Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Discrete-to-continuum limits of particles with an annihilation rule   In the recent trend of extending discrete-to-continuum limit passages for
gradient flows of single-species particle systems with singular and nonlocal
interactions to particles of opposite sign, any annihilation effect of
particles with opposite sign has been side-stepped. We present the first
rigorous discrete-to-continuum limit passage which includes annihilation. This
result paves the way to applications such as vortices, charged particles, and
dislocations. In more detail, the discrete setting of our discrete-to-continuum
limit passage is given by particles on the real line. Particles of the same
type interact by a singular interaction kernel; those of opposite sign interact
by a regular one. If two particles of opposite sign collide, they annihilate,
i.e., they are taken out of the system. The challenge for proving a
discrete-to-continuum limit is that annihilation is an intrinsically discrete
effect where particles vanish instantaneously in time, while on the continuum
scale the mass of the particle density decays continuously in time. The proof
contains two novelties: (i) the observation that empirical measures of the
discrete dynamics (with annihilation rule) satisfy the continuum evolution
equation that only implicitly encodes annihilation, and (ii) the fact that, by
imposing a relatively mild separation assumption on the initial data, we can
identify the limiting particle density as a solution to the same continuum
evolution equation.

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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph