Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global well-posedness of the periodic cubic fourth order NLS in negative
  Sobolev spaces   We consider the Cauchy problem for the cubic fourth order nonlinear
Schr\"odinger equation (4NLS) on the circle. In particular, we prove global
well-posedness of the renormalized 4NLS in negative Sobolev spaces
$H^s(\mathbb{T})$, $s > -\frac{1}{3}$, with enhanced uniqueness. The proof
consists of two separate arguments. (i) We first prove global existence in
$H^s(\mathbb{T})$, $s > -\frac{9}{20}$, via the short-time Fourier restriction
norm method. By following the argument in Guo-Oh for the cubic NLS, this also
leads to non-existence of solutions for the (non-renormalized) 4NLS in negative
Sobolev spaces. (ii) We then prove enhanced uniqueness in $H^s(\mathbb{T})$, $s
> -\frac{1}{3}$, by establishing an energy estimate for the difference of two
solutions with the same initial condition. For this purpose, we perform an
infinite iteration of normal form reductions on the $H^s$-energy functional,
allowing us to introduce an infinite sequence of correction terms to the
$H^s$-energy functional in the spirit of the $I$-method. In fact, the main
novelty of this paper is this reduction of the $H^s$-energy functionals (for a
single solution and for the difference of two solutions with the same initial
condition) to sums of infinite series of multilinear terms of increasing
degrees.

13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics