Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measure solutions to a system of continuity equations driven by
  Newtonian nonlocal interactions   We prove global-in-time existence and uniqueness of measure solutions of a
nonlocal interaction system of two species in one spatial dimension. For
initial data including atomic parts we provide a notion of gradient-flow
solutions in terms of the pseudo-inverses of the corresponding cumulative
distribution functions, for which the system can be stated as a gradient flow
on the Hilbert space $L^2(0,1)^2$ according to the classical theory by
Br\'ezis. For absolutely continuous initial data we construct solutions using a
minimising movement scheme in the set of probability measures. In addition we
show that the scheme preserves finiteness of the $L^m$-norms for all $m\in
[1,+\infty]$ and of the second moments. We then provide a characterisation of
equilibria and prove that they are achieved (up to time subsequences) in the
large time asymptotics. We conclude the paper constructing two examples of
non-uniqueness of measure solutions emanating from the same (atomic) initial
datum, showing that the notion of gradient flow solution is necessary to single
out a unique measure solution.

12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org