Invariant measures and long time behaviour for the Benjamin-Ono equation II

TL;DR

This paper extends measure invariance results for the Benjamin-Ono equation, focusing on conservation laws linked to fractional Sobolev norms of order ≥ 5/2, using orthogonality of Gaussian variables.

Contribution

It introduces new invariance results for measures associated with higher-order fractional Sobolev norms, advancing understanding of the Benjamin-Ono equation's long-term behavior.

Findings

Proved measure invariance for conservation laws with fractional Sobolev norms ≥ 5/2.

Utilized orthogonality relations of multilinear products of Gaussian variables.

Provided partial results for lower regularity conservation laws.

Abstract

As a continuation of our previous work on the subject, we prove new measure invariance results for the Benjamin-Ono equation. The measures are associated with conservation laws whose leading term is a fractional Sobolev norm of order larger or equal than 5/2. The new ingredient, compared with the case of conservation laws whose leading term is an integer Sobolev norm of order larger or equal than 3 (that has been studied in our previous work), is the use of suitable orthogonality relations satisfied by multilinear products of centered complex independent gaussian variables. We also give some partial results for the measures associated with the two remaining conservation laws at lower regularity. We plan to complete the proof of their invariance in a separated article which will be the final in the series. Finally in an appendix, we give a brief comparing of the recurrence properties of the flows of Benjamin-Ono and KdV equations.