April  2019, 39(4): 2021-2057. doi: 10.3934/dcds.2019085

Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received  March 2018 Revised  August 2018 Published  January 2019

In this paper, we are concerned with the compressible viscoelastic flows in whole space $ \mathbb{R}^n $ with $ n\geq2 $. We aim at extending the global existence in energy spaces (see [18] by Hu & Wang and [30] by Qian & Zhang) such that it holds in more general $ L^p $ critical spaces, which allows to the case of large highly oscillating initial velocity. Precisely, We define "two effective velocities" which are used to eliminate the coupling between the density, velocity and deformation tensor. Consequently, the global existence in the $ L^p $ critical framework is constructed by elementary energy approaches. In addition, the optimal time-decay estimates of strong solutions are firstly shown in the $ L^p $ framework, which improve recent decay efforts for compressible viscoelastic flows.

Citation: 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
References:
[1]

H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l'espace critique, Rev. Mat. Iberoam., 23 (2007), 537-586. doi: 10.4171/RMI/505.

[2]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, 343. Springer, Heidelberg, 2011. xvi+523 pp. doi: 10.1007/978-3-642-16830-7.

[3]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x.

[4]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.

[5]

Q. ChenC. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325.

[6]

Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960.

[7]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90. doi: 10.1007/s00205-016-1067-y.

[8]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[9]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.

[10]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.

[11]

R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2.

[12]

R. Danchin, A Lagrangian approach for the compressible Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 64 (2014), 753-791. doi: 10.5802/aif.2865.

[13]

W. ET. Li and P. Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. Math. Phys., 248 (2004), 409-427. doi: 10.1007/s00220-004-1102-y.

[14]

M. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, 158, New York-London, 1981.

[15]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013.

[16]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[17]

D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x.

[23]

Z. LeiC. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. doi: 10.4310/CMS.2007.v5.n3.a5.

[24]

Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813.

[25]

F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.

[26]

F. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.

[27]

P. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170.

[28]

C. Liu and N. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252. doi: 10.1007/s002050100158.

[29]

J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234. doi: 10.1016/j.na.2009.12.022.

[30]

J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5.

[31]

V. Sohinger and R. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $ {\mathbb R} ^n_x$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[32]

J. Xu and S. Kawashima, The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218 (2015), 275-315. doi: 10.1007/s00205-015-0860-3.

[33]

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]

H. Abidi, Équation de Navier-Stokes avec densité et viscosité variables dans l'espace critique, Rev. Mat. Iberoam., 23 (2007), 537-586. doi: 10.4171/RMI/505.

[2]

H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, 343. Springer, Heidelberg, 2011. xvi+523 pp. doi: 10.1007/978-3-642-16830-7.

[3]

F. Charve and R. Danchin, A global existence result for the compressible Navier-Stokes equations in the critical Lp framework, Arch. Ration. Mech. Anal., 198 (2010), 233-271. doi: 10.1007/s00205-010-0306-x.

[4]

J. Chemin and N. Masmoudi, About lifespan of regular solutions of equations related to viscoelastic fluids, SIAM J. Math. Anal., 33 (2001), 84-112. doi: 10.1137/S0036141099359317.

[5]

Q. ChenC. Miao and Z. Zhang, Global well-posedness for compressible Navier-Stokes equations with highly oscillating initial velocity, Comm. Pure Appl. Math., 63 (2010), 1173-1224. doi: 10.1002/cpa.20325.

[6]

Y. Chen and P. Zhang, The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions, Comm. Partial Differential Equations, 31 (2006), 1793-1810. doi: 10.1080/03605300600858960.

[7]

R. Danchin and J. Xu, Optimal time-decay estimates for the compressible Navier-Stokes equations in the critical $L^p$ framework, Arch. Ration. Mech. Anal., 224 (2017), 53-90. doi: 10.1007/s00205-016-1067-y.

[8]

R. Danchin, Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614. doi: 10.1007/s002220000078.

[9]

R. Danchin, Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233. doi: 10.1081/PDE-100106132.

[10]

R. Danchin, Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 1311-1334. doi: 10.1017/S030821050000295X.

[11]

R. Danchin, On the uniqueness in critical spaces for compressible Navier-Stokes equations, Nonlinear Differential Equations Appl., 12 (2005), 111-128. doi: 10.1007/s00030-004-2032-2.

[12]

R. Danchin, A Lagrangian approach for the compressible Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), 64 (2014), 753-791. doi: 10.5802/aif.2865.

[13]

W. ET. Li and P. Zhang, Well-posedness for the dumbbell model of polymeric fluids, Comm. Math. Phys., 248 (2004), 409-427. doi: 10.1007/s00220-004-1102-y.

[14]

M. Gurtin, An introduction to continuum mechanics, Mathematics in Science and Engineering, 158, New York-London, 1981.

[15]

B. Haspot, Well-posedness in critical spaces for the system of compressible Navier-Stokes in larger spaces, J. Differential Equations, 251 (2011), 2262-2295. doi: 10.1016/j.jde.2011.06.013.

[16]

B. Haspot, Existence of global strong solutions in critical spaces for barotropic viscous fluids, Arch. Ration. Mech. Anal., 202 (2011), 427-460. doi: 10.1007/s00205-011-0430-2.

[17]

D. Hoff, Uniqueness of weak solutions of the Navier-Stokes equations of multidimensional, compressible flow, SIAM J. Math. Anal., 37 (2006), 1742-1760. doi: 10.1137/040618059.

[18]

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.

[19]

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.

[20]

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.

[21]

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.

[22]

Z. LeiC. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Ration. Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x.

[23]

Z. LeiC. Liu and Y. Zhou, Global existence for a 2D incompressible viscoelastic model with small strain, Commun. Math. Sci., 5 (2007), 595-616. doi: 10.4310/CMS.2007.v5.n3.a5.

[24]

Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813.

[25]

F. Lin and P. Zhang, On the initial-boundary value problem of the incompressible viscoelastic fluid system, Comm. Pure Appl. Math., 61 (2008), 539-558. doi: 10.1002/cpa.20219.

[26]

F. LinC. Liu and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471. doi: 10.1002/cpa.20074.

[27]

P. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B, 21 (2000), 131-146. doi: 10.1142/S0252959900000170.

[28]

C. Liu and N. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252. doi: 10.1007/s002050100158.

[29]

J. Qian, Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Anal., 72 (2010), 3222-3234. doi: 10.1016/j.na.2009.12.022.

[30]

J. Qian and Z. Zhang, Global well-posedness for compressible viscoelastic fluids near equilibrium, Arch. Ration. Mech. Anal., 198 (2010), 835-868. doi: 10.1007/s00205-010-0351-5.

[31]

V. Sohinger and R. Strain, The Boltzmann equation, Besov spaces, and optimal time decay rates in $ {\mathbb R} ^n_x$, Adv. Math., 261 (2014), 274-332. doi: 10.1016/j.aim.2014.04.012.

[32]

J. Xu and S. Kawashima, The optimal decay estimates on the framework of Besov spaces for generally dissipative systems, Arch. Ration. Mech. Anal., 218 (2015), 275-315. doi: 10.1007/s00205-015-0860-3.

[33]

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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