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

2020/07/30 by Bhattacharya, Debdeep
#35B44 #35C07 #35Q53 #37K40 #37L50 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.15773

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 H¹(\R²) 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(\R²) norm for (17)/(18) < s < 1, we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range (17)/(18) < s ≤ 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 H¹(\R²) functions.

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