On the determining form for the three-dimensional incompressible Leray-$\alpha $ models
Volume 133 / 2024
Abstract
We study a determining form for the three-dimensional incompressible Leray-$\alpha $ models with periodic boundary conditions. We will show that the long-time dynamics (the global attractor) of this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) in an infinite-dimensional Banach space of trajectories called a determining form. We also point out that there is a one-to-one correspondence between the trajectories in the global attractor of the Leray-$\alpha $ equations and the steady state solutions of the determining form, and the determining form is a true ODE in the sense that its vector field is Lipschitz. In particular, the solution of the determining form is a convex combination of the initial trajectory and the fixed steady state, with a dynamical convexity parameter $\lambda $, which will be called the characteristic determining parameter. This gives us a remarkable separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter $\lambda $ which is governed by a one-dimensional ODE. This one-dimensional ODE is investigated to receive one-sided convergence rate estimates as the solution to the determining form converges to the fixed state. Remarkably, it is shown that the zeros of the scalar function that governs the dynamics of $\lambda $, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the Leray-$\alpha $ equations.
