The Cauchy problem for the generalized KdV equation in the Sobolev space $H^{s}(\mathbf{R})$
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912904247246848 |
|---|---|
| author | Yan, Xiangqian Li, Yongsheng Huang, Juan Huang, Jianhua Yan, Wei |
| author_facet | Yan, Xiangqian Li, Yongsheng Huang, Juan Huang, Jianhua Yan, Wei |
| contents | 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*} |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13364 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Cauchy problem for the generalized KdV equation in the Sobolev space $H^{s}(\mathbf{R})$ Yan, Xiangqian Li, Yongsheng Huang, Juan Huang, Jianhua Yan, Wei Analysis of PDEs 35Q53, 35R60, 42B37 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*} |
| title | The Cauchy problem for the generalized KdV equation in the Sobolev space $H^{s}(\mathbf{R})$ |
| topic | Analysis of PDEs 35Q53, 35R60, 42B37 |
| url | https://arxiv.org/abs/2602.13364 |