 96194 Gregory L. Eyink and Jack Xin
 Existence and Uniqueness of $L^2$Solutions at ZeroDiffusivity
in the Kraichnan Model of a Passive Scalar
(95K, LaTeX)
May 15, 96

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Abstract. We study Kraichnan's model of a turbulent scalar, passively advected by
a Gaussian random velocity field deltacorrelated in time, for every
space dimension $d\geq 2$ and eddydiffusivity (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 singularelliptic, 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, $p$thorder
hyperdiffusivities $\kappa_p>0$, $p\geq 1$, exist when $\kappa_p
\rightarrow 0$ and coincide with the unique zerodiffusivity solutions.
These results follow from a lower estimate on the minimum eigenvalue of
the $N$particle eddydiffusivity 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) timedependent
weak solutions to the PDE's for $N$point correlation functions with
$L^2$ initial data.
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