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December  2017, 37(12): 6437-6470. doi: 10.3934/dcds.2017279

Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter

Ruizhao Zi ,

School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

Corresponding author: Ruizhao Zi

Received  April 2017 Published  August 2017

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].

Keywords: Compressible Oldroyd-B model, critical Besov space, global well-posedness.
Mathematics Subject Classification: Primary: 76A10, 76N10.
Citation: Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497. Google Scholar

[22]

X. P. Hu and D. H. Wang, Formation of sigularity for compressible viscoelasticity, arXiv: 1109.1332v1 [math. AP] 7 Sep 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

R. Talhouk, Analyse Mathématique de Quelques Écoulements de Fluides Viscoélastiques, Thèse, Université Paris-Sud, 1994. Google Scholar

[40]

F. Weissler, The Navier-Stokes initial value problem in $L^p $, Arch. Rational Mech. Anal., 74 (1980), 219-230.  doi: 10.1007/BF00280539.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

X. P. Hu and F. H. Lin, On the Cauchy problem for two dimensional incompressible viscoelastic flows, arXiv: 1601.03497. Google Scholar

[22]

X. P. Hu and D. H. Wang, Formation of sigularity for compressible viscoelasticity, arXiv: 1109.1332v1 [math. AP] 7 Sep 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

R. Talhouk, Analyse Mathématique de Quelques Écoulements de Fluides Viscoélastiques, Thèse, Université Paris-Sud, 1994. Google Scholar

[40]

F. Weissler, The Navier-Stokes initial value problem in $L^p $, Arch. Rational Mech. Anal., 74 (1980), 219-230.  doi: 10.1007/BF00280539.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389

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