Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Large-time rescaling behaviors for large data to the Hele-Shaw problem   This paper addresses a rescaling behavior of some classes of global solutions
to the zero surface tension Hele-Shaw problem with injection at the origin,
$\{\Omega(t)\}_{t\geq 0}$. Here $\Omega(0)$ is a small perturbation of
$f(B_{1}(0),0)$ if $f(\xi,t)$ is a global strong polynomial solution to the
Polubarinova-Galin equation with injection at the origin and we prove the
solution $\Omega(t)$ is global as well. We rescale the domain $\Omega(t)$ so
that the new domain $\Omega^{'}(t)$ always has area $\pi$ and we consider
$\partial\Omega^{'}(t)$ as the radial perturbation of the unit circle centered
at the origin for $t$ large enough. It is shown that the radial perturbation
decays algebraically as $t^{-\lambda}$. This decay also implies that the
curvature of $\partial\Omega^{'}(t)$ decays to 1 algebraically as
$t^{-\lambda}$. The decay is faster if the low Richardson moments vanish. We
also explain this work as the generalization of Vondenhoff's work which deals
with the case that $f(\xi,t)=a_{1}(t)\xi$.

26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin