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.

Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity / Barbato, David; Morandin, Francesco. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 20:3(2013), pp. 1105-1123. [10.1007/s00030-012-0200-3]

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

#### Abstract

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.
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Positive and non-positive solutions for an inviscid dyadic model: well-posedness and regularity / Barbato, David; Morandin, Francesco. - In: NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS. - ISSN 1021-9722. - 20:3(2013), pp. 1105-1123. [10.1007/s00030-012-0200-3]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2480437
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