# 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 & 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. Google Scholar [2] X. Hu, Wellposedness of self-gravitating Hookean elastodynamics, preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. Google Scholar [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. Google Scholar [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. Google Scholar [17] Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math. , doi: 10.1002/cpa.21633. Google Scholar [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. Google Scholar [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. Google Scholar [20] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] S. -M. Zheng, Nonlinear Evolution Equations, CRC Press, New York, 2004.Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [2] X. Hu, Wellposedness of self-gravitating Hookean elastodynamics, preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] Z. Lei, On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. Google Scholar [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. Google Scholar [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. Google Scholar [17] Z. Lei, Global well-posedness of incompressible elastodynamics in two dimensions, Comm. Pure Appl. Math. , doi: 10.1002/cpa.21633. Google Scholar [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. Google Scholar [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. Google Scholar [20] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] S. -M. Zheng, Nonlinear Evolution Equations, CRC Press, New York, 2004.Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Bernard Ducomet, Eduard Feireisl, Hana Petzeltová, Ivan Straškraba. Global in time weak solutions for compressible barotropic self-gravitating fluids. Discrete & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Applied Analysis, 2009, 8 (4) : 1451-1467. doi: 10.3934/cpaa.2009.8.1451 [6] Dehua Wang. Global solution for the mixture of real compressible reacting flows in combustion. Communications on Pure & Applied Analysis, 2004, 3 (4) : 775-790. doi: 10.3934/cpaa.2004.3.775 [7] Jincheng Gao, Zheng-An Yao. Global existence and optimal decay rates of solutions for compressible Hall-MHD equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3077-3106. doi: 10.3934/dcds.2016.36.3077 [8] Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127 [9] Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001 [10] Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503 [11] Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001 [12] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [13] Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513 [14] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [15] Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 [16] Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085 [17] Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083 [18] Zaynab Salloum. Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 625-642. doi: 10.3934/cpaa.2010.9.625 [19] Colette Guillopé, Abdelilah Hakim, Raafat Talhouk. Existence of steady flows of slightly compressible viscoelastic fluids of White-Metzner type around an obstacle. Communications on Pure & Applied Analysis, 2005, 4 (1) : 23-43. doi: 10.3934/cpaa.2005.4.23 [20] Ammar Khemmoudj, Yacine Mokhtari. General decay of the solution to a nonlinear viscoelastic modified von-Kármán system with delay. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3839-3866. doi: 10.3934/dcds.2019155

2018 Impact Factor: 0.925