# American Institute of Mathematical Sciences

• Previous Article
Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction
• CPAA Home
• This Issue
• Next Article
Biharmonic systems involving multiple Rellich-type potentials and critical Rellich-Sobolev nonlinearities
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 and 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.
 [1] Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete and Continuous Dynamical Systems, 2004, 11 (1) : 113-130. doi: 10.3934/dcds.2004.11.113 [2] W. Wei, Yin Li, Zheng-An Yao. Decay of the compressible viscoelastic flows. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1603-1624. doi: 10.3934/cpaa.2016004 [3] René Pinnau, Oliver Tse. On a regularized system of self-gravitating particles. Kinetic and Related Models, 2014, 7 (3) : 591-604. doi: 10.3934/krm.2014.7.591 [4] Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $L^p$ critical spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085 [5] Yulan Xu, Yanping Dou. Large BV solutions to Euler equations in the isothermal self-gravitating gases with damping. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1451-1467. doi: 10.3934/cpaa.2009.8.1451 [6] Lan Huang, Zhiying Sun, Xin-Guang Yang, Alain Miranville. Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1595-1620. doi: 10.3934/cpaa.2022033 [7] Dehua Wang. Global solution for the mixture of real compressible reacting flows in combustion. Communications on Pure and Applied Analysis, 2004, 3 (4) : 775-790. doi: 10.3934/cpaa.2004.3.775 [8] Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021041 [9] Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077 [10] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [11] Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure and Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 [12] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [13] Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513 [14] Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001 [15] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [16] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [17] Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 [18] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [19] Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083 [20] Ruiying Wei, Yin Li, Zheng-an Yao. Global existence and convergence rates of solutions for the compressible Euler equations with damping. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2949-2967. doi: 10.3934/dcdsb.2020047

2020 Impact Factor: 1.916