Trajectory attractor for the 2d dissipative Euler equations and its relation to the Navier-Stokes system with vanishing viscosity

2007 ◽  
Vol 76 (3) ◽  
pp. 856-860
Author(s):  
M. I. Vishik ◽  
V. V. Chepyzhov
Keyword(s):  
Euler Equations ◽  
Stokes System ◽  
Navier Stokes ◽  
Vanishing Viscosity ◽  
Trajectory Attractor

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 Vanishing viscosity limit for the compressible Navier–Stokes system via measure-valued solutions

2020 ◽  
Vol 27 (6) ◽  
Author(s):  
Danica Basarić
Keyword(s):  
Strong Convergence ◽  
Strong Solution ◽  
Stokes System ◽  
Spatial Domain ◽  
Euler System ◽  
Navier Stokes ◽  
Strong Uniqueness ◽  
Vanishing Viscosity ◽  
Vanishing Viscosity Limit ◽  
Viscosity Limit

AbstractWe identify a class of measure-valued solutions of the barotropic Euler system on a general (unbounded) spatial domain as a vanishing viscosity limit for the compressible Navier–Stokes system. Then we establish the weak (measure-valued)–strong uniqueness principle, and, as a corollary, we obtain strong convergence to the Euler system on the lifespan of the strong solution.

scholarly journals Trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bo You
Keyword(s):  
Stokes System ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Time Behavior ◽  
Contact Lines ◽  
Trajectory Attractor ◽  
Moving Contact ◽  
Moving Contact Lines ◽  
Long Time ◽  
Statistical Solutions

<p style='text-indent:20px;'>The objective of this paper is to consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines. As we know, it is very difficult to obtain the uniqueness of an energy solution for this system even in two dimensions caused by the presence of the strong coupling at the boundary. Thus, we first prove the existence of a trajectory attractor for such system, which is a minimal compact trajectory attracting set for the natural translation semigroup defined on the trajectory space. Furthermore, based on the abstract results (trajectory attractor approach) developed in [<xref ref-type="bibr" rid="b38">38</xref>], we construct trajectory statistical solutions for the Cahn-Hilliard-Navier-Stokes system with moving contact lines.</p>

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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