A note on the Ostrovsky equation in weighted Sobolev spaces

TL;DR

This paper investigates the well-posedness of the Ostrovsky equation's initial value problem within specific weighted Sobolev spaces, extending understanding of solution behavior in these function spaces.

Contribution

It establishes well-posedness results for the Ostrovsky equation in weighted Sobolev spaces with fractional regularity, a novel analysis in this context.

Findings

Well-posedness is proven for s in (3/4, 1] in weighted Sobolev spaces.

The study extends previous results to include weighted spaces with fractional derivatives.

Results contribute to the understanding of solution regularity and decay properties.

Abstract

In this work we consider the initial value problem (IVP) associated to the Ostrovsky equations $$\left. \begin{array}{rl} u_t+\partial_x^3 u\pm \partial_x^{-1}u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad x\in\mathbb R,\; t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}=u_0(x). \end{array} \right\}$$ We study the well-posedness of the IVP in the weighted Sobolev spaces $$Z_{s,\frac{s}2}:=\{u\in H^s(\mathbb R):D_x^{-s} u\in L^2(\mathbb R)\}\cap L^2(|x|^s dx ),$$ with $\frac34<s\leq 1$.