论文标题
具有粗略数据和随机数据的广义KDV方程的Cauchy问题
The Cauchy problem for the generalized KdV equation with rough data and random data
论文作者
论文摘要
在本文中，我们考虑了具有粗略数据和随机数据的广义KDV方程的库奇问题。首先，我们证明$ u（x，t）\ longrightarrow u（x，0）$ as $ t \ t \ longrightArrow0 $ for A.E. $ x \ in \ mathbb {r} $带有$ u（x，0）\ in H^{s}（\ Mathbb {r}）（s> \ frac {1} {2} {2} - \ frac {2} {2} {2} {k} {k \ geq8），k \ geq8）。 e^{ - t \ partial_ {x}^{3}} u（x，0）$ as $ t \ longrightArrow0 $ for a.e.。 $ x \ in \ Mathbb {r} $带有$ u（x，0）\ in H^{s}（\ Mathbb {r}）（s> \ frac {1} {2} {2} - \ frac {2} {2} {2} {k} {k \ geq8），k \ geq8），k \ geq8）。 0} \ left \ | u（x，x，t）-e^{ - t \ partial_ {x}^{3}} u（x，0）\ right \ | _ {l_ {l_ {x}^{\ infty}} = 0，带有$ u（x，0）\ in h^{s}（\ Mathbb {r}）（s> \ frac {1} {2} {2} - \ frac {2} {k+1}，k \ geq5）$。第四，通过使用strichartz估计，概率的strichartz估计值，我们在$ h^{s}中建立了概率良好（\ mathbb {r}）\ left（s> {\ rm max}） \ left \ {\ frac {1} {k+1} \ left（\ frac {1} {2} - \ frac {2} {2} {k} {k} \ right），\ frac {1} {1} {6} {6} - \ frac {2} {2} {k} {k} {k} {k} {k} {k} {k} {k} {k} {k} {k} {ranty \ right）我们的结果改善了Hwang，Kwak（Proc。Amer。Math。Soc。146（2018），267-280。）。第五，我们证明$ \forallε> 0，$ $ \ forallω\ inω_{t}，$ $ \ lim \ limits_ {t \ longrightArrow0} \ left \ | u（x，x，t）-e^{ - t \ partial_ {x}^{3}} u^ω（x，0）\ right \ | _ {l_ {l_ {l_ {x}}}^{x}^{\ infty} { h^{s}（\ mathbb {r}）（s> \ frac {1} {6} {6}，k \ geq6），$其中$ {\ rm p} \ rm p}（ω__{t}） \ left（ - \ frac {c} {t^{\fracε{48K}} \ | u（x，0）\ | _ {h^{h^{s}}}}^{2}}}}}}} \ right）$ and $ u^ω（x，0）$是$ u（x，0）$的随机化。最后，我们证明$ \forallε> 0，$ $ \ forallω\ inω_{t}，\ lim \ limits_ { p}（ω__{t}）\ geq 1- c_ {1} {\ rm exp} \ left（ - \ frac {c} {t^{t^{\fracε{48K}}} \ | u（x，0） h^{s}（\ mathbb {r}）（s> \ frac {1} {6}，k \ geq6）$。
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 $\forall ε>0,$ $\forall ω\in Ω_{T},$ $\lim\limits_{t\longrightarrow0}\left\|u(x,t)-e^{-t\partial_{x}^{3}}u^ω(x,0)\right\|_{L_{x}^{\infty}}=0$ with $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6),$ where ${\rm P}(Ω_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\fracε{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u^ω(x,0)$ is the randomization of $u(x,0)$. Finally, we prove that $\forall ε>0,$ $\forall ω\in Ω_{T}, \lim\limits_{t\longrightarrow0}\left\|u(x,t)-u^ω(x,0)\right\|_{L_{x}^{\infty}}=0$ with ${\rm P}(Ω_{T})\geq 1- C_{1}{\rm exp} \left(-\frac{C}{T^{\fracε{48k}}\|u(x,0)\|_{H^{s}}^{2}}\right)$ and $u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.
