Local and global well-posedness for the 2D generalized Zakharov–Kuznetsov equation

Abstract

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov equation, namely, { u t + ∂ x Δ u + u k u x = 0 , ( x , y ) ∈ R 2 , t > 0 , u ( x , y , 0 ) = u 0 ( x , y ) . For 2 ⩽ k ⩽ 7 , the IVP above is shown to be locally well posed for data in H s ( R 2 ) , s > 3 / 4 . For k ⩾ 8 , local well-posedness is shown to hold for data in H s ( R 2 ) , s > s k , where s k = 1 − 3 / ( 2 k − 4 ) . Furthermore, for k ⩾ 3 , if u 0 ∈ H 1 ( R 2 ) and satisfies ‖ u 0 ‖ H 1 ≪ 1 , then the solution is shown to be global in H 1 ( R 2 ) . For k = 2 , if u 0 ∈ H s ( R 2 ) , s > 53 / 63 , and satisfies ‖ u 0 ‖ L 2 < 3 ‖ φ ‖ L 2 , where φ is the corresponding ground state solution, then the solution is shown to be global in H s ( R 2 ) .