Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase transition approach to detecting singularities of PDEs   We present a mesh refinement algorithm for detecting singularities of
time-dependent partial differential equations. The main idea behind the
algorithm is to treat the occurrence of singularities of time-dependent partial
differential equations as phase transitions. We show how the mesh refinement
algorithm can be used to calculate the blow-up rate as we approach the
singularity. This calculation can be done in three different ways: i) the
direct approach where one monitors the blowing-up quantity as it approaches the
singularity and uses the data to calculate the blow-up rate ; ii) the "phase
transition" approach (\`a la Wilson) where one treats the singularity as a
fixed point of the renormalization flow equation and proceeds to compute the
blow-up rate via an analysis in the vicinity of the fixed point and iii) the
"scaling" approach (\`a la Widom-Kadanoff) where one postulates the existence
of scaling laws for different quantities close to the singularity, computes the
associated exponents and then uses them to estimate the blow-up rate. Our
algorithm allows a unified presentation of these three approaches. The inviscid
Burgers equation and the supercritical Schrodinger equation are used as
instructive examples to illustrate the constructions.

32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs