# American Institute of Mathematical Sciences

October  2013, 6(5): 1307-1313. doi: 10.3934/dcdss.2013.6.1307

## Remarks on the theory of Oldroyd-B fluids in exterior domains

 1 Fachbereich Mathematik, Angewandte Analysis, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany

Received  December 2011 Revised  March 2012 Published  March 2013

Consider the set of equations describing Oldroyd-B fluids with finite Weissenberg numbers in exterior domains. In this note, we describe the main ideas of the proofs of two recent results on global existence for this set of equations on exterior domains subject to Dirichlet boundary conditions. The methods described here are quite different from the techniques used in the Lagrangian approach which is often used in the case of infinite Weissenberg numbers.
Citation: Matthias Hieber. Remarks on the theory of Oldroyd-B fluids in exterior domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1307-1313. doi: 10.3934/dcdss.2013.6.1307
##### References:
 [1] 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. [2] D. Fang, M. Hieber and R. Zi, Global existence results for Oldroyd-B fluids on exterior domains with non small coupling parameter, preprint, (2011). [3] E. Fernández-Cara, F. Guillén and R. Ortega, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 1-29. [4] P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems," Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. [5] V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [6] C. Guillopé and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modél. Math. Anal. Numér. 24 (1990), 369-401. [7] 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. [8] C. Guillopé and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model. Math. Anal. Numer., 24 (1990), 369-401. [9] M. Hieber, Y. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains, J. Diff. Equations, 252 (2012), 2617-2629. doi: 10.1016/j.jde.2011.09.001. [10] O. Kreml and M. Pokorný, On the local strong solutions to the FENE-dumbbell model, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 311-324. doi: 10.3934/dcdss.2010.3.311. [11] 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. [12] Z. Lei, On 2D viscoelasticity with small strain, Arch. Rational Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. [13] 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. [14] Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the compressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813. [15] 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. [16] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Rational Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. [17] 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. [18] 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. [19] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [20] L. Molinet and R. Talhouk, On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, Nonl. Diff. Equ. Appl., 11 (2004), 349-359. doi: 10.1007/s00030-004-1073-x. [21] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London Ser. A, 245 (1958), 278-297. [22] R. Talhouk, Existence locale et unicité d'écoulement de fluids viscoélastiques dans des domains non bornés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 87-92. doi: 10.1016/S0764-4442(99)80160-8.

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##### References:
 [1] 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. [2] D. Fang, M. Hieber and R. Zi, Global existence results for Oldroyd-B fluids on exterior domains with non small coupling parameter, preprint, (2011). [3] E. Fernández-Cara, F. Guillén and R. Ortega, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 1-29. [4] P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems," Second edition, Springer Monographs in Mathematics, Springer, New York, 2011. doi: 10.1007/978-0-387-09620-9. [5] V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [6] C. Guillopé and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Modél. Math. Anal. Numér. 24 (1990), 369-401. [7] 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. [8] C. Guillopé and J.-C. Saut, Global existence and one-dimensional nonlinear stability of shearing motions of viscoelastic fluids of Oldroyd type, RAIRO Model. Math. Anal. Numer., 24 (1990), 369-401. [9] M. Hieber, Y. Naito and Y. Shibata, Global existence results for Oldroyd-B fluids in exterior domains, J. Diff. Equations, 252 (2012), 2617-2629. doi: 10.1016/j.jde.2011.09.001. [10] O. Kreml and M. Pokorný, On the local strong solutions to the FENE-dumbbell model, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 311-324. doi: 10.3934/dcdss.2010.3.311. [11] 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. [12] Z. Lei, On 2D viscoelasticity with small strain, Arch. Rational Mech. Anal., 198 (2010), 13-37. doi: 10.1007/s00205-010-0346-2. [13] 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. [14] Z. Lei and Y. Zhou, Global existence of classical solutions for the two-dimensional Oldroyd model via the compressible limit, SIAM J. Math. Anal., 37 (2005), 797-814. doi: 10.1137/040618813. [15] 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. [16] Z. Lei, C. Liu and Y. Zhou, Global solutions for incompressible viscoelastic fluids, Arch. Rational Mech. Anal., 188 (2008), 371-398. doi: 10.1007/s00205-007-0089-x. [17] 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. [18] 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. [19] A. Matsumura and T. Nishida, Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464. [20] L. Molinet and R. Talhouk, On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, Nonl. Diff. Equ. Appl., 11 (2004), 349-359. doi: 10.1007/s00030-004-1073-x. [21] J. G. Oldroyd, Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids, Proc. Roy. Soc. London Ser. A, 245 (1958), 278-297. [22] R. Talhouk, Existence locale et unicité d'écoulement de fluids viscoélastiques dans des domains non bornés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 87-92. doi: 10.1016/S0764-4442(99)80160-8.
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