Dispersive blow-up for solutions to three dimensional generalized Zakharov-Kuznetsov equations

TL;DR

This paper investigates dispersive blow-up phenomena in solutions to three-dimensional generalized Zakharov-Kuznetsov equations, showing how solutions can lose regularity at rational times but remain smooth at irrational times.

Contribution

It constructs initial data leading to solutions that are not $C^{1}$ at rational times but are smooth at generic irrational times, combining linear solution construction with nonlinear smoothing estimates.

Findings

Solutions can fail to be $C^{1}$ at rational times

Solutions are $C^{1}$ at generic irrational times

Linear solutions exhibit dispersive blow-up phenomena

Abstract

We illustrate the dispersive blow up phenomena of the solutions of three dimensional generalized Zakharov-Kuznetsov equations. In particular, we construct smooth initial data such that, the associated global solutions fail to be $C^{1}$ at time $t$ in a null set containing all rational numbers, but are $C^{1}$ at all times $t$ which are generic irrational numbers. The key ingredient are to construct linear solutions which exhibit such phenomena and to prove nonlinear smoothing estimates for the full nonlinear model.