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

@article{Herr2020SubcriticalWR, 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}, year={2020} }

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.

## 11 Citations

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

- Mathematics
- 2020

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

- MathematicsSIAM J. Math. Anal.
- 2021

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

- Mathematics
- 2020

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

- Mathematics
- 2020

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

- Mathematics, Computer Science
- 2020

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.

### Numerical study of soliton stability, resolution and interactions in the 3D Zakharov-Kuznetsov equation

- MathematicsArXiv
- 2020

### Existence of solutions for the surface electromigration equation

- Mathematics
- 2020

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

- MathematicsArchive 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

- Mathematics
- 2022

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

- MathematicsJournal of Evolution Equations
- 2021

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

## References

SHOWING 1-10 OF 32 REFERENCES

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

- MathematicsSIAM J. Math. Anal.
- 2012

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

- Mathematics
- 2019

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

- MathematicsFunkcialaj Ekvacioj
- 2022

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

- MathematicsSIAM J. Math. Anal.
- 2009

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.

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

- MathematicsJournal of Differential Equations
- 2020

### On the 2D Zakharov system with L2 Schrödinger data

- Mathematics
- 2009

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…

### A NOTE ON THE 2D GENERALIZED ZAKHAROV-KUZNETSOV EQUATION: LOCAL, GLOBAL, AND SCATTERING RESULTS

- Mathematics
- 2011

### The Cauchy problem for the 3D Zakharov-Kuznetsov equation

- Mathematics
- 2009

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 > $\frac{9}{8}$.