Related papers: A note on the 2D generalized Zakharov-Kuznetsov eq…

In this note we study the generalized 2D Zakharov-Kuznetsov equations $\partial_tu+\Delta\partial_xu+u^k\partial_xu=0$ for $k\ge 2$. By an iterative method we prove the local well-posedness of these equations in the Sobolev spaces…

We consider the Cauchy problem associated with the modified Zakharov-Kuznetsov equation over $\mathbb{R}^2$. Taking into consideration the associated dispersive effects, we introduce, for $s,a\ge 0$, a two-parameter space…

The Cauchy problem for the generalized Zakharov-Kuznetsov equation $$\partial_t u +\partial_x\Delta u=\partial_x u^{k+1}, \qquad \qquad u(0)=u_0$$ is considered in space dimensions $n=2$ and $n=3$ for integer exponents $k \ge 3$. For data…

We consider the Zakharov-Kuznetsov equation in space dimension 3: \[ \left\{ \begin{array}{l} \partial_t u + \partial_x \Delta u + \partial_x \frac{u^2}{2} = 0 \\ u(t = 0) = u_0 \end{array} \right. \] where $u : (t, x, y) \in \mathbb{R}…

This article is concerned with the Zakharov-Kuznetsov equation {equation} \label{ZK0} \partial_tu+\partial_x\Delta u+u\partial_xu=0 . {equation} We prove that the associated initial value problem is locally well-posed in $H^s(\mathbb R^2)$…

We study the modified Zakharov-Kuznetsov equation in dimension $2$ : \[ \partial_t u + \partial_x \left( \Delta u + u^3 \right) = 0 \] where $u : (t, (x, y)) \in \mathbb{R} \times \mathbb{R}^2 \mapsto u(t, x, y) \in \mathbb{R}$ and $\Delta…

The Cauchy problem for the Kadomtsev-Petviashvili-II equation (u_t+u_{xxx}+uu_x)_x+u_{yy}=0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space \dot…

This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov-Kuznetsov equation, namely, \{equation*} \quad \left\{\{array}{lll} {\displaystyle u_t+\partial_x \Delta u+u^ku_x =…

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{tt}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the…

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\R^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\R^3)$. The proof is…

This paper is concerned with the Cauchy problem of the $2$D Zakharov-Kuznetsov equation. We prove bilinear estimates which imply local in time well-posedness in the Sobolev space $H^s({\mathbb{R}}^2)$ for $s > -1/4$, and these are optimal…

We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $(u(0), n(0), \partial_t n(0)) \in H^k(\mathbb{R}^d) \times \dot{H}^l(\mathbb{R}^d)\times \dot{H}^{l-1}(\mathbb{R}^d)$. According to…

This paper is a continuation of our previous study arXiv:2507.01288 on the scattering problem for the Zakharov-Kuznetsov equation (ZK). When the space dimension is three, we construct a global solution to (ZK) which scatters to a given free…

We prove local well-posedness for the $L^2$ critical generalized Zakharov-Kuznetsov equation in $H^s, \, s \in (3/4,1).$ We also prove that the equation is "almost well-posedness" for initial data $u_0 \in H^s, \, s \in [1,2),$ in the sense…

We show the global well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^{s}({\mathbb{R}^2})$ when $\frac{11}{13}<s<1$ via the I-method. Additionally, local well-posedness for the symmetrized ZK equation in $…

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…

In this article, we address the Cauchy problem associated with the $k$-generalized Zakharov-Kuznetsov equation posed on $\mathbb{R} \times \mathbb{T}$. By establishing an almost optimal linear $L^4$-estimate, along with a family of bilinear…

In the present paper, we consider the Cauchy problem of the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation, which has the dissipative term $-\partial_x^2u$. This is known that the 2D Zakharov-Kuznetsov equation is well-posed in…

In this paper, we consider the defocusing cubic nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ in the energy-supercritical regime, in dimensions $d\geq 6$, with no radial assumption on the initial data. We prove that if a solution…

We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in…