Periodic Cauchy problem for one two-dimensional generalization of the Benjamin–Ono equation in Sobolev spaces of low regularity. (November 2019) - Bustamante, Eddye; Jiménez Urrea, José; Mejía, Jorge
Abstract: In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin–Ono equation u t + H Δ u + u u x = 0, ( x, y ) ∈ T 2, t ∈ R, u ( x, y, 0 ) = u 0 ( x, y ), where H denotes the Hilbert transform with respect to the variable x and Δ is the Laplacian with respect to the spatial variables x and y, is locally well-posed in the periodic Sobolev space H s ( T 2 ), with s > 7 ∕ 4 .