Long-time behavior of a family of incompressible three-dimensional Leray-$\alpha$-like models
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We study the long-term dynamics for a family of incompressible three-dimensional Leray-$\alpha$-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-$\alpha$ model when varying two nonnegative parameters $\theta_1$ and $\theta_2$. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-$\alpha$-like models into a finite-dimensional space is determining if it annihilates the difference of two ``nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.
Abstract
We study the long-term dynamics for a family of incompressible three-dimensional Leray-$\alpha$-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-$\alpha$ model when varying two nonnegative parameters $\theta_1$ and $\theta_2$. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-$\alpha$-like models into a finite-dimensional space is determining if it annihilates the difference of two ``nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.
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