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 k n = 2βn. 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.
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Barbato, D., Morandin, F. Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity. Nonlinear Differ. Equ. Appl. 20, 1105–1123 (2013). https://doi.org/10.1007/s00030-012-0200-3
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DOI: https://doi.org/10.1007/s00030-012-0200-3