Incompressible Euler as a limit of complex fluid models with Navier boundary conditions. (English) Zbl 1232.35122

The authors study an incompressible second-grade fluid equation of two or three space dimensions. This a model for visco-elastic fluid, where the stress tensor is described by two parameters: the elastic response \(\alpha\) and the viscosity \(\nu\). When \(\nu= 0\), this equation is called \(\alpha\)-Euler equation, and when \(\alpha =\nu =0\), this is the Euler equation. They study the limiting process as \(\alpha ,\nu \to 0\), from a second-grade fluid equation to the Euler equation. First they discuss the boundary conditions and show that perfect-slip boundary condition of Navier is an admissible condition for an incompressible second-grade fluid equation. In this way they formulate an initial-boundary problem, and give a definition of weak solutions for this problem. They prove that if the initial data is smooth, their weak solutions strongly converge in \(L^2\) to a solution of incompressible Euler equation as \(\alpha ,\nu \to 0\). They also prove an existence theorem of global solutions for the axisymmetric flow equations without swirl.


35Q31 Euler equations
76A10 Viscoelastic fluids
35D30 Weak solutions to PDEs
35B07 Axially symmetric solutions to PDEs
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