Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity

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 $k_n=2^{\beta n}$. Some regularity results are proved for positive solutions, namely $\sup_n n^{-\alpha}k_n^{\frac13}X_n(t)<\infty$ for a.e.\ $t$ and $\sup_n k_n^{\frac13-\frac1{3\beta}}X_n(t)\leq Ct^{-1/3}$ for all $t$. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.