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

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\{ω:\lim_{t\rightarrow0}\|u(t,x)-U(t)f^ω(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\{ω: \forall t\in I_ω, \lim_{|x|\rightarrow \infty}\left(u(t,x)-U(t)f^ω(x)\right)=0\right\}\right)=1. \end{eqnarray*}

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5