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

abstract

  • © 2015, Springer Basel. The second-grade 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 second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-α model ($${
    u = 0}$$ν=0), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes 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 second-grade 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 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 $${
    u = mathcal{O}(alpha^{6/5})}$$ν=O(α6/5), $${
    u/alpha^{2} o infty}$$ν/α2→∞ as α → 0. This amounts to a convergence criterion similar to the 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 $${alpha = mathcal{O}(
    u^{3/2})}$$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.

altmetric score

  • 0.75

author list (cited authors)

  • Lopes Filho, M. C., Nussenzveig Lopes, H. J., Titi, E. S., & Zang, A.

citation count

  • 11

publication date

  • April 2015