## 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.

### MSC:

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