Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and regularity theorems of one-dimensional Brakke flows   Given a closed countably $1$-rectifiable set in $\mathbb R^2$ with locally
finite $1$-dimensional Hausdorff measure, we prove that there exists a Brakke
flow starting from the given set with the following regularity property. For
almost all time, the flow locally consists of a finite number of embedded
curves of class $W^{2,2}$ whose endpoints meet at junctions with angles of
either 0, 60 or 120 degrees.

37
The categories of this paper are: supr-con
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th