The Cauchy problem for the generalized KdV equation with rough data and random data

TL;DR

This paper investigates the well-posedness and asymptotic behavior of solutions to the generalized KdV equation with rough and random initial data, improving existing results through probabilistic and Strichartz estimates.

Contribution

It establishes new well-posedness results for the generalized KdV with rough and random data, extending previous work by incorporating probabilistic methods and refined estimates.

Findings

Proves pointwise convergence of solutions as time approaches zero.

Establishes probabilistic well-posedness in lower regularity spaces.

Improves previous results on the KdV equation with random initial data.

Abstract

In this paper, we consider the Cauchy problem for the generalized KdV equation with rough data and random data. Firstly, we prove that $u(x,t)\longrightarrow u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Secondly, we prove that $u(x,t)\longrightarrow e^{-t\partial_{x}^{3}}u(x,0)$ as $t\longrightarrow0$ for a.e. $x\in \mathbb{R}$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k},k\geq8).$ Thirdly, we prove that $\lim\limits_{t\longrightarrow 0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u(x,0)\right\|_{L_{x}^{\infty}}=0$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{2}-\frac{2}{k+1},k\geq5)$. Fourthly, by using Strichartz estimates, probabilistic Strichartz estimates, we establish the probabilistic well-posedness in $H^{s}(\mathbb{R})\left(s>{\rm max} \left\{\frac{1}{k+1}\left(\frac{1}{2}-\frac{2}{k}\right), \frac{1}{6}-\frac{2}{k}\right\}\right)$ with random data. Our result improves the result of Hwang, Kwak (Proc. Amer. Math. Soc. 146(2018), 267-280.). Fifthly, we prove that $\forall \epsilon>0,$ $\forall \omega \in \Omega_{T},$ $\lim\limits_{t\longrightarrow0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u^{\omega}(x,0)\right\|_{L_{x}^{\infty}}=0$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6),$ where ${\rm P}(\Omega_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\frac{\epsilon}{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u^{\omega}(x,0)$ is the randomization of $u(x,0)$. Finally, we prove that $\forall \epsilon>0,$ $\forall \omega \in \Omega_{T}, \lim\limits_{t\longrightarrow0}\left\|u(x,t)-u^{\omega}(x,0)\right\|_{L_{x}^{\infty}}=0$ with ${\rm P}(\Omega_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\frac{\epsilon}{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.