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

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 than or equal to 5/2. The new ingredient, compared with the case of conservation laws whose leading term is an integer Sobolev norm of order larger than or equal to 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.

RésuméCet article est la suite des nos travaux sur des mesures invariantes pour l'équation de Benjamin–Ono. On démontre de nouveaux résultats d'existence de mesures invariantes. Ces mesures sont construites à partir de lois de conservation dont les parties dominantes sont des normes de Sobolev d'indice demi-entier fractionnaire, d'ordre supérieur ou égal à 5/2. De nouveaux arguments d'orthogonalité sont utilisés. Finalement dans une annexe, on compare brièvement les propriétés de récurrence de KdV et de l'équation de Benjamin–Ono.