THE CAUCHY PROBLEM FOR A FAMILY OF TWO-DIMENSIONAL FRACTIONAL BENJAMIN-ONO EQUATIONS

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation u(t) + D(x)(alpha)u(x) + Hu(yy) +uu(x) = 0 (x, y) is an element of R-2, t is an element of R,}, u(x, y, 0) = u(0)(x, y), where 0 < alpha <= 1, D-x(alpha) denotes the operator defined through the Fourier transform by (D(x)(alpha)f)boolean AND(xi, eta) :=vertical bar xi vertical bar(alpha)(f) over cap(xi, eta), (0.1) and H denotes the Hilbert transform with respect to the variable x, is locally well posed in the Sobolev space H-s (R-2) with s > 3/2 + 1/4 (1 - alpha).