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

Abstract

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