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Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation
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 |
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 [
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. Chen, C. 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. E, T. 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. Lei, C. 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. Lei, C. 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. Lin, C. 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. Chen, C. 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. E, T. 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. Lei, C. 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. Lei, C. 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. Lin, C. 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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