On the periodic Schrödinger-Debye equation
Alexander Arbieto Carlos Matheus
We study local and global well-posedness of the initial value problem for the Schrödinger-Debye equation in the periodic case. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new a priori estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this a priori estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$ without any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.
keywords: Schrödinger-Debye system Bourgain's method. well-posedness
Uniqueness of equilibrium states for some partially hyperbolic horseshoes
Alexander Arbieto Luciano Prudente
In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.
keywords: Partially hyperbolic horseshoes Equilibrium States maximal entropy measure.

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