The Cauchy problem for the generalized KdV equation in the Sobolev space $H^{s}(\mathbf{R})$

TL;DR

This paper establishes almost sure local well-posedness, nonlinear smoothing, and asymptotic decay results for the generalized KdV equation with rough and random initial data in Sobolev spaces.

Contribution

It introduces new probabilistic techniques and auxiliary spaces to improve well-posedness and decay results for the generalized KdV equation with rough data.

Findings

Almost sure local well-posedness in Sobolev spaces.

Nonlinear smoothing effect for rough data.

Asymptotic decay of solutions at spatial infinity.

Abstract

In this paper, we are concerned with the Cauchy problem for the generalized KdV equation with random data and rough data. Firstly, when $s\in\mathbf{R}$, by using the initial value randomization technique introduced by Shen et al. (arXiv:2111.11935) and the construction of appropriate auxiliary spaces, we establish the almost sure local well-posedness of the generalized KdV equation in $H^{s}(\mathbf{R})$, which improves Theorem 1.3 of Hwang and Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.) and Theorem 1.5 of Yan et al.(arXiv:2011.07128.). Secondly, by using the well-posedness results proved in Theorem 1.1, for $f\in H^{s}(\mathbf{R}),\, s\in\mathbf{R}$, we obtain \begin{eqnarray*} &&\mathbb{P}\left(\left\{\omega:\lim_{t\rightarrow0}\|u(t,x)-U(t)f^{\omega}(x)\|_{L_{x}^{\infty}}=0\right\}\right)=1, \end{eqnarray*} which improves Theorem 1.6 of Yan et al.(arXiv:2011.07128.). Thirdly, by using the dyadic decomposition and constructing appropriate function spaces, we establish nonlinear smoothing for the generalized KdV equation with rough data. Furthermore, by using this estimate, when data $f\in H^{s}(\mathbf{R})\cap\hat{L}^{\infty}(\mathbf{R}),\, s>\frac{1}{2}-\frac{2}{k+1},\, k\geq4$, we obtain \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}u(t,x)=0,\quad t\in[0, T]. \end{eqnarray*} In particular, for $f(x)\in H^{s}(\mathbf{R}),\,s>\frac{1}{2}-\frac{2}{k+1},\,k\geq4$, we prove \begin{eqnarray*} &&\lim_{|x|\rightarrow \infty}(u(t,x)-U(t)f(x))=0. \end{eqnarray*} Finally, by using Theorem 1.1, when $f\in H^{s}(\mathbf{R}),\, s\in\mathbf{R}$, we obtain \begin{eqnarray*} &&\mathbb{P}\left(\left\{\omega: \forall t\in I_{\omega}, \lim_{|x|\rightarrow \infty}\left(u(t,x)-U(t)f^{\omega}(x)\right)=0\right\}\right)=1. \end{eqnarray*}