加权随机矢量场的谱表示:湍流和不粘性极限中的异常耗散问题的潜在应用

Steven D Miller
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If\n$\\mathfrak{G}$ contains incompressible fluid of viscosity $\\nu$ with velocity\n$u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds\nfunction' $\\mathsf{RE}(x,t)=\\tfrac{\\|u_{a}(x,t)\\|L}{\\nu} $ then aspects of a\nturbulent flow with $\\mathsf{RE}(x,t)\\gg \\mathsf{RE}_{*}$, a critical Reynolds\nnumber, might be represented by the 'weighted' random field\n$\\mathscr{U}_{a}(x,t)=\nu_{a}(x,t)+\\mathrm{A}u_{a}(x,t)\\big(\\mathsf{RE}(x,t)-\\mathsf{RE}_{*}\\big)^{\\beta}\\sum_{I=1}^{\\infty}\n\\mathrm{Z}^{1/2}_{I}f_{I}(x)\\otimes\\mathscr{Z}_{I}$ where random fluctuations\nand amplitude scale nonlinearly with $\\mathsf{RE}(x,t)$, with mean\n$\\mathbf{E}[{\\mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can\nprove an anomalous dissipation-type law \\begin{align} \\lim_{\\nu\\rightarrow\n0}\\bigg(\\lim_{u_{a}(x,t)\\rightarrow {u}_{a}}\\sup~\\nu\n\\int_{\\mathfrak{G}}\\int_{0}^{T}{\\mathbf{E}}\\bigg[\\bigg|{\\nabla}_{a}{\\mathscr{U}}_{a}(x,s)\\bigg|^{2}\\bigg]d\\mathcal{V}(x)\nds\\bigg)>0 \\end{align} iff $\\beta=\\tfrac{1}{2}$ and\n$\\sum_{I=1}^{\\infty}\\mathrm{Z}_{I}\\int_{{\\mathfrak{G}}}{\\nabla}_{a}f_{I}(x){\\nabla}^{a}f_{I}(x)d\\mathcal{V}(x)>0$.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"103 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Spectral Representation of a Weighted Random Vectorial Field: Potential Applications to Turbulence and the Problem of Anomalous Dissipation in the Inviscid Limit\",\"authors\":\"Steven D Miller\",\"doi\":\"arxiv-2409.10636\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let ${\\\\mathfrak{G}}\\\\subset\\\\mathbb{R}^{3}$ with $vol(\\\\mathfrak{G})\\\\sim L^{3}$.\\nLet ${\\\\mathscr{T}}(x)$ be a Gaussian random field $\\\\forall~x\\\\in\\\\mathfrak{G}$\\nwith expectation $\\\\mathbf{E}[{\\\\mathscr{T}}(x)]=0$ and correlation\\n$\\\\mathbf{E}[{\\\\mathscr{T}}(x)\\\\otimes{\\\\mathscr{T}}(y)]=K(x,y;\\\\lambda)$, an\\nisotropic and regulated kernel with correlation length $\\\\lambda$. The field has\\na Karhunen-Loeve spectral representation\\n${\\\\mathscr{T}}(x)=\\\\sum_{I=1}^{\\\\infty}\\\\mathrm{Z}^{1/2}_{I}f_{I}(x)\\\\otimes\\\\mathscr{Z}_{I}$,\\nwith eigenvalues $\\\\lbrace\\\\mathrm{Z}_{I}\\\\rbrace$, eigenfunctions $\\\\lbrace\\nf_{I}(x)\\\\rbrace $ and Gaussian random variables $\\\\mathscr{Z}_{I}$ with\\n$\\\\mathbf{E}[\\\\mathscr{Z}_{I}]=0$ and\\n$\\\\mathbf{E}[\\\\mathscr{Z}_{I}\\\\otimes\\\\mathscr{Z}_{J}]=\\\\delta_{IJ}$. If\\n$\\\\mathfrak{G}$ contains incompressible fluid of viscosity $\\\\nu$ with velocity\\n$u_{a}(x,t)$ that evolves via the Navier-Stokes equations with a high 'Reynolds\\nfunction' $\\\\mathsf{RE}(x,t)=\\\\tfrac{\\\\|u_{a}(x,t)\\\\|L}{\\\\nu} $ then aspects of a\\nturbulent flow with $\\\\mathsf{RE}(x,t)\\\\gg \\\\mathsf{RE}_{*}$, a critical Reynolds\\nnumber, might be represented by the 'weighted' random field\\n$\\\\mathscr{U}_{a}(x,t)=\\nu_{a}(x,t)+\\\\mathrm{A}u_{a}(x,t)\\\\big(\\\\mathsf{RE}(x,t)-\\\\mathsf{RE}_{*}\\\\big)^{\\\\beta}\\\\sum_{I=1}^{\\\\infty}\\n\\\\mathrm{Z}^{1/2}_{I}f_{I}(x)\\\\otimes\\\\mathscr{Z}_{I}$ where random fluctuations\\nand amplitude scale nonlinearly with $\\\\mathsf{RE}(x,t)$, with mean\\n$\\\\mathbf{E}[{\\\\mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$. In the inviscid limit one can\\nprove an anomalous dissipation-type law \\\\begin{align} \\\\lim_{\\\\nu\\\\rightarrow\\n0}\\\\bigg(\\\\lim_{u_{a}(x,t)\\\\rightarrow {u}_{a}}\\\\sup~\\\\nu\\n\\\\int_{\\\\mathfrak{G}}\\\\int_{0}^{T}{\\\\mathbf{E}}\\\\bigg[\\\\bigg|{\\\\nabla}_{a}{\\\\mathscr{U}}_{a}(x,s)\\\\bigg|^{2}\\\\bigg]d\\\\mathcal{V}(x)\\nds\\\\bigg)>0 \\\\end{align} iff $\\\\beta=\\\\tfrac{1}{2}$ and\\n$\\\\sum_{I=1}^{\\\\infty}\\\\mathrm{Z}_{I}\\\\int_{{\\\\mathfrak{G}}}{\\\\nabla}_{a}f_{I}(x){\\\\nabla}^{a}f_{I}(x)d\\\\mathcal{V}(x)>0$.\",\"PeriodicalId\":501245,\"journal\":{\"name\":\"arXiv - MATH - Probability\",\"volume\":\"103 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Probability\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10636\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10636","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

让 ${mathfrak{G}}\subset\mathbb{R}^{3}$ 带有 $vol(\mathfrak{G})\sim L^{3}$。让 ${mathscr{T}}(x)$ 是一个高斯随机场 $/forall~x\inmathfrak{G}$,期望值为 $mathbf{E}[{/mathscr{T}}(x)]=0$,相关性为 $mathbf{E}[{/mathscr{T}}(x)\otimes{/mathscr{T}}(y)]=K(x,y;\lambda)$,各向异性且具有相关长度 $\lambda$ 的调节核。该场具有卡尔胡宁-洛夫谱表示${mathscr{T}}(x)=\sum_{I=1}^{infty}\mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$,特征值为$\lbrace\mathrm{Z}_{I}\rbrace$、特征函数 $\lbracef_{I}(x)\rbrace $ 和高斯随机变量 $\mathscr{Z}_{I}$ with$\mathbf{E}[\mathscr{Z}_{I}]=0$ and$\mathbf{E}[\mathscr{Z}_{I}\otimes\mathscr{Z}_{J}]=\delta_{IJ}$.如果$mathfrak{G}$包含不可压缩流体,其粘度为$\nu$,速度为$u_{a}(x,t)$,通过纳维-斯托克斯方程以高 "雷诺函数"$\mathsf{RE}(x、t)=\tfrac{|u_{a}(x,t)\|L}{/nu}$,那么湍流的某些方面具有 $\mathsf{RE}(x,t)\gg \mathsf{RE}_{*}$,即临界雷诺数、可以用 "加权 "随机场$\mathscr{U}_{a}(x,t)=u_{a}(x,t)+\mathrm{A}u_{a}(x,t)\big(\mathsf{RE}(x、t)-(mathsf{RE}_{*}\big)^{\beta}\sum_{I=1}^{infty}\mathrm{Z}^{1/2}_{I}f_{I}(x)\otimes\mathscr{Z}_{I}$,其中随机波动和振幅与 $\mathsf{RE}(x. t)$非线性扩展、t)$,平均值$\mathbf{E}[{\mathscr{U}}_{a}(x,t)] =u_{a}(x,t)$ 。在不粘性极限中,我们可以证明一个反常耗散型定律 \begin{align}\lim_{\nu\rightarrow0}\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$\sum_{I=1}^{\infty}\mathrm{Z}_{I}\int_{{\mathfrak{G}}}{\nabla}_{a}f_{I}(x){\nabla}^{a}f_{I}(x)d\mathcal{V}(x)>0$.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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 ${\mathscr{T}}(x)$ 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 $\sum_{I=1}^{\infty}\mathrm{Z}_{I}\int_{{\mathfrak{G}}}{\nabla}_{a}f_{I}(x){\nabla}^{a}f_{I}(x)d\mathcal{V}(x)>0$.
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