Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Cauchy problem for the generalized KdV equation with rough data and
  random data   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
$u(x,0)\in H^{s}(\mathbb{R})(s>\frac{1}{6},k\geq6)$.

8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc