A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit

Let $\mathfrak{G}\subset {\mathbb{R}}^3$ with $vol(\mathfrak{G})\sim L^3$. Let $\mathcal{T}(x)$ be a Gaussian random field $\mathrm{\forall }x\in \mathfrak{G}$ with expectation $\mathbf{E}[\mathcal{T}(x)]=0$ and correlation $\mathbf{E}[\mathcal{T}(x)\otimes \mathcal{T}(y)]=K(x,y;\lambda )$, an isotropic and regulated kernel with correlation length $\lambda$. The field has a Karhunen-Loeve spectral representation $\mathcal{T}(x)={\sum }_{I=1}^{\mathrm{\infty }}{\mathrm{Z}}_I^{1\mathrm{/}2}{f}_I(x)\otimes {\mathcal{Z}}_I$, with eigenvalues $\{{\mathrm{Z}}_I\}$, eigenfunctions $\{f_I(x)\}$ and Gaussian random variables ${\mathcal{Z}}_I$ with $\mathbf{E}[{\mathcal{Z}}_I]=0$ and $\mathbf{E}[{\mathcal{Z}}_I\otimes {\mathcal{Z}}_J]={\delta }_{IJ}$. If $\mathfrak{G}$ contains incompressible fluid of viscosity $\nu$ with velocity $u_a(x,t)$ that evolves via the Navier-Stokes equations with a high ‘Reynolds function’ $\mathsf{R}\mathsf{E}(x,t)=\frac{\mathrm{\parallel }{u}_a(x,t)\mathrm{\parallel }L}{\nu }$ then aspects of a turbulent flow with $\mathsf{R}\mathsf{E}(x,t)\gg {\mathsf{R}\mathsf{E}}_{\ast }$, a critical Reynolds number, might be represented by the ‘weighted’ random field ${\mathcal{U}}_a(x,t)=u_a(x,t)+\mathrm{A}{u}_a(x,t)(\mathsf{R}\mathsf{E}(x,t)-{\mathsf{R}\mathsf{E}}_{\ast }{)}^{\beta }{\sum }_{I=1}^{\mathrm{\infty }}{\mathrm{Z}}_I^{1\mathrm{/}2}{f}_I(x)\otimes {\mathcal{Z}}_I$ where random fluctuations and amplitude scale nonlinearly with $\mathsf{R}\mathsf{E}(x,t)$, with mean $\mathbf{E}[{\mathcal{U}}_a(x,t)]=u_a(x,t)$. In the inviscid limit one can prove an anomalous dissipation-type law \begin{align} \lim_{ν\rightarrow 0}\bigg(\lim_{u_{a}(x,t)\rightarrow {u}{a}}\sup~ν\int{\mathfrak{G}}\int_{0}^{T}{$\mathbf{E}$}\bigg[\bigg|{\nabla}{a}{\mathscr{U}}{a}(x,s)\bigg|^{2}\bigg]d$\mathcal{V}$(x) ds\bigg)>0 \end{align} iff $\beta =\frac{1}{2}$ and ${\sum }_{I=1}^{\mathrm{\infty }}{\mathrm{Z}}_I{\int }_{\mathfrak{G}}{\mathrm{\nabla }}_a{f}_I(x){\mathrm{\nabla }}^a{f}_I(x)d\mathcal{V}(x)>0$.