The Fourier restriction norm method for the Zakharov-Kuznetsov equation

Abstract

The Cauchy problem for the Zakharov-Kuznetsov equation is shown to be locally well-posed in H^s(R^2) for all s>1/2 by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities

Global well-posedness for low regularity data in the 2d modified Zakharov-Kuznetsov equation

“…It is clear that if u is a solution to this equation, then u| [0,δ] is a solution to (51). We will use the following two Lemmas (from Grünrock and Herr [15] and Colliander, Keel, Staffilani, Takaoka and Tao [9]) to establish a local existence result for I N u in H 1 (R 2 ) space.…”

“…To prove the above result we deal with the symmetrized version of the mZK equation. For that we make a linear change of variables x → ax + by and y → ax − by with a = 2 − 2 3 and b = 3 1 2 2 − 2 3 , following Grünrock and Herr [15], to obtain…”

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

“…Moreover, results for the Zakharov-Kuznetsov equation (m = 1) and the modified Zakharov-Kuznetsov equation (m = 2) are also obtained. See, for example, [1,3,9,20] for the Zakharov-Kuznetsov equation and [1,4,16,17,22] for the modified Zakharov-Kuznetsov equation.…”

Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D

“…The Zakharov-Kuznetsov equation can be seen as a multi-dimensional extension of the KdV equation The aim of the paper is to establish well-posedness of (1.1) in 2 dimensions for low regularity initial data. In the 2D case, Grünrock and Herr in [6], and Molinet and Pilod in [14] obtained independently the well-posedness of (1.1) in H s (R 2 ) for s > 1/2 by using the Fourier restriction norm method. The scaling critical index s = −1 of (1.1) for n = 2 suggests well-posedness in the range −1 ≤ s. In particular, in view of the conservation of mass, it is a natural question whether L 2 -well-posedness holds true.…”

“…The scaling critical index s = −1 of (1.1) for n = 2 suggests well-posedness in the range −1 ≤ s. In particular, in view of the conservation of mass, it is a natural question whether L 2 -well-posedness holds true. Before stating the main result, we introduce the symmetrized equation of (1.1) by performing a linear change of variables as in [6]. Put x = 4 −1/3 x 1 + √ 34 −1/3 x 2 , y = 4 −1/3 x 1 − √ 34 −1/3 x 2 and v(t, x, y) := u(t, x 1 , x 2 ), v 0 (x, y) := u 0 (x 1 , x 2 ).…”