Abstract
In this paper, we prove local well-posedness for compressible viscoelastic fluids of the Oldroyd model under the assumption that the initial density is bounded away from zero and global well-posedness near equilibrium. The proof of global well-posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic–parabolic system with convection terms.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chemin J.-Y.: Localization in Fourier space and Navier-Stokes system. Phase Space Anal. Partial Differ. Equ. 1, 53–136 (2004)
Chemin J.-Y., Lerner N.: Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes. J. Differ. Equ. 121, 314–328 (1992)
Chemin J.-Y., Masmoudi N.: About lifespan of regular solutions of equations related to viscoelastic fluids. SIAM J. Math. Anal. 33, 84–112 (2001)
Chen Q., Miao C., Zhang Z.: On the well-posedness for the viscous shallow water equations. SIAM J. Math. Anal. 40, 443–474 (2008)
Chen Q., Miao C., Zhang Z.: Well-posedness in critical space for the compressible Navier-Stokes equations with density dependent viscosities. Revista Matemática Iberoamericana 26, 915–946 (2010)
Chen Y., Zhang P.: The global existence of small solutions to the incompressible viscoelastic fluid system in 2 and 3 space dimensions. Comm. Part. Differ. Equ. 31, 1793–1810 (2006)
Danchin R.: Global existence in critical space for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000)
Danchin R.: Local theory in critical space for compressible viscous and heat- conductive gases. Comm. Part. Differ. Equ. 26, 1183–1233 (2001)
Danchin R.: Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A 133, 1311–1334 (2003)
Danchin R.: Fouries analysis methods for PDEs. http://perso-math.univ-mlv.fr/users/danchin.raphael
E W., Li T., Zhang P.W.: Well-Posedness for the Dumbbell model of polymeric fluids. Comm. Math. Phys. 248, 409–427 (2004)
Gurtin M.: An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, Vol. 158. Academic Press, New York (1981)
Hukuhara M.: Sur l’existence des points invariants d’une transformation dans l’espace fonctionanel. Jpn. J. Math. 20, 1–4 (1950)
Larson G.: The Structure and Rheology of Complex Fluids. Topics in chemical Engeering, Vol. 1. Oxford University Press, New York (1998)
Lei Z., Zhou Y.: Global existence of classical solution for the two-dimentional Oldroyd model via the incompressible limits. SIAM J. Math. Anal. 37, 797–814 (2005)
Lei Z., Liu C., Zhou Y.: Global solutions of incompressible viscoelastic fluids. Arch. Rational Mech. Anal. 188, 371–398 (2008)
Lei, Z., Liu, C., Zhou, Y.: Global solutions for compressible viscoelastic fluids with small initial data. Preprint
Lin F.H., Liu C., Zhang P.: On hydrodynamics of viscoelastic fluids. Comm. Pure Appl. Math. 58, 1437–1471 (2005)
Lin F.H., Zhang P.: On the initial-boundery value problem of the incompressible viscolastic fluid system. Commu. Pure Appl. Math. 61, 539–558 (2008)
Lions P.L., Masmoudi N.: Global solutions for some Oldroyd models of non- Newtonian flows. Chin. Ann. Math. Ser. B 21, 131–146 (2000)
Liu C., Walkington N.J.: An Eulerian description of fluids containing visco- hyperelastic particles. Arch. Rational Mech. Anal. 159, 229–252 (2001)
Qian J.: Well-posedness in critical spaces for incompressible viscoelastic fluids system. Nonlinear Anal. Theory Methods Appl. 72, 3222–3234 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. Le Bris
Rights and permissions
About this article
Cite this article
Qian, J., Zhang, Z. Global Well-Posedness for Compressible Viscoelastic Fluids near Equilibrium. Arch Rational Mech Anal 198, 835–868 (2010). https://doi.org/10.1007/s00205-010-0351-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-010-0351-5