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

We improve regularity 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 β of the scaling coefficients kn =2βn . Some regularity results are proved for positive solutions, namely sup_n n^{−α} k_n^{1/3} X_n (t) < ∞ for a.e. t and sup_n k_n^{1/3-1/3β} Xn (t) ≤ Ct^{−1/3} for all t. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.