Show that the Duchon-Robert smoothness term \(D(\mathbf{v})\) in Sect. 22.3 is zero for the following class of velocity fields \(\mathbf{v}(\mathbf{x}, t)\) :

\[ \int_{\mathcal{D}} d \mathbf{w}|\mathbf{v}(\mathbf{x}+\mathbf{w}, t)-\mathbf{v}(\mathbf{x}, t)|^{3} \leq C(t)|\mathbf{w}| \sigma(|\mathbf{w}|) \]

where \(\int_{0}^{T} d t C(t)arrow 0} \sigma(|\mathbf{w}|)=0\).

Sect. 22.3

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It would be highly desirable to describe the emergence of singularities of the second kind in weak solutions of the Navier-Stokes pdes explicitly as term in the strong form of the mechanical energy equation. This has been proposed by Duchon and Robert [4], who constructed a term in the mechanical energy pde describing the lack of smoothness in the velocity field. They consider a cube with periodic boundary conditions as flow domain D for mathematical convenience. The term is constructed with the aid of an infinitely differentiable, symmetric, positive mollifier, i.e. a scalar function (x) 0 with compact support in D and finite integral scaling with > 0 according to 1 4e (w) = 34(), lim (x) = 8(x) E0 ((x) is the Dirac pseudo-function). The limit e 0 of (22.38) D[v] = lim De[v] (22.39) E0 defines a functional of v independent of the 4, where De[v] = dwV.(w) Av(Av) (22.40)