Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hopf bifurcation of viscous shock wave solutions of compressible
  Navier-Stokes equations and MHD   Extending our previous results for artificial viscosity systems, we show,
under suitable spectral hypotheses, that shock wave solutions of compressible
Navier--Stokes (cNS) and magnetohydrodynamics (MHD) equations undergo Hopf
bifurcation to nearby time-periodic solutions. The main new difficulty
associated with physical viscosity and the corresponding absence of parabolic
smoothing is the need to show that the difference between nonlinear and
linearized solution operators is quadratically small in $H^s$ for data in
$H^s$. We accomplish this by a novel energy estimate carried out in Lagrangian
coordinates; interestingly, this estimate is false in Eulerian coordinates. At
the same time, we greatly sharpen and simplify the analysis of the previous
work.

5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn