Positive and non-positive solutions for an inviscid dyadic model. Well-posedness and regularity

TL;DR

This paper establishes improved well-posedness and regularity results for the inviscid dyadic model, demonstrating global solutions for positive cases and conditions under which solutions become positive over time.

Contribution

It proves global well-posedness for positive solutions across all growth rates and introduces new regularity results, including solutions becoming positive after finite time.

Findings

Positive solutions are globally well-posed for all growth rates.

Regularity results include boundedness of scaled solutions over time.

Solutions tend to become positive after finite time under general conditions.

Abstract

We improve regolarity and uniqueness results from the literature for the inviscid dyadic model. We show that positive dyadic is globally well-posed for every rate of growth $\beta$ of the scaling coefficients k_n = 2^{bn}. Some regularity results are proved for positive solutions, namely \sup_n n^{-a} k_n^{1/3} X_n(t) < \infty for a.e. t and \sup_n k_n^{1/3-1/(3b)} X_n(t) \leq C t^{-1/3}$ for all $t$. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.