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On the minimal Blow-up rate for the 2D modified Zakharov-Kuznetsov model

Jessica Trespalacios

TL;DR

The authors address the blow-up behavior of the 2D modified Zakharov-Kuznetsov equation in $H^s$ with $s>3/4$, deriving a quantitative lower bound on the blow-up rate. They combine linear estimates for the dispersive propagator, refined local well-posedness theory, and a Weissler–Colliander approach to obtain explicit time powers that feed into a blow-up-rate bound. The main result shows a lower bound of the form $\|u(t)\|_{H^s} \gtrsim (T^*-t)^{-7/48}$, revealing a gap from the conjectured near-self-similar rates. This work advances understanding of finite-time blow-up in mass-critical dispersive equations and provides a framework for more precise blow-up rate analyses in higher-dimensional ZK-type models.

Abstract

In this note we consider the modified $L^2$ critical Zakharov-Kuznetsov equation in $\mathbb R^2$, for initial conditions in the Sobolev space $H^s$ with $s>3/4.$ Assuming that there is a blow up solution at finite time $T^{*}$, we obtain a lower bound for the blow up rate of that solution, expressed in terms of a lower bound for the $H^s$ norm of the solution. It is found a nontrivial gap between conjectured blow up rates and our results. The analysis is based on properly quantifying the linear estimates given by Faminskii, as well as, the local well-posedness theory of Linares and Pastor, combined with an argument developed by Weissler for study singular solutions of the semilinear heat equations and also used by Colliander-Czuback-Sulem in the context of the Zakharov system.

On the minimal Blow-up rate for the 2D modified Zakharov-Kuznetsov model

TL;DR

The authors address the blow-up behavior of the 2D modified Zakharov-Kuznetsov equation in with , deriving a quantitative lower bound on the blow-up rate. They combine linear estimates for the dispersive propagator, refined local well-posedness theory, and a Weissler–Colliander approach to obtain explicit time powers that feed into a blow-up-rate bound. The main result shows a lower bound of the form , revealing a gap from the conjectured near-self-similar rates. This work advances understanding of finite-time blow-up in mass-critical dispersive equations and provides a framework for more precise blow-up rate analyses in higher-dimensional ZK-type models.

Abstract

In this note we consider the modified critical Zakharov-Kuznetsov equation in , for initial conditions in the Sobolev space with Assuming that there is a blow up solution at finite time , we obtain a lower bound for the blow up rate of that solution, expressed in terms of a lower bound for the norm of the solution. It is found a nontrivial gap between conjectured blow up rates and our results. The analysis is based on properly quantifying the linear estimates given by Faminskii, as well as, the local well-posedness theory of Linares and Pastor, combined with an argument developed by Weissler for study singular solutions of the semilinear heat equations and also used by Colliander-Czuback-Sulem in the context of the Zakharov system.
Paper Structure (8 sections, 5 theorems, 91 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Setting
  3. Modified ZK
  4. Main Result
  5. Linear estimates
  6. Preliminaries
  7. Nonlinear estimates
  8. Lower bound for the rate of blow-up

Key Result

Theorem 1.1

Consider the IVP mZK with initial conditions $u_0\in H^s(\mathbb R^2)$ with $s>3/4$. Assume that the solution $u(t,x,y)$ blows up in a finite time $T^*$ in $H^s(\mathbb R^2)$ with a rate at least $7/48\sim \text{0.1458}$. Indeed, we have the following lower bound for the blow-up rate:

Figures (1)

  • Figure 1: Recent proposed blow-up rates in mZK.

Theorems & Definitions (9)

  • Definition 1.1: Blow-up solution
  • Theorem 1.1
  • Lemma 1.1
  • Theorem 1.2: Theorem 1.1 in Linares2009
  • Lemma 2.1
  • proof : Proof of the Lemma \ref{['lemma_aux']}
  • Lemma 2.2
  • proof
  • proof : Proof of the Theorem \ref{['LWP']}