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Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case
Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China |
This paper is dedicated to the global well-posedness issue of the compressible Oldroyd-B model in the whole space $\mathbb{R}^d$ with $d≥2 $. By exploiting the intrinsic structure of the system, we prove that if the initial data is small enough (depending on the coupling parameter), this set of equations admits a unique global solution in a certain critical Besov space. This result partially improves the previous work by Fang and the author [J. Differential Equations, 256(2014), 2559-2602].
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[3] |
M. Cannone,
A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.
doi: 10.4171/RMI/229. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions Autosimilaires des Équations de Navier-Stokes, in Séminaire "Équations aux Dérivées Partielles" de l'École Polytechnique, Exposé Ⅷ, 1993-1994, Palaiseau, 1994.
doi: 10.1108/09533239410052824. |
[5] |
J.-Y. Chemin,
Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, Pisa, (2004), 53-135.
|
[6] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-228.
doi: 10.1006/jdeq.1995.1131. |
[7] |
J. Y. 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. |
[8] |
Q. L. Chen and C. X. Miao,
Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Analysis, 68 (2008), 1928-1939.
doi: 10.1016/j.na.2007.01.042. |
[9] |
P. Constantin and M. Kliegl,
Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Rational Mech. Anal., 206 (2012), 725-740.
doi: 10.1007/s00205-012-0537-0. |
[10] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[11] |
R. Danchin,
A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations, Sci. China Math., 55 (2012), 245-275.
doi: 10.1007/s11425-011-4357-8. |
[12] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up critirion for 3D compressible viscoelasticity, arXiv: 1202.3693v1 [math. AP] 16 Feb 2012. |
[13] |
D.Y. Fang, M. Hieber and R. Z. Zi,
Global existence results for Oldroyd-B Fluids in exterior domains: The case of non-small coupling parameters, Math. Ann., 357 (2013), 687-709.
doi: 10.1007/s00208-013-0914-5. |
[14] |
D. Y. Fang and R. Z. Zi,
Strong solutions of 3D compressible Oldroyd-B fluids, Math. Meth. Appl. Sci., 36 (2013), 1423-1439.
doi: 10.1002/mma.2695. |
[15] |
D. Y. Fang and R. Z. Zi,
Incompressible limit of Oldroyd-B fluids in the whole space, J. Differential Equations, 256 (2014), 2559-2602.
doi: 10.1016/j.jde.2014.01.017. |
[16] |
E. Fernández-Cara, F. Guillén and R. Ortega,
Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 1-29.
|
[17] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[18] |
C. Guillopé and J. C. Saut,
Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869.
doi: 10.1016/0362-546X(90)90097-Z. |
[19] |
C. Guillopé, Z. Salloum and R. Talhouk,
Regular flows of weakly compressible viscoelastic fluids and the incompressible limit, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1001-1028.
doi: 10.3934/dcdsb.2010.14.1001. |
[20] |
M. Hieber, Y. Naito and Y. Shibata,
Global existence results for Oldroyd-B fluids in exterior domains, J. Differential Equations, 252 (2012), 2617-2629.
doi: 10.1016/j.jde.2011.09.001. |
[21] |
X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497. |
[22] |
X. P. Hu and D. H. Wang, Formation of sigularity for compressible viscoelasticity, arXiv: 1109.1332v1 [math. AP] 7 Sep 2011. |
[23] |
T. Kato,
Strong $L^p $-solutions of the Navier-Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[24] |
H. Koch and D. Tataru,
Well-posedness for the Navier--Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[25] |
R. Kupferman, C. Mangoubi and E. S. Titi,
A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime, Commun. Math. Sci., 6 (2008), 235-256.
doi: 10.4310/CMS.2008.v6.n1.a12. |
[26] |
Z. Lei,
Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[27] |
Z. Lei,
On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[28] |
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. |
[29] |
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. |
[30] |
Z. Lei, N. Masmoudi and Y. Zhou,
Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.
doi: 10.1016/j.jde.2009.07.011. |
[31] |
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. |
[32] |
F. H. Lin, C. Liu and P. Zhang,
On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[33] |
F. H. 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. |
[34] |
P. L. 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. |
[35] |
L. Molinet and R. Talhouk,
On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, Nonlinear Diff. Equations Appl., 11 (2004), 349-359.
doi: 10.1007/s00030-004-1073-x. |
[36] |
J. G. Oldroyd,
Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London., 245 (1958), 278-297.
doi: 10.1098/rspa.1958.0083. |
[37] |
J. Z. Qian,
Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Analysis, 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[38] |
J. Z. Qian and Z. F. Zhang,
Global well-posedness for compressible viscoelastic fluids near equilibrum, Arch. Rational Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[39] |
R. Talhouk,
Analyse Mathématique de Quelques Écoulements de Fluides Viscoélastiques, Thèse, Université Paris-Sud, 1994. |
[40] |
F. Weissler,
The Navier-Stokes initial value problem in $L^p $, Arch. Rational Mech. Anal., 74 (1980), 219-230.
doi: 10.1007/BF00280539. |
[41] |
T. Zhang and D. Y. 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. |
[42] |
R. Z. Zi, D. Y. Fang and T. Zhang,
Global solution to the incompressible Oldroyd-B model in the critical $ L^p$ framework: the case of the non-small coupling parameter, Arch. Rational Mech. Anal., 213 (2014), 651-687.
doi: 10.1007/s00205-014-0732-2. |
show all references
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour équations aux dérivées partielles nonlinéaires, Annales Scinentifiques de l'école Normale Supérieure, 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[3] |
M. Cannone,
A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamericana, 13 (1997), 515-541.
doi: 10.4171/RMI/229. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions Autosimilaires des Équations de Navier-Stokes, in Séminaire "Équations aux Dérivées Partielles" de l'École Polytechnique, Exposé Ⅷ, 1993-1994, Palaiseau, 1994.
doi: 10.1108/09533239410052824. |
[5] |
J.-Y. Chemin,
Localization in Fourier space and Navier-Stokes system, Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, Pisa, (2004), 53-135.
|
[6] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, J. Differential Equations, 121 (1995), 314-228.
doi: 10.1006/jdeq.1995.1131. |
[7] |
J. Y. 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. |
[8] |
Q. L. Chen and C. X. Miao,
Global well-posedness of viscoelastic fluids of Oldroyd type in Besov spaces, Nonlinear Analysis, 68 (2008), 1928-1939.
doi: 10.1016/j.na.2007.01.042. |
[9] |
P. Constantin and M. Kliegl,
Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress, Arch. Rational Mech. Anal., 206 (2012), 725-740.
doi: 10.1007/s00205-012-0537-0. |
[10] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[11] |
R. Danchin,
A survey on Fourier analysis methods for solving the compressible Navier-Stokes equations, Sci. China Math., 55 (2012), 245-275.
doi: 10.1007/s11425-011-4357-8. |
[12] |
Y. Du, C. Liu and Q. T. Zhang, A blow-up critirion for 3D compressible viscoelasticity, arXiv: 1202.3693v1 [math. AP] 16 Feb 2012. |
[13] |
D.Y. Fang, M. Hieber and R. Z. Zi,
Global existence results for Oldroyd-B Fluids in exterior domains: The case of non-small coupling parameters, Math. Ann., 357 (2013), 687-709.
doi: 10.1007/s00208-013-0914-5. |
[14] |
D. Y. Fang and R. Z. Zi,
Strong solutions of 3D compressible Oldroyd-B fluids, Math. Meth. Appl. Sci., 36 (2013), 1423-1439.
doi: 10.1002/mma.2695. |
[15] |
D. Y. Fang and R. Z. Zi,
Incompressible limit of Oldroyd-B fluids in the whole space, J. Differential Equations, 256 (2014), 2559-2602.
doi: 10.1016/j.jde.2014.01.017. |
[16] |
E. Fernández-Cara, F. Guillén and R. Ortega,
Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 1-29.
|
[17] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem Ⅰ, Arch. Ration. Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[18] |
C. Guillopé and J. C. Saut,
Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal., 15 (1990), 849-869.
doi: 10.1016/0362-546X(90)90097-Z. |
[19] |
C. Guillopé, Z. Salloum and R. Talhouk,
Regular flows of weakly compressible viscoelastic fluids and the incompressible limit, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1001-1028.
doi: 10.3934/dcdsb.2010.14.1001. |
[20] |
M. Hieber, Y. Naito and Y. Shibata,
Global existence results for Oldroyd-B fluids in exterior domains, J. Differential Equations, 252 (2012), 2617-2629.
doi: 10.1016/j.jde.2011.09.001. |
[21] |
X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497. |
[22] |
X. P. Hu and D. H. Wang, Formation of sigularity for compressible viscoelasticity, arXiv: 1109.1332v1 [math. AP] 7 Sep 2011. |
[23] |
T. Kato,
Strong $L^p $-solutions of the Navier-Stokes equation in $ \mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[24] |
H. Koch and D. Tataru,
Well-posedness for the Navier--Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[25] |
R. Kupferman, C. Mangoubi and E. S. Titi,
A Beale-Kato-Majda breakdown criterion for an Oldroyd-B fluid in the creeping flow regime, Commun. Math. Sci., 6 (2008), 235-256.
doi: 10.4310/CMS.2008.v6.n1.a12. |
[26] |
Z. Lei,
Global existence of classical solutions for some Oldroyd-B model via the incompressible limit, Chinese Ann. Math. Ser. B, 27 (2006), 565-580.
doi: 10.1007/s11401-005-0041-z. |
[27] |
Z. Lei,
On 2D viscoelasticity with small strain, Arch. Ration. Mech. Anal., 198 (2010), 13-37.
doi: 10.1007/s00205-010-0346-2. |
[28] |
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. |
[29] |
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. |
[30] |
Z. Lei, N. Masmoudi and Y. Zhou,
Remarks on the blowup criteria for Oldroyd models, J. Differential Equations, 248 (2010), 328-341.
doi: 10.1016/j.jde.2009.07.011. |
[31] |
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. |
[32] |
F. H. Lin, C. Liu and P. Zhang,
On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 (2005), 1437-1471.
doi: 10.1002/cpa.20074. |
[33] |
F. H. 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. |
[34] |
P. L. 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. |
[35] |
L. Molinet and R. Talhouk,
On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, Nonlinear Diff. Equations Appl., 11 (2004), 349-359.
doi: 10.1007/s00030-004-1073-x. |
[36] |
J. G. Oldroyd,
Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London., 245 (1958), 278-297.
doi: 10.1098/rspa.1958.0083. |
[37] |
J. Z. Qian,
Well-posedness in critical spaces for incompressible viscoelastic fluid system, Nonlinear Analysis, 72 (2010), 3222-3234.
doi: 10.1016/j.na.2009.12.022. |
[38] |
J. Z. Qian and Z. F. Zhang,
Global well-posedness for compressible viscoelastic fluids near equilibrum, Arch. Rational Mech. Anal., 198 (2010), 835-868.
doi: 10.1007/s00205-010-0351-5. |
[39] |
R. Talhouk,
Analyse Mathématique de Quelques Écoulements de Fluides Viscoélastiques, Thèse, Université Paris-Sud, 1994. |
[40] |
F. Weissler,
The Navier-Stokes initial value problem in $L^p $, Arch. Rational Mech. Anal., 74 (1980), 219-230.
doi: 10.1007/BF00280539. |
[41] |
T. Zhang and D. Y. 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. |
[42] |
R. Z. Zi, D. Y. Fang and T. Zhang,
Global solution to the incompressible Oldroyd-B model in the critical $ L^p$ framework: the case of the non-small coupling parameter, Arch. Rational Mech. Anal., 213 (2014), 651-687.
doi: 10.1007/s00205-014-0732-2. |
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