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- W2161163690 abstract "In this paper we establish the local Lyapunov property of certain L p and Besov norm of the vorticity fields. We have resolved in part, certain open problem posed by Tosio Kato for the three dimensional Navier-Stokes equation by studying the vorticity equation. The local dissipativity of the sum of linear and non-linear operators of the vorticity equation is established. One of the main techniques used here is the Littlewood-Paley analysis. Stability and control of a dynamical system is often studied using Lyapunov functions (12, 16, 17). The local Lyapunov property we study in this paper can thus be of interest to the understanding, control and stabilization of turbulent fields (25). This property also sheds some light towards the research on global Navier-Stokes solutions in super-critical spaces (for definitions and examples of these spaces see (7) and (9)). Weak solutions of the Navier-Stokes equation satisfy the energy inequality which in turn implies that the L 2 -norm of velocity decreases in time (15). This idea was generalized by Tosio Kato (13) to prove that for every solution of Navier-Stokes equation in R m (m ≥ 3), there exist a large number of Lyapunov functions, which decrease monotonically in time if the solution have small L m (R m )- norm. More specifically Kato proved that the local Lyapunov property in L p -norm for 1 0, 2 ≤ p < ∞. He also noted that for any Lyapunov function L(u) and any monotone increasing function �, �(L(u)) will again be a Lyapunov function. Moreover Kato also proved the local dissipativity of the sum of the linear and nonlinear operators of the Navier-Stokes equation in" @default.
- W2161163690 created "2016-06-24" @default.
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- W2161163690 date "2007-01-01" @default.
- W2161163690 modified "2023-10-16" @default.
- W2161163690 title "Lyapunov functionals and local dissipativity for the vorticity equation in $L^p$ and Besov spaces" @default.
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