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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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