Random data Cauchy problem for a generalized KdV equation in the supercritical case

TL;DR

This paper demonstrates that for a generalized KdV equation with random initial data, almost sure local well-posedness can be established at regularity levels below the known deterministic critical threshold, using probabilistic methods.

Contribution

It introduces a probabilistic approach to prove local well-posedness for the generalized KdV in lower regularity spaces than previously known.

Findings

Almost sure local well-posedness for s > 17/112

Lower regularity threshold below deterministic critical s > 3/14

Use of Wiener randomization and probabilistic Strichartz estimates

Abstract

We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem is globally well-posed in H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc. 53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the time of existence T and the continuous dependence cannot be expressed in terms of the size of the data in the H^{\frac{3}{14}}-norm. In this present paper, we prove that almost sure local in time well-posedness holds in H^{s}(\R) with s>\frac{17}{112}, whose lower bound is below \frac{3}{14}. The key ingredients are the Wiener randomization of the initial data and probabilistic Strichartz estimates together with some important embedding Theorems.