10.1002/zamm.201500095. [27] T. Zhang and D. 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. doi: 10.1137/110851742.
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