The probabilistic convergence problem of density functions related to ${\partial }_x^3+{\partial }_x^{-1}$

Abstract

In this article, by using full randomization introduced by Hadama and Yamamoto (Probabilistic Strichartz estimates in Schatten classes and their applications to Hartree equation, J. Math. Phys. 67(2026), 35pp) and high-low frequency technique as well as the property of ${\mathfrak{S}}^2$, we establish the probabilistic convergence of the density function related to ${\partial }_x^3+{\partial }_x^{-1}$ on $\mathbb{R}$, which extends the Theorem 1.3 of Yan et al. (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.).