On internal regularity of solutions to the initial value problem for the zakharov-kuznetsov equation
The initial value problem is considered for the Zakharov-Kuznetsov equation in two spatial dimensions, which generalizes the Korteweg-de Vries equation for description of wave propagation in dispersive media on the plane. An initial function is assumed to be irregular, namely, from the spaces L 2 or H 1. Results on gain of internal regularity for corresponding weak solutions depending on the decay rate of the initial function at infinity are established. Existence of both Sobolev and continuous derivatives of any prescribed order is proved. One of important items of the study is the investigation of the fundamental solution to the corresponding linearized equation. The obtained properties are to some extent similar to the ones of the Airy function.