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In this paper, we consider the pointwise convergence problem of freeOstrovsky equation with rough data and random data. Firstly, we show the almosteverywhere pointwise convergence of free Ostrovsky equation in$H^{s}(\mathbb{R})$ with $s\geq \frac{1}{4}$ with rough data. Secondly, wepresent counterexamples showing that the maximal function estimate related tothe free Ostrovsky equation can fail if $s<\frac{1}{4}$. Finally, for every$x\in \mathbb{R}$, we show the almost surely pointwise convergence of freeOstrovsky equation in $L^{2}(\mathbb{R})$ with random data. The main tools arethe density theorem, high-low frequency idea, Wiener decomposition and Lemmas2.1-2.6 as well as the probabilistic estimates of some random series which arejust Lemmas 3.2-3.4 in this paper. The main difficulty is that zero is thesingular point of the phase functions of free Ostrovsky equation. We usehigh-low frequency idea to conquer the difficulties.

Pointwise convergence problem of Ostrovsky equation with rough data and random data