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

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 \(H^1(\mathbb {R}^{2})\) 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 the blow-up time. For finite-time blow-up solutions in the \(H^s(\mathbb {R}^2)\) 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 \(H^1(\mathbb {R}^{2})\) functions.