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

Abstract

We consider the Cauchy problem for the generalized Zakharov-Kuznetzov equation ∂ t u + ∂ x u = ∂ 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 (R 2 ) for m ≥ 4. Moreover, for the quartic case (namely, m = 3), the local well-posedness in M 1/4 2,1 (R 2 ) is given. The well-posedness on three dimensions is also considered.

Regular solutions to initial-boundary value problems in a half-strip for two-dimensional Zakharov–Kuznetsov equation

“…For ZK equation there is a lot of literature, devoted to the initial value and initial-boundary value problems, where the variable y is considered on the whole line (see, for example, bibliography in [10,11] and recent papers [13,23]). In particular, global well-posedness in Sobolev spaces H k of arbitrary large regularity to the initial value problem and the initial-boundary value problem, posed on R 2 + = {(x, y) : x > 0} and (0, 1) × R , was established (see, for example, [6,7]).…”

Complex valued semi-linear heat equations in super-critical spaces $E^s_σ$

“…Let us denote ∇ s = F −1 ξ s F , 2 s|∇| = F −1 2 s|ξ| F . We will use the frequency-uniform decomposition techniques, which were first applied to nonlinear PDE in [53], see also some recent works [3,6,25,27,28,40,41,48,50,52] and their references in the study for a variety of nonlinear evolution equations. For any k ∈ Z d , we denote…”

On Inner Regularity of Solutions of Two-Dimensional Zakharov–Kuznetsov Equation

“…In [5,22,23,25,30,41,42,47], the Cauchy problem is studied for equations of type (1.1) with nonlinearities of degree higher than two.…”