题目：Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions
报告摘要：The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: α>0, corresponding to the elastic response, and 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 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 (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 ν = O(α^2), 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 v= O(α^6/5), ν/α^2→∞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 α=O(v^3/2), as ν→0. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.
报告人简介：臧爱彬，博士，宜春学院教授。2011年博士毕业于香港中文大学，2013年至2014年巴西里约热内卢联邦大学访问学者。主要从事非线性分析以及流体力学中的偏微分方程组的研究，特别是粘性消失极限问题；论文发表在JMFM, Phys.D, Nonlinearity, JDE等。