• Corpus ID: 210911809

Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher

  title={Subcritical well-posedness results for the Zakharov-Kuznetsov equation in dimension three and higher},
  author={Sebastian Herr and Shinya Kinoshita},
  journal={arXiv: Analysis of PDEs},
The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered. It is proved that the Cauchy problem is locally well-posed in $H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal up to the endpoint. As a corollary, global well-posedness in $L^2(\mathbb{R}^3)$ and, under a smallness condition, in $H^1(\mathbb{R}^4)$, follow. 

The Cauchy problem for the L 2–critical generalized Zakharov-Kuznetsov equation in dimension 3

Abstract We prove local well-posedness for the L 2 critical generalized Zakharov-Kuznetsov equation in We also prove that the equation is “almost well-posedness” for initial data in the sense that

Maximal Function Estimates and Local Well-Posedness for the Generalized Zakharov-Kuznetsov Equation

We prove a high-dimensional version of the Strichartz estimates for the unitary group associated to the free Zakharov--Kuznetsov equation. As a by--product, we deduce maximal estimates which allow us

On local energy decay for large solutions of the Zakharov-Kuznetsov equation

Abstract We consider the Zakharov-Kutznesov (ZK) equation posed in with d = 2 and 3. Both equations are globally well-posed in In this article, we prove local energy decay of global solutions: if

Asymptotic stability of solitary waves of the 3D quadratic Zakharov-Kuznetsov equation

We consider the quadratic Zakharov-Kuznetsov equation $$ \partial_t u + \partial_x \Delta u + \partial_x u^2 =0 $$ on $\mathbb{R}^3$. A solitary wave solution is given by $Q(x-t,y,z)$, where $Q$ is

On the propagation of regularity for solutions of the Zakharov-Kuznetsov equation

New localization formulas are presented that allow us to portray the regularity of the solution of the Zakharov-Kuznetsov-(ZK) equation on a certain class of subsets of the euclidean space.

Existence of solutions for the surface electromigration equation

We consider a model that describes electromigration in nanoconductors known as surface electromigration (SEM) equation. Our purpose here is to establish local well-posedness for the associated

Partial Differential Equations with Quadratic Nonlinearities Viewed as Matrix-Valued Optimal Ballistic Transport Problems

  • D. Vorotnikov
  • Mathematics
    Archive for Rational Mechanics and Analysis
  • 2022
We study a rather general class of optimal “ballistic” transport problems for matrix-valued measures. These problems naturally arise, in the spirit of Brenier (Commun Math Phys 364(2):579–605, 2018),

Review on long time asymptotics of large data in some nonintegrable dispersive models

A BSTRACT . In this short note we review recent results concerning the long time dynamics of large data solutions to several dispersive models. Starting with the KdV case and ending with the KP

The Zakharov–Kuznetsov equation in high dimensions: small initial data of critical regularity

The Zakharov–Kuznetsov equation in spatial dimension d≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}



Well-Posedness Results for the Three-Dimensional Zakharov-Kuznetsov Equation

It is proved that the local well-posedness of the three-dimensional Zakharov--Kuznetsov equation $\partial_tu+Delta\partial_xu+ u\partial-xu=0$ in the Sobolev spaces and in the Besov space.

Global well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D

Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation

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

Well-Posedness for the Two-Dimensional Modified Zakharov-Kuznetsov Equation

It is proved that the initial value problem for the two-dimensional modified Zakharov–Kuznetsov equation is locally well-posed for data in H^s(\mathbb{R}^2)$, and a sharp maximal function estimate is established.

On the 2D Zakharov system with L2 Schrödinger data

We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L2 × H−1/2 × H−3/2. This is the space of optimal regularity in the sense that the

The Cauchy problem for the 3D Zakharov-Kuznetsov equation

We prove that the Cauchy problem for the three-dimensional Zakharov-Kuznetsov equation is locally well-posed for data in $H^s(\R^3)$, s > $\frac{9}{8}$.