A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit (2409.10636v1)
Abstract: Let ${\mathfrak{G}}\subset\mathbb{R}{3}$ with $vol(\mathfrak{G})\sim L{3}$. Let ${\mathscr{T}}(x)$ be a Gaussian random field $\forall~x\in\mathfrak{G}$ with expectation $\mathbf{E}[{\mathscr{T}}(x)]=0$ and correlation $\mathbf{E}[{\mathscr{T}}(x)\otimes{\mathscr{T}}(y)]=K(x,y;\lambda)$, an isotropic and regulated kernel with correlation length $\lambda$. The field has a Karhunen-Loeve spectral representation ${\mathscr{T}}(x)=\sum_{I=1}{\infty}\mathrm{Z}{1/2}{I}f{I}(x)\otimes\mathscr{Z}_{I}$, with eigenvalues $\lbrace\mathrm{Z}{I}\rbrace$, eigenfunctions $\lbrace f{I}(x)\rbrace $ and Gaussian random variables $\mathscr{Z}{I}$ with $\mathbf{E}[\mathscr{Z}{I}]=0$ and $\mathbf{E}[\mathscr{Z}{I}\otimes\mathscr{Z}{J}]=\delta_{IJ}$. 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{RE}(x,t)=\tfrac{|u_{a}(x,t)|L}{\nu} $ then aspects of a turbulent flow with $\mathsf{RE}(x,t)\gg \mathsf{RE}{*}$, a critical Reynolds number, might be represented by the 'weighted' random field $\mathscr{U}{a}(x,t)= u_{a}(x,t)+\mathrm{A}u_{a}(x,t)\big(\mathsf{RE}(x,t)-\mathsf{RE}{*}\big){\beta}\sum{I=1}{\infty} \mathrm{Z}{1/2}{I}f{I}(x)\otimes\mathscr{Z}_{I}$ where random fluctuations and amplitude scale nonlinearly with $\mathsf{RE}(x,t)$, with mean $\mathbf{E}[{\mathscr{U}}{a}(x,t)] =u{a}(x,t)$. In the inviscid limit one can prove an anomalous dissipation-type law \begin{align} \lim_{\nu\rightarrow 0}\bigg(\lim_{u_{a}(x,t)\rightarrow {u}{a}}\sup~\nu \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 $\beta=\tfrac{1}{2}$ and $\sum_{I=1}{\infty}\mathrm{Z}{I}\int{{\mathfrak{G}}}{\nabla}{a}f{I}(x){\nabla}{a}f_{I}(x)d\mathcal{V}(x)>0$.