Approximation of 2D Euler Equations by the SecondGrade Fluid Equations with Dirichlet Boundary Conditions
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© 2015, Springer Basel. The secondgrade fluid equations are a model for viscoelastic fluids, with two parameters: α > 0, corresponding to the elastic response, and $${
u > 0}$$ν>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 $${alpha,
u o 0}$$α,ν→0 of solutions of the secondgrade fluid system, in a smooth, bounded, twodimensional domain with noslip boundary conditions. This class of problems interpolates between the Eulerα model ($${
u = 0}$$ν=0), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the NavierStokes case (α = 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 secondgrade model to those of the Euler equations provided $${
u = mathcal{O}(alpha^2)}$$ν=O(α2), as α → 0, extending the main result in (Lopes Filho et al., Physica D 292(293):51–61, 2015). Second, we prove equivalence between convergence (of the secondgrade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $${
u = mathcal{O}(alpha^{6/5})}$$ν=O(α6/5), $${
u/alpha^{2} o infty}$$ν/α2→∞ as α → 0. This amounts to a convergence criterion similar to the wellknown Kato criterion for the vanishing viscosity limit of the NavierStokes equations to the Euler equations. Finally, we obtain an extension of Kato’s classical criterion to the secondgrade fluid model, valid if $${alpha = mathcal{O}(
u^{3/2})}$$α=O(ν3/2), as $${
u o 0}$$ν→0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.
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Lopes Filho, M. C., Nussenzveig Lopes, H. J., Titi, E. S., & Zang, A.
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Boundary Layer

Euler Equations

Secondgrade Complex Fluid

Vanishing Viscosity Limit
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http://dx.doi.org/10.1007/s0002101502078