Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation

Abstract

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on R^d. If d = 2, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space H^s(R²) for s ≥ 1/4. If d ≥ 3, by employing U^p and V^p spaces, we establish the small data global well-posedness in the scaling critical Sobolev space H^s_c(R^d) where s_c = d/2-1.