Modified energies for the periodic generalized KdV equation and applications

TL;DR

This paper develops modified energies for the periodic generalized KdV equation, enabling analysis of measure invariance, regularity, and Sobolev norm growth, advancing understanding of the equation's long-term behavior.

Contribution

It introduces a novel construction of modified energies for the generalized KdV, leading to new results on measure quasi-invariance and solution regularity.

Findings

Quasi-invariance of high order Gaussian measures

$L^p$ regularity of Radon-Nikodym densities

Bounds on Sobolev norm growth

Abstract

We construct modified energies for the generalized KdV equation. As a consequence, we obtain quasi-invariance of the high order Gaussian measures along with $L^p$ regularity on the corresponding Radon-Nykodim density, as well as new bounds on the growth of the Sobolev norms of the solutions.