Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the advection-diffusion equation with rough coefficients: weak
  solutions and vanishing viscosity   In the first part of the paper, we study the Cauchy problem for the
advection-diffusion equation $\partial_t v + \text{div }(v\boldsymbol{b} ) =
\Delta v$ associated with a merely integrable, divergence-free vector field
$\boldsymbol{b}$ defined on the torus. We first introduce two notions of
solutions (distributional and parabolic), recalling the corresponding available
results of existence and uniqueness. Then, we establish a regularity criterion,
which in turn provides uniqueness for distributional solutions. This is
motivated by the recent results in [31] where the authors showed non-uniqueness
of distributional solutions to the advection-diffusion equation despite the
parabolic one is unique. In the second part of the paper, we precisely describe
the vanishing viscosity scheme for the transport/continuity equation drifted by
$\boldsymbol{b}$, i.e. $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$.
Under Sobolev assumptions on $\boldsymbol{b} $, we give two independent proofs
of the convergence of such scheme to the Lagrangian solution of the transport
equation. The first proof slightly generalizes the original one of [21]. The
other one is quantitative and yields rates of convergence. This offers a
completely general selection criterion for the transport equation (even beyond
the distributional regime) which compensates the wild non-uniqueness phenomenon
for solutions with low integrability arising from convex integration schemes,
as shown in recent works [10, 31, 32, 33], and rules out the possibility of
anomalous dissipation.

22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
5
The categories of this paper are: gr-qc