scholarly journals Vanishing Viscosity Limit of the Navier–Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

2019 ◽  
Vol 234 (2) ◽  
pp. 727-775 ◽  
Author(s):  
Yongcai Geng ◽  
Yachun Li ◽  
Shengguo Zhu
Keyword(s):  
Fluid Flow ◽  
Euler Equations ◽  
Compressible Fluid ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit ◽  
Compressible Fluid Flow

Viscous profiles of vortex patches

2013 ◽  
Vol 14 (1) ◽  
pp. 1-68 ◽  
Author(s):  
Franck Sueur
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Euler Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Navier Stokes Equations ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit ◽  
Sharp Variation

AbstractWe deal with the incompressible Navier–Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

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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