The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

Details The geometry and analysis of the averaged Euler equations and a new diffeomorphism group The geometry and analysis of the averaged Euler equations and a new diffeomorphism group

1999-08-19

arxiv.org/abs/math/9908103v1

Mathematics

Analysis of PDEs

Scientific paper

We present a geometric analysis of the incompressible averaged Euler equations for an ideal inviscid fluid. We show that solutions of these equations are geodesics on the volume-preserving diffeomorphism group of a new weak right invariant pseudo metric. We prove that for precompact open subsets of ${\mathbb R}^n$, this system of PDEs with Dirichlet boundary conditions are well-posed for initial data in the Hilbert space $H^s$, $s>n/2+1$. We then use a nonlinear Trotter product formula to prove that solutions of the averaged Euler equations are a regular limit of solutions to the averaged Navier-Stokes equations in the limit of zero viscosity. This system of PDEs is also the model for second-grade non-Newtonian fluids.

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