Incompressible Euler As A Limit Of Complex Fluid Models With Navier Boundary Conditions

In this article we study the limit α→0 of solutions of the α-Euler equations and the limit α,ν→0 of solutions of the second grade fluid equations in a bounded domain, both in two and in three space dimensions. We prove that solutions of the complex fluid models converge to solutions of the incompressible Euler equations in a bounded domain with Navier boundary conditions, under the hypothesis that there exists a uniform time of existence for the approximations, independent of α and ν This additional hypothesis is not necessary in 2D, where global existence is known, and for axisymmetric flows without swirl, for which we prove global existence. Our conclusion is strong convergence in L2 to a solution of the incompressible Euler equations, assuming smooth initial data. © 2011 Elsevier Inc.2521624640Abboud, H., Sayah, T., Upwind discretization of a time-dependent two-dimensional grade-two fluid model (2009) Comput. Math. Appl., 57, pp. 1249-1264Ahmad, I., Sajid, M., Hayat, T., Heat transfer in unsteady axisymmetric second grade fluid (2009) Appl. Math. Comput., 215, pp. 1685-1695Bardos, C., Linshiz, J., Titi, E., Global regularity for a Birkhoff-Rott-α approximation of the dynamics of vortex sheets of the 2D Euler equations (2008) Phys. D, 237, pp. 1905-1911Clopeau, T., Mikelic, A., Robert, R., On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions (1998) Nonlinearity, 11, pp. 1625-1636Busuioc, A.V., Iftimie, D., A non-Newtonian fluid with Navier boundary conditions (2006) J. Dynam. Differential Equations, 18 (2), pp. 357-379Busuioc, A.V., Ratiu, T., The second grade fluid and averaged Euler equations with Navier-slip boundary conditions (2003) Nonlinearity, 16, pp. 1119-1149Busuioc, A.V., Ratiu, T., Some remarks on a certain class of axisymmetric fluids of differential type (2004) Phys. D, 191, pp. 106-120Dunn, J.E., Fosdick, R.L., Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade (1974) Arch. Ration. Mech. Anal., 56, pp. 191-252Dunn, J.E., Rajagopal, K.R., Fluids of differential type - critical review and thermodynamic analysis (1995) Internat. J. Engrg. Sci., 33, pp. 689-729Fan, J., Ozawa, T., On the regularity criteria for the generalized Navier-Stokes equations and Lagrangian averaged Euler equations (2008) Differential Integral Equations, 21, pp. 443-457Holm, D., Marsden, J., Ratiu, T., Euler-Poincaré models of ideal fluids with nonlinear dispersion (1998) Phys. Rev. Lett., 80, pp. 4173-4176Hou, T., Congming, L., On global well-posedness of the Lagrangian averaged Euler equations (2006) SIAM J. Math. Anal., 38, pp. 782-794Iftimie, D., Remarques sur la limite α→0 pour les fluides de grade 2 (2002) Stud. Math. Appl., 31. , ElsevierIftimie, D., Planas, G., Inviscid limits for the Navier-Stokes equations with Navier friction boundary conditions (2006) Nonlinearity, 19, pp. 899-918Khan, M., Hyder Ali, S., Qi, H., Exact solutions for some oscillating flows of a second grade fluid with a fractional derivative model (2009) Math. Comput. Modelling, 49, pp. 1519-1530le Roux, C., Existence and uniqueness of the flow of second-grade fluids with slip boundary conditions (1999) Arch. Ration. Mech. Anal., 148, pp. 309-356Linshiz, J., Titi, E., On the convergence rate of the Euler-α inviscid second-grade complex fluid, model to the Euler equations (2010) J. Stat. Phys., 138, pp. 305-332Liu, X., Jia, H., Local existence and blowup criterion of the Lagrangian averaged Euler equations in Besov spaces (2008) Commun. Pure Appl. Anal., 7, pp. 845-852Liu, X., Wang, M., Zhang, Z., A note on the blowup criterion of the Lagrangian averaged Euler equations (2007) Nonlinear Anal., 67, pp. 2447-2451Lopes Filho, M.C., Nussenzveig Lopes, H.J., Planas, G., On the inviscid limit for 2D incompressible flow with Navier friction condition (2005) SIAM J. Math. Anal., 36, pp. 1130-1141Massoudi, M., Tran, P.X., Wulandana, R., Convection-radiation heat transfer in a nonlinear fluid with temperature-dependent viscosity (2009) Math. Probl. Eng., , article ID 232670Navier, C.L.M.H., Mémoire sur les lois du mouvement des fluides (1823) Mém. Acad. Royale Sci. Inst. Fr., 6, pp. 389-440Xiao, Y., Xin, Z., On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition (2007) Comm. Pure Appl. Math., 60, pp. 1027-105