Abstract
We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions that does not vanish as t → ∞, and which is independent of the Rivlin–Ericksen material parameter α. Applications to the damped incompressible Euler equations are also given.
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Alinhac S., Métivier G.: Propagation de l’analyticité locale pour les solutions de l’équation d’Euler. Arch. Ration. Mech. Anal. 92(4), 287–296 (1986)
Babin A.V., Vishik M.I.: Attractors of Evolutionary Equations. North-Holland, Amsterdam (1989)
Bardos C., Benachour S.: Domaine d’analycité des solutions de l’équation d’Euler dans un ouvert de \({\mathbb{R}^n}\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4(4), 647–687 (1977)
Bardos C., Benachour S., Zerner M.: Analycité des solutions périodiques de l’équation d’Euler en deux dimensions. C. R. Acad. Sci. Paris 282, 995–998 (1976)
Bardos C., Titi E.S.: Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations. Discrete Contin. Dyn. Syst. 3(2), 185–197 (2010)
Benachour S.: Analycité des solutions périodiques de l’équation d’Euler en trois dimension. C. R. Acad. Sci. Paris 283, 107–110 (1976)
Bony J.-M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Annales Scientifiques de l’École Normale Supérieure 14, 209–246 (1981)
Brézis H., Gallouet T.: Nonlinear Shrödinger evolution equation. Nonlinear Anal. 4(4), 677–681 (1980)
Cao Y., Lunasin E.M., Titi E.S.: Global well-posedness of three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4(4), 823–848 (2006)
Chemin J.-Y.: Fluides parfaits incompressibles. Astérisque 230, 177 (1995)
Chemin J.-Y.: Le système de Navier-Stokes incompressible soixante dix ans après Jean Leray. Séminaire et Congrès 9, 99–123 (2004)
Chemin, J.-Y., Gallagher, I., Paicu, M.: Global regularity for some classes of large solutions to the Navier-Stokes equations. Ann. Math. (accepted)
Cioranescu D., Girault V.: Weak and classical solutions of a family of second-grade fluids. Int. J. Non-Linear Mech. 32(2), 317–335 (1997)
Cioranescu, D., Ouazar, E.H.: Existence and uniqueness for fluids of second-grade. In: Nonlinear Partial Differential Equations and Their Applications. Collége de France seminar, vol. VI (Paris, 1982/1983), pp. 178–197. Res. Notes in Math., vol. 109. Pitman, Boston (1984)
Cockburn B., Jones D., Titi E.S.: Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems. Math. Comput. 66, 1073–1087 (1997)
Constantin, P., Foias, C.: Navier–Stokes equations. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)
DiPerna R., Majda A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys. 108(4), 667–689 (1987)
Dunn J.E., Fosdick R.L.: Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second-grade. Arch. Ration. Mech. Anal. 56, 191–252 (1974)
Ferrari A.B., Titi E.S.: Gevrey regularity for nonlinear analytic parabolic equations. Comm. Partial Differ. Equ. 23(1–2), 1–16 (1998)
Foias C., Holm D., Titi E.S.: The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14(1), 1–35 (2002)
Foias, C., Holm, D., Titi, E.S.: The Navier-Stokes-alpha model of fluid turbulence. Advances in nonlinear mathematics and science. Phy. D 152/153, 505–519 (2001)
Foias C., Prodi G.: Sur le comportement global des solutions non stationnaires des équations de Navier-Stokes en dimension deux. Rend. Sem. Mat. Univ. Padova 39, 1–34 (1967)
Foias C., Temam R.: Gevrey class regularity for the solutions of the Navier–Stokes equations. J. Funct. Anal. 87, 359–369 (1989)
Galdi G.P., Sequeira A.: Further existence results for classical solutions of the equations of second-grade fluids. Arch. Ration. Mech. Anal. 128, 297–312 (1994)
Galdi G.P., Grobbelaar-van Dalsen M., Sauer N.: Existence and uniqueness of classical-solutions of the equations of motion for second-grade fluids. Arch. Ration. Mech. Anal. 124, 221–237 (1993)
Ghidaglia J.-M., Temam R.: Regularity of the solutions of second order evolution equations and their attractors. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14(3), 485–511 (1987)
Hale J.K., Raugel G.: Regularity, determining modes and Galerkin method. J. Math. Pures Appl. 82(9), 1075–1136 (2003)
Henshaw W.D., Kreiss H.-O., Reyna L.G.: Smallest scale estimates for the Navier-Stokes equations for incompressible fluids. Arch. Ration. Mech. Anal. 112(1), 21–44 (1990)
Iftimie D.: Remarques sur la limite α→ 0 pour les fluides de grade 2. C. R. Acad. Sci. Paris Sér. I Math. 334(1), 83–86 (2002)
Jones D., Titi E.S.: Upper bounds on the number of determining modes, nodes, and volume elements for the Navier-Stokes equations. Indiana Math. J. 42, 875–887 (1993)
Kukavica I.: On the dissipative scale for the Navier-Stokes equation. Indiana Univ. Math. J. 48(3), 1057–1081 (1999)
Kukavica I., Vicol V.: On the radius of analyticity of solutions to the three-dimensional Euler equations. Proc. Am. Math. Soc. 137, 669–677 (2009)
Kukavica I., Vicol V.: The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete Contin. Dyn. Syst. Ser. A 29, 285–303 (2011)
Larios A., Titi E.S.: On the Higher-Order Global Regularity of the Inviscid Voigt-Regularization of Three-Dimensional Hydrodynamic Models. Discrete Contin. Dyn. Syst. 14, 603–627 (2010)
Lemarié–Rieusset P.G.: Une remarque sur l’analyticité des solutions milds des équations de Navier–Stokes dans R 3. C. R. Acad. Sci. Paris Sér. I Math. 330(3), 183–186 (2000)
Levermore C.D., Oliver M.: Analyticity of solutions for a generalized Euler equation. J. Differ. Equ. 133(2), 321–339 (1997)
Linshiz J.S., Titi E.S.: On the convergence rate of the Euler-α, an inviscid second-grade complex fluid, model to the Euler equations. J. Stat. Phys. 138, 305–332 (2010)
Majda, A.J., Bertozzi, A.L.: Vorticity and incompressible flow. In: Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)
Moise I., Rosa R.: On the regularity of the global attractor of a weakly damped, forced Korteweg-de Vries equation. Adv. Differ. Equ. 2, 257–296 (1997)
Moise I., Rosa R., Wang X.: Attractors for non-compact semigroups via energy equations. Nonlinearity 11, 1369–1393 (1998)
Ngo, V.S.: Thèse de l’Université Paris-Sud (2009)
Nussbaum R.: Periodic solutions of analytic functional differential equations are analytic. Mich. Math. J. 20, 249–255 (1973)
Oliver M., Titi E.S.: Analyticity of the attractor and the number of determining nodes for a weakly damped driven nonlinear Schrödinger equation. Indiana Univ. Math. J. 47, 49–73 (1998)
Oliver M., Titi E.S.: On the domain of analyticity of solutions of second order analytic nonlinear differential equations. J. Differ. Equ. 174(1), 55–74 (2001)
Paicu, M., Raugel, G., Rekalo, A.: Regularity of the global attractor and finite-dimensional behaviour for the second grade fluid equations. J. Differ. Equ. (accepted)
Paicu, M., Zhang, Z.: Global Regularity for the Navier-Stokes equations with large, slowly varying initial data in the vertical direction. Preprint (2009)
Raugel, G.: Global attractors in partial differential equations. In: Handbook of Dynamical Systems, vol. 2, pp. 885–982. North-Holland, Amsterdam (2002)
Rivlin R.S., Ericksen J.L.: Stress-deformation relations for isotropic materials. J. Ration. Mech. Anal. 4, 323–425 (1955)
Temam R.: Navier-Stokes Equations, Theory and Numerical Analysis, 3rd edn. North-Holland, Amsterdam (2001)
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Paicu, M., Vicol, V. Analyticity and Gevrey-Class Regularity for the Second-Grade Fluid Equations. J. Math. Fluid Mech. 13, 533–555 (2011). https://doi.org/10.1007/s00021-010-0032-z
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DOI: https://doi.org/10.1007/s00021-010-0032-z