scholarly journals A Note on the Vanishing Viscosity Limit in the Yudovich Class

2020 ◽  
pp. 1-11
Author(s):  
Christian Seis
Keyword(s):  
Stokes Equations ◽  
Velocity Fields ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit ◽  
The Difference

Abstract We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu $ . Here, we provide an order $ (\nu /|\log \nu | )^{\frac 12\exp (-Ct)}$ bound, which slightly improves upon earlier results by Chemin.

scholarly journals Strong Convergence of the Vorticity for the 2D Euler Equations in the Inviscid Limit

2021 ◽  
Vol 240 (1) ◽  
pp. 295-326
Author(s):  
Gennaro Ciampa ◽  
Gianluca Crippa ◽  
Stefano Spirito
Keyword(s):  
Euler Equations ◽  
Stokes Equations ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Lagrangian Solution ◽  
Viscosity Limit ◽  
2D Euler Equations

AbstractIn this paper we prove the uniform-in-time $$L^p$$ L p convergence in the inviscid limit of a family $$\omega ^\nu $$ ω ν of solutions of the 2D Navier–Stokes equations towards a renormalized/Lagrangian solution $$\omega $$ ω 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 $$\omega ^{\nu }$$ ω ν to $$\omega $$ ω in $$L^p$$ L p . Finally, we show that solutions of the Euler equations with $$L^p$$ 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.

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