Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions

Abstract

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 [19]. 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.

20pages,1figure

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Cited by in corpus (5)

• Weak solutions for the $\alpha$-Euler equations and convergence to Euler
• Inviscid Limit for Stochastic Second-Grade Fluid Equations
• Global Well-Posedness of 2D Second Grade Fluid Equations in Exterior Domain
• The Limit Behavior of the Second Grade Fluid
• Singular limit of 2D second grade fluid past an obstacle