The Cauchy problem for the generalized KdV equation with rough data and random data (2011.07128v2)

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 $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.