Existence and Uniqueness of $L^2$-Solutions at Zero-Diffusivity in the Kraichnan Model of a Passive Scalar

Abstract

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 $d\ge 2$ 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 ${\kappa }_p>0$, $p\ge 1$, exist when ${\kappa }_p\to 0$ 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) Hölder 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.

40 pages, LaTeX (Version 2.09), submitted to Communications in Mathematical Physics

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• ANOMALOUS SCALING OF THE PASSIVE SCALAR
• Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar
• Anomalous Scaling in the N-Point Functions of Passive Scalar
• Universality in Turbulence: an Exactly Soluble Model

Cited by in corpus (3)

• Renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow
• Passive Advection and the Degenerate Elliptic Operators $M_n$
• Noise based on vortex structures in 2D and 3D