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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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