Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Scaling Invariance and Type-I Singularities for the Compressible
  Navier-Stokes Equations   We find a new scaling invariance of the barotropic compressible Navier-Stokes
equations. Then it is shown that type I singularities of solutions with
$$\limsup_{t \nearrow T}|{\rm div} u(t, x)|(T - t) \leq \kappa,$$ can never
happen at time $T$ for all adiabatic number $\gamma \geq 1$. Here $\kappa > 0$
doesn't depend on the initial data. This is achieved by proving the regularity
of solutions under $$\rho(t, x) \leq \frac{M}{(T - t)^\kappa},\quad M <
\infty.$$ This new scaling invariance also motivates us to construct an
explicit type II blowup solution for $\gamma > 1$.

21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex