Mass-concentration of low-regularity blow-up solutions to the focusing 2D modified Zakharov-Kuznetsov equation

TL;DR

This paper investigates the mass concentration behavior of finite-time blow-up solutions to the focusing 2D modified Zakharov-Kuznetsov equation, establishing conditions under which solutions concentrate a significant portion of their mass at blow-up.

Contribution

It proves mass concentration results for blow-up solutions in low regularity spaces, extending previous understanding to solutions with less regularity and under specific blow-up rate conditions.

Findings

Solutions blow up with at least ground state mass concentration.

Mass concentration holds for solutions in $H^s$ with $17/18 < s extless 1$.

Stronger concentration results are valid under additional blow-up rate bounds.

Abstract

We consider the focusing modified Zakharov-Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the $H^1(\R^{2})$ norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at blow-up time. For finite-time blow-up solutions in the $H^s(\R^2)$ norm for $\frac{17}{18} < s < 1$, we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range $ \frac{17}{18} < s \le 1$ under an additional assumption on the upper bound of the blow-up rate of the solution. The main tools used here are the $I$-method and a profile decomposition theorem for a bounded family of $H^1(\R^{2})$ functions.