FRACTIONAL BROWNIAN VECTOR FIELDS 1. Introduction. A one ...

FRACTIONAL BROWNIAN VECTOR FIELDS 5
standard Cartesian coordinates) of some related vector differential operators. 3
grad
div
curl
graddiv
curlcurl
∆
E
(−∆) γ ξ
←→
jω;
←→ (jω) T ;
←→
0 −jω3 jω 2
jω 3 0 −jω 1
−jω 2 jω 1 0
←→ −ωω T ;
←→ ‖ω‖ 2 I − ωω T ;
←→ −‖ω‖ 2 I;
;
←→ ωω T /‖ω‖ 2 ;
←→ ˆΦ
γ
(ω) = ξ ‖ω‖2γ e ξ irr ωωT + e ξ ‖ω‖ 2 sol
I − ωωT
‖ω‖ 2
.
(2.8)
The penultimate operator (E) and its complement (Id − E) project a vector field respectively
onto its curl-free and divergence-free components. In other words, together they afford a
Helmholtz decomposition of the vector field on which they act (these operators appear
prominently in fluid dynamics literature [8–10, 46]). This is because
In addition, one has
div (Id − E) = 0 and curl E = 0.
E grad = grad and E curl = 0.
(Id − E) is known as the Leray projector in turbulence literature.
Our notation for the fractional vector Laplacian (−∆) γ is motivated by the observation
ξ
that it can be factorized as
(−∆) γ ξ = (−∆)γ 0
e
ξ irrE + e
ξ sol (Id − E)
.
In view of the properties of the operator E, this factorization means that the operator
(−∆) γ combines a coordinate-wise fractional Laplacian with a re-weighting of the curl- and
ξ
divergence-free components of the operand.
2.2. Some properties of ˆΦ γ . Let us now take a closer look at the family of matrixvalued
functions ˆΦ γ , γ ∈ , ξ eξ irr ,e
ξ
ξ sol ∈ . They, of course, satisfy the required invariances:
ˆΦ γ ξ (Ωω) = ΩˆΦ γ ξ (ω)ΩT ;
ˆΦ γ ξ (σω) = σ 2γ ˆΦγ
ξ (ω).
But they exhibit, in addition, the following properties.
(ˆΦ1) Closedness under multiplication: We have
ˆΦ γ 1
ξ 1
(ω)ˆΦ γ 2
ξ 2
(ω) = ˆΦ γ 1 +γ 2
ξ 1 +ξ 2
(ω),
3 Note that, while the curl operator is classically defined in three dimensions, the equivalents of graddiv,
curlcurl, and ∆ = graddiv−curlcurl can be defined in any number of dimensions, for instance by their
Fourier symbols. More precisely, for arbitrary d, the equivalents of −curlcurl and −graddiv that appear
in the definition of the vector Laplacian correspond, respectively, to the product of d-dimensional curl and
divergence with their adjoints.