Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Vlasov--Poisson--Landau system in the weakly collisional regime   Consider the Vlasov--Poisson--Landau system with Coulomb potential in the
weakly collisional regime on a $3$-torus, i.e. $$\begin{aligned} \partial_t
F(t,x,v) + v_i \partial_{x_i} F(t,x,v) + E_i(t,x) \partial_{v_i} F(t,x,v) = \nu
Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta^{-1} (\int_{\mathbb R^3} F(t,x,v)\,
\mathrm{d} v - \frac{1}{(2\pi)^3}\int_{\mathbb T^3} \int_{\mathbb R^3}
F(t,x,v)\, \mathrm{d} v \, \mathrm{d} x), \end{aligned}$$ with $\nu\ll 1$. We
prove that for $\epsilon>0$ sufficiently small (but independent of $\nu$),
initial data which are $O(\epsilon \nu^{1/3})$-Sobolev space perturbations from
the global Maxwellians lead to global-in-time solutions which converge to the
global Maxwellians as $t\to \infty$. The solutions exhibit uniform-in-$\nu$
Landau damping and enhanced dissipation.
  Our main result is analogous to an earlier result of Bedrossian for the
Vlasov--Poisson--Fokker--Planck equation with the same threshold. However,
unlike in the Fokker--Planck case, the linear operator cannot be inverted
explicitly due to the complexity of the Landau collision operator. For this
reason, we develop an energy-based framework, which combines Guo's weighted
energy method with the hypocoercive energy method and the commuting vector
field method. The proof also relies on pointwise resolvent estimates for the
linearized density equation.

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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org