Well-Posedness for the Generalized Zakharov–Kuznetsov Equation on Modulation Spaces

Abstract

We consider the Cauchy problem for the generalized Zakharov–Kuznetzov equation \(\partial _t u + \partial _x \Delta u = \partial _x ( u^{m+1} )\) on two or three space dimensions. We mainly study the two dimensional case and give the local well-posedness and the small data global well-posedness in the modulation space \(M_{2,1}(\mathbb {R}^2)\) for \(m \ge 4\). Moreover, for the quartic case (namely, \(m = 3\)), the local well-posedness in \( M_{2,1}^{1/4}(\mathbb {R}^2)\) is given. The well-posedness on three dimensions is also considered.