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摘要
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 -> 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-a 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 = O(alpha(2)), as alpha -> 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 nu = O(alpha(6/5)), nu/alpha(3) -> infinity as alpha -> 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 = O(nu(3/2)), as nu -> 0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.
- 出版日期2015-6
- 单位宜春学院
