On the modified Zakharov--Kuznetsov equation on the cylinder: A trilinear Bourgain-space estimate and its application to local well-posedness

Abstract

We study the modified Zakharov--Kuznetsov equation on $\mathbb R \times \mathbb T$. We establish a trilinear Bourgain-space estimate on the full time line for the derivative cubic term, with input exponent $\frac12+\delta$ and output exponent $-\frac12+3\delta$, and use it to recover local well-posedness in $H^s(\mathbb R \times \mathbb T)$ for $s>1$. After time localization, this output exponent yields a factor $T^{2\delta}$ in the contraction argument. Lower Sobolev regularity thresholds are already known; the purpose here is to give a self-contained dyadic proof of this different modulation balance.