Global Attractor for the Periodic Generalized Korteweg-de Vries Equation Through Smoothing

TL;DR

This paper proves a smoothing property for the periodic generalized KdV equation with damping and forcing, leading to the existence of a compact global attractor in higher Sobolev spaces.

Contribution

It establishes a new smoothing result for the gKdV on the torus with polynomial non-linearity, damping, and forcing, and proves the existence of a global attractor.

Findings

Smoothing level matches that of gKdV at H^1.

Existence of a compact global attractor in H^s for s in (1,2).

Enhanced understanding of long-term dynamics of gKdV with damping and forcing.

Abstract

We establish a smoothing result for the generalized KdV (gKdV) on the torus with polynomial non-linearity, damping, and forcing that matches the smoothing level for the gKdV at $H^1$. As a consequence, we establish the existence of a global attractor for this equation as well as its compactness in $H^s(\mathbb{T})$, $s\in (1,2).$