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

摘要

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 k_n = 2~(βn). 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_nk_n ~(1/3-1/3β)X_n(t) ≤ Ct~(-1/3) for all t. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.