The uniform time of existence of the smooth solution for 3D Euler-$\alpha$ equations with Dirichlet boundary conditions

TL;DR

This paper proves that smooth solutions to 3D Euler-$eta$ equations with Dirichlet boundary conditions exist uniformly in time, and these solutions converge to Euler solutions as $eta$ approaches zero, under certain regularity conditions.

Contribution

It establishes uniform existence time for 3D Euler-$eta$ equations with Dirichlet boundary conditions and demonstrates convergence to Euler solutions as $eta$ tends to zero.

Findings

Solutions exist in a uniform time interval independent of $eta$.

Euler-$eta$ solutions converge to Euler solutions in $L^2$ as $eta o 0$.

Higher regularity solutions exist in fixed sub-intervals for small $eta$.

Abstract

After reformulate the incompressible Euler-$\alpha$ equations in 3D smooth domain with Drichlet data, we obtain the unique classical solutions to Euler-$\alpha$ equations exist in uniform time interval independent of $\alpha$. We also show the solution of the Euler-$\alpha$ converge to the corresponding solution of Euler equation in $L^2$ in space, uniformly in time. In the sequel, it follows that the $H^s$ $(s>\frac{n}{2}+1)$ solutions of Euler-$\alpha$ equations exist in any fixed sub-interval of the maximum existent interval for the Euler equations provided that initial is regular enough and $\alpha$ is small sufficiently.