Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit
Gennaro Ciampa, Gianluca Crippa, Stefano Spirito
Abstract
Abstract In this paper we prove the uniform-in-time $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> convergence in the inviscid limit of a family $$\omega ^\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ω</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:math> of solutions of the 2 D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> 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 $$\omega ^{\nu }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ω</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:math> to $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> in $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> . Finally, we show that solutions of the Euler equations with $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> 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.