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,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/(2k − 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 ).

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

“…Indeed, we prove the inequality (17). In the case (C), we don't lose generality if we change k 1 + 1 and k 2 + 2 to k 1 and k 2 in the estimate (17), respectively.…”

“…For the two dimensional case, Linares and Pastor [17] studied the local well-posedness in the classical Sobolev space H s (R 2 ) for…”

“…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.…”

The Improved Riccati Equation Method and Exact Solutions to mZK Equation

“…The mZK equation has attracted the attention of many researchers in the past few years. For instance, from the mathematical point of view, local and global existence for the Cauchy problem was studied in [16][17][18] .…”

On the Support of Solutions to a Two-Dimensional Nonlinear Wave Equation

“…For = = = 1 and = 0 in (1), one obtains the classical Zakharov-Kuznetsov (ZK) equation [5], which is a mathematical model to describe the propagation of nonlinear ion-acoustic waves in magnetized plasma. Solitary wave solutions and the Cauchy problem to ZK equation have extensively been studied in the literature ( [6][7][8][9][10][11]). Panthee [12] proved that if a sufficiently smooth solution to the initial value problem associated with the ZK equation is supported compactly in a nontrivial time interval, then it vanishes identically.…”