Periodic Cauchy problem for one two-dimensional generalization of the Benjamin–Ono equation in Sobolev spaces of low regularity

In this work we prove that the initial value problem (IVP) associated to the two-dimensional Benjamin–Ono equation ut+HΔu+uux=0,(x,y)∈T2,t∈R,u(x,y,0)=u0(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 Hs(T2), with s>7∕4.