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Análise / Equações Diferenciais Parciais

On some properties of propagation of regularity in some nonlocal dispersive models
Argenis J. Mendez

Quinta-feira, 7 de março de 2019, 15:30
Sala 232

We consider properties of solutions of the IVP associated to the following nonlocal dispersive model

$$ \partial_t u-D^{\alpha}_x\partial_x u+u\partial_x u=0, \;\;\; x,t\in \mathbb{R}, \;\;\alpha\in (0,2).  $$

 Isaza, Linares, and Ponce   showed that solutions  of the IVP associated to the KdV and Benjamin-Ono  equations enjoy the following property: for  initial data $u_0$ in a suitable Sobolev space   $H^s(\mathbb{R})$ whose restriction belongs to $H^m((b,\infty))$ for some $m\in \mathbb{Z}^{+}$ and $b\in\mathbb{R}$ then the restriction of the corresponding solution belongs to $H^m((\beta,\infty))$ for any $\beta\in \mathbb{R}$ and any   $t\in (0,T)$.

Since in the  cases  $\alpha=1$ and $\alpha=2$ this  property is satisfied, it was an open problem  to  extend this result to solutions of the  equation above. By means of the  techniques introduced by Izasa, Linares and Ponce and Kenig, Linares, Ponce and Vega, combined with  the  commutator expansions of Ginibre and Velo, we obtain  the same properties of propagation of regularity  for  solutions of the IVP  associated to the equation (1).