## Approximation of 2D Euler equations by the second-grade fluid equations with Dirichlet boundary conditions.(English)Zbl 1328.35153

Summary: The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: $$\alpha>0$$, corresponding to the elastic response, and $$\nu>0$$, corresponding to viscosity. Formally setting these parameters to 0 reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $$\alpha,\nu\to 0$$ of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-$$\alpha$$ model $$(\nu=0)$$, for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case $$(\alpha=0)$$, for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $$\nu=\mathcal O(\alpha^2)$$, as $$\alpha\to 0$$, extending the main result in [M. C. Lopes Filho et al., “Convergence of the 2D Euler-$$\alpha$$ to Euler equations in the Dirichlet case: Indifference to boundary layers”, Physica D 292–293, 51–61 (2015; doi:10.1016/j.physd.2014.11.001)]. Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $$\nu=\mathcal O(\alpha^{6/5})$$, $$\nu/\alpha^2\to\infty$$ as $$\alpha\to 0$$. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato’s classical criterion to the second-grade fluid model, valid if $$\alpha=\mathcal O(\nu^{3/2})$$, as $$\nu\to 0$$. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 76D10 Boundary-layer theory, separation and reattachment, higher-order effects 76A10 Viscoelastic fluids 35Q31 Euler equations
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### References:

 [1] Bardos, C., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42-76 (2013) · Zbl 1117.35063 [2] Busuioc, A.V.; Iftimie, D.; Lopes Filho, M.C.; Nussenzveig Lopes, H.J., Incompressible Euler as a limit of complex fluid models with Navier boundary conditions, J. Differ. Equ., 252, 624-640, (2012) · Zbl 1232.35122 [3] Cioranescu, D., El Hacène, O. (1984) Existence and uniqueness for fluids of second grade. In: Brezis, H., Lions, J.L. (eds.) Pitman Research Notes in Mathematics, Boston, vol. 109, pp. 178-197 (1984) · Zbl 0309.35061 [4] Clopeau, T.; Mikelić, A.; Robert, R., On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary condition, Nonlinearity, 11, 1625-1636, (1998) · Zbl 0911.76014 [5] Constantin, P., Note on loss of regularity for solutions of the 3-D incompressible Euler and related equations, Commun. Math. Phys., 104, 311-326, (1986) · Zbl 0655.76041 [6] Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: Cannone, M., Miyakawa, T. (eds.) Mathematical Foundation of Turbulent Viscous Flows. Lectures given at the C.I.M.E. Summer School, Martina Franca, Italy. Springer Lecture Notes in Mathematics, vol. 1871, pp. 1-43 (2005) · Zbl 1364.35277 [7] Constantin, P., On the Euler equations of incompressible fluids, Bull. Am. Math. Soc., 44, 603-621, (2007) · Zbl 1132.76009 [8] Constantin P., Foias C.: Navier-Stokes equations. University of Chicago Press, Chicago (1988) · Zbl 0687.35071 [9] Galdi G.P.: An introduction to the mathematical theory of the Navier-Stokes equations steady-state problem, 2nd edn. Springer, New York (2011) · Zbl 1245.35002 [10] Galdi, G.P.; Sequeira, A., Further existence results for classical solutions of the equations of a second-grade fluid, Arch. Rational Mech. Anal., 128, 297-312, (1994) · Zbl 0833.76005 [11] Iftimie, D.; Lopes Filho, M.C.; Nussenzveig Lopes, H.J., Incompressible flow around a small obstacle and the vanishing viscosity limit, Commun. Math. Phys., 289, 99-115, (2009) · Zbl 1173.35628 [12] Kato T. (1984) Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In: Chern, S.S. (ed.) Seminar on Nonlinear Partial Differential Differential Equations, pp. 85-98. Mathematical Sciences Research institute Publications, New York (1984). · Zbl 1173.35628 [13] Kato, T.; Lai, C.Y., Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56, 15-28, (1984) · Zbl 0545.76007 [14] Kelliher, J., On kato’s conditions for vanishing viscosity, Indiana Univ. Math. J., 56, 1711-1721, (2007) · Zbl 1125.76014 [15] Ladyzhensakya, O.A., Global solvability of a boundary value problem for the Navier-Stokes equations in the case of two spatial variables, Proc. Acad. Sci. USSR, 123, 427-429, (1958) [16] Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press (Dover edition) (1945) · Zbl 1132.76009 [17] Linshiz, J.S., Titi, E.S.: On the convergence rate of the Euler-$$α$$, an inviscid second-grade complex fluid, model to Euler equations, vol. 138, pp. 305-332 (2010) · Zbl 1375.35348 [18] Lopes Filho, M.C.; Nussenzveig Lopes, H.J.; Planas, G., On the inviscid limit for 2D incompressible flow with Navier friction condition, SIAM J. Math. Anal., 36, 1130-1141, (2005) · Zbl 1084.35060 [19] Lopes Filho, M.C.; Nussenzveig Lopes, H.J.; Titi, E.S.; Zang, A., Convergence of the 2D Euler-$$α$$ to Euler equations in the Dirichlet case: indifference to boundary layers, Physica D, 292, 51-61, (2015) · Zbl 1364.35277 [20] Masmoudi, N., Remarks about the inviscid limit of the Navier-Stokes system, Commun. Math. Phys., 270, 777-788, (2007) · Zbl 1118.35030 [21] Masmoudi, N.; Rousset, F., Uniform regularity for the Navier-Stokes equation with Navier boundary condition, Arch. Rational Mech. Anal., 203, 529-575, (2012) · Zbl 1286.76026 [22] Temam, R., On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20, 32-43, (1975) · Zbl 0309.35061 [23] Wang, X.: A Kato type theorem on zero viscosity limit of Navier-Stokes flows. Indiana Univ. Math. J. 50, 223-241 (2001). (Dedicated to Professors Ciprian Foias and Roger Temam) (Bloomington, IN, 2000) · Zbl 0991.35059 [24] Wang, L.; Xin, Z.; Zang, A., Vanishing viscous limits for 3D Navier-Stokes equations with a Navier-slip boundary condition, J. Math. Fluid Mech., 14, 791-825, (2012) · Zbl 1256.35068 [25] Xiao, Y.; Xin, Z., On the vanishing viscosity limit for the 3D navierstokes equations with a slip boundary condition, Commun. Pure Appl. Math., 60, 1027-1055, (2007) · Zbl 1117.35063
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