On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations

Abstract

We study the convergence rate of the solutions of the incompressible Euler-$\alpha$, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter $\alpha$ approaches zero. First we show the convergence in $H^s$, $s>n/2+1$, in the whole space, and that the smooth Euler-$\alpha$ solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.

References in corpus (4)

• Analytical Study of Certain Magnetohydrodynamic-alpha Models
• Three regularization models of the Navier-Stokes equations
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations

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• Global generalized solutions for Maxwell-alpha and Euler-alpha equations
• Optimal control of steady second grade fluids with a Navier-slip boundary condition
• Optimal control of non-stationary second grade fluids with Navier-slip boundary conditions
• A priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type
• Singular limit of 2D second grade fluid past an obstacle
• Uniform time of existence for the alpha Euler equations
• Some results on the well-posedness of Euler-Voigt and Navier-Stokes-Voigt models