ON KATO'S SMOOTHING EFFECT FOR A FRACTIONAL VERSION OF THE ZAKHAROV-KUZNETSOV EQUATION

In this work, we study some regularity properties associated with the initial value problem (IVP) "otu -ox1 (-Delta)alpha{2u + uox1u = 0, 0 < alpha 5 2, (1) u(x, 0) = u0(x), x = (x1, x2, . .., xn) E Rn, n 2, t E R,where (-Delta)alpha{2 denotes the n-dimensional fractional Laplacian.We show that solutions to the IVP (1) with initial data in a suitable Sobolev space exhibit a local smoothing effect in the spatial variable of alpha 2 derivatives, al-most everywhere in time. One of the main difficulties that emerge when trying to obtain this regularizing effect underlies that the operator in consideration is non-local, and the property we are trying to describe is local, so new ideas are required. Nevertheless, to avoid these problems, we use a perturbation argu-ment replacing (-Delta) alpha 2 by (I-Delta) alpha 2 , that through the use of pseudo-differential calculus allows us to show that solutions become locally smoother by alpha 2 of a derivative in all spatial directions.As a by-product, we use this particular smoothing effect to show that the extra regularity of the initial data on some distinguished subsets of the Eu-clidean space is propagated by the flow solution with infinity speed.