Smoothing and Global Attractors for the Majda-Biello System on the Torus

TL;DR

This paper studies the Majda-Biello system on the torus, demonstrating smoothing effects for solutions, analyzing the influence of number-theoretic properties, and establishing the existence and triviality of global attractors under damping.

Contribution

It provides new smoothing estimates for the Majda-Biello system and links these to number-theoretic properties, also proving the existence of a global attractor and conditions for triviality.

Findings

Smoothing effects depend on number-theoretic properties of the coupling parameter.

Existence of a global attractor in the energy space.

Large damping leads to a trivial attractor.

Abstract

In this paper, we consider the Majda-Biello system, a coupled KdV-type system, on the torus. In the first part of the paper, it is shown that, given initial data in a Sobolev space, the difference between the linear and the nonlinear evolution almost always resides in a smoother space. The smoothing index depends on number-theoretic properties of the coupling parameter in the system which control the behavior of the resonant sets. In the second part of the paper, we consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also show that when the damping is large in relation to the forcing terms, the attractor is trivial.