Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Localization, Smoothness, and Convergence to Equilibrium for a Thin Film
  Equation   We investigate the long-time behavior of weak solutions to the thin-film type
equation $$v_t =(xv - vv_{xxx})_x\ ,$$ which arises in the Hele-Shaw problem.
We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium
solution, which has the form $\frac{1}{24}(C^2-x^2)^2_+$, in the norm $$|\!|\!|
f |\!|\!|_{m,1}^2 = \int_{\R}(1+ |x|^{2m})|f(x)|^2\dd x +
\int_{\R}|f_x(x)|^2\dd x\ .$$ We obtain exponential convergence in the $|\!|\!|
\cdot |\!|\!|_{m,1}$ norm for all $m$ with $1\leq m< 2$, thus obtaining rates
of convergence in norms measuring both smoothness and localization. The
localization is the main novelty, and in fact, we show that there is a close
connection between the localization bounds and the smoothness bounds:
Convergence of second moments implies convergence in the $H^1$ Sobolev norm. We
then use methods of optimal mass transportation to obtain the convergence of
the required moments. We also use such methods to construct an appropriate
class of weak solutions for which all of the estimates on which our convergence
analysis depends may be rigorously derived. Though our main results on
convergence can be stated without reference to optimal mass transportation,
essential use of this theory is made throughout our analysis.

22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex