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Strong convergence of the vorticity for the 2D Euler equations in the inviscid limit. (English) Zbl 1462.35264

Summary: In this paper we prove the uniform-in-time \(L^p\) convergence in the inviscid limit of a family \(\omega^\nu\) of solutions of the \(2D\) Navier-Stokes equations towards a renormalized/Lagrangian solution \(\omega\) of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of \(\omega^{\nu }\) to \(\omega\) in \(L^p\). Finally, we show that solutions of the Euler equations with \(L^p\) vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.

MSC:

35Q31 Euler equations
35Q30 Navier-Stokes equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35D30 Weak solutions to PDEs

Software:

SimEuler
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References:

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