Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and Uniqueness of $L^2$-Solutions at Zero-Diffusivity in the
  Kraichnan Model of a Passive Scalar   We study Kraichnan's model of a turbulent scalar, passively advected by a
Gaussian random velocity field delta-correlated in time, for every space
dimension $d\geq 2$ and eddy-diffusivity (Richardson) exponent $0<\zeta<2$. We
prove that at zero molecular diffusivity, or $\kappa = 0$, there exist unique
weak solutions in $L^2\left(\Omega^{\otimes N}\right)$ to the
singular-elliptic, linear PDE's for the stationary $N$-point statistical
correlation functions, when the scalar field is confined to a bounded domain
$\Omega$ with Dirichlet b.c. Under those conditions we prove that the $N$-body
elliptic operators in the $L^2$ spaces have purely discrete, positive spectrum
and a minimum eigenvalue of order $L^{-\gamma}$, with $\gamma =2-\zeta$ and
with $L$ the diameter of $\Omega$. We also prove that the weak $L^2$-limits of
the stationary solutions for positive, $p$th-order hyperdiffusivities
$\kappa_p>0$, $p\geq 1$, exist when $\kappa_p \rightarrow 0$ and coincide with
the unique zero-diffusivity solutions. These results follow from a lower
estimate on the minimum eigenvalue of the $N$-particle eddy-diffusivity matrix,
which is conjectured for general $N$ and proved in detail for $N=2,3,4$. Some
additional issues are discussed: (1) H\"{o}lder regularity of the solutions;
(2) the reconstruction of an invariant probability measure on scalar fields
from the set of $N$-point correlation functions, and (3) time-dependent weak
solutions to the PDE's for $N$-point correlation functions with $L^2$ initial
data.

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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: math
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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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: eess
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th