Blow-up in finite or infinite time of the 2D cubic Zakharov-Kuznetsov equation

TL;DR

This work establishes finite or infinite forward-time blow-up for near-threshold negative-energy solutions of the 2D cubic focusing Zakharov-Kuznetsov equation, an $L^2$-critical model. The authors adapt the Merle–Martel framework to the 2D ZK setting, constructing near-threshold global solutions, performing a modulation around the ground state $Q$, and proving a nonlinear Liouville theorem via a limiting process that produces a nontrivial solution to the linearized equation. A key novelty is a direct monotonicity-based decay argument and a rotated monotonicity scheme that yield uniform exponential spatial localization of the renormalized remainder, together with an $L^1$-type invariance that controls the scaling parameter. The analysis culminates in a linear virial estimate for the adjoint linearized problem and a linear Liouville property, which together rule out nontrivial limiting profiles and thereby prove blow-up, advancing the understanding of dispersion, instability, and singularity formation for multi-dimensional KdV-type models.

Abstract

We prove that near-threshold negative energy solutions to the 2D cubic ($L^2$-critical) focusing Zakharov-Kuznetsov (ZK) equation blow-up in finite or infinite time. The proof consists of several steps. First, we show that if the blow-up conclusion is false, there are negative energy solutions arbitrarily close to the threshold that are globally bounded in $H^1$ and are spatially localized, uniformly in time. In the second step, we show that such solutions must in fact be exact remodulations of the ground state, and hence, have zero energy, which is a contradiction. This second step, a nonlinear Liouville theorem, is proved by contradiction, with a limiting argument producing a nontrivial solution to a (linear) linearized ZK equation obeying uniform-in-time spatial localization. Such nontrivial linear solutions are excluded by a local-viral space-time estimate. The general framework of the argument is modeled on Merle [29] and Martel & Merle [24], who treated the 1D problem of the $L^2$-critical gKdV equation. Several new features are introduced here to handle the 2D ZK case.

Figures (8)

• Figure 6.1: Here, $x_0>0$. The function $u(x+x(t_0),y,t)$ has the soliton component centered at position $(x(t)-x(t_0),y(t))$. In this figure, only the $x$ spatial direction depicted horizontally, and time vertically. At time $t=t_0$ the soliton is centered in this frame of reference at position $x(t)-x(t_0)=0$, and the soliton trajectory is the line $x(t)-x(t_0)$ with slope approximately $1$. For the functional $I_+$, with increasing weight $\phi_+$, we can estimate forward in time from $t_{-1}$ to $t_0$ or from $t_0$ to $t_1$. In the case of $t_{-1}$ to $t_0$, the weight $\phi_+$ has transition centered along the right half-slope line $x_0+ \frac{1}{2}(x(t)-x(t_0))$ and in the case of $t_0$ to $t_1$, the weight $\phi_+$ has transition centered along the left half-slope line $-x_0+\frac{1}{2}(x(t)-x(t_0))$. In either case, the essential aspect is that the soliton and weight center trajectories are separating in time as we move forward or backward. For $I_-$, with decreasing weight $\phi_-$, the trajectories are the same except the estimates are backwards in time from $t_1$ to $t_0$ or from $t_0$ to $t_{-1}$.
• Figure 6.2: Region $\Omega_0$, see \ref{['E:mon-30-gen-1']}, the set of all $(x,y)$ for which $x>r_0$ and $|\alpha|<\frac{3\pi}{8}$ when polar coordinates $(x,y)=(r\cos\alpha, r\sin\alpha)$ are used.
• Figure 6.3: Regions $\Omega_\pm$, see \ref{['E:mon-30-gen-1']}, defined analogously to $\Omega^0$ but in the $(\bar{x}^\theta, \bar{y}^\theta)$ coordinates introduced in \ref{['E:rotated-coords']} for $\theta=\pm\frac{\pi}{4}$ respectively. Specifically $\Omega_\theta$ is the set of all $(\bar{x}^\theta,\bar{y}^\theta)$, for which $\bar{x}^\theta>r_0$ and $|\alpha|<\frac{3\pi}{8}$ when polar coordinates $(\bar{x}^\theta ,\bar{y}^\theta)=(r\cos\alpha, r\sin\alpha)$ are used.
• Figure 6.4: Region $\Omega$, see \ref{['E:mon-30-gen-2']}, which is the set of all $(x,y)$ such that $x^2+y^2>16r_0^2$ and $x>-x_0$.
• Figure 6.5: Estimate \ref{['E:tilde-decay']} for $\theta=0$ gives a bound by $e^{-x_0/4}$ outside $|x|<x_0$, while estimate \ref{['E:tilde-decay']} for $\theta=\frac{\pi}{24}$ gives a bound of $e^{-x_0/4}$ outside $|\bar{x}|<x_0$, where $\bar{x} = x \cos \theta - y\sin \theta$. The combination gives decay $e^{-x_0/4}$ outside radius $r=x_0/\sin(|\theta|/2)$.

Theorems & Definitions (77)

• Theorem 1.1: main theorem
• Proposition 1.2
• Proposition 1.3: nonlinear Liouville property
• Lemma 3.1: linear homogeneous estimates
• proof
• Lemma 3.2: linear inhomogeneous estimates
• proof
• Theorem 3.3
• Lemma 3.4
• Lemma 3.5