We study the BO-ZK operator in the two-dimensional case. This operator contains both local and nonlocal terms and appears in a model describing the electromigration in thin nanoconductors on a dielectric substrate. It is also introduced in the half-wave-Schr\"{o}dinger equation. We first prove the associated anisotropic Gagliardo-Nirenberg inequality and its best constant. Next, as an application, by considering the related evolution equations and their Cauchy problems, we show the uniform bound of solutions in the energy space.