The Cauchy problem for the generalized KdV equation with rough data and random 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 $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.