[Tex学习笔记]一个数学公式

你的名字 2022-03-31 06:24 266阅读 0赞

\\begin\{equation\*\} \\begin\{aligned\} &\\quad\\int |\\nabla(T\_1-\\overline\{T\})^+|^2 \\rm dx-\\int \\frac\{3m\_1\}\{2\}\\nabla \\psi\_1\\nabla T\_1(T\_1-\\overline\{T\})^+ \\rm dx +\\int \\nabla m\_1\\nabla \\overline\{\\psi\_1\}\\nabla T\_1(T\_1-\\overline\{T\})^+dx\\\\ &=\\int \\frac\{\\varepsilon m\_1\}\{2\}(\\nabla \\psi\_1)^2 T\_1(T\_1-\\overline\{T\})^+ \\rm dx-\\int \\frac\{3m\_1\}\{2\\tau\}(T\_1-T\_\{1L\})(T\_1-\\overline\{T\})^+ dx, \\end\{aligned\} \\end\{equation\*\} \\begin\{equation\*\} \\begin\{aligned\} &\\quad\\int |\\nabla(T\_2-\\overline\{T\})^+ |^2 \\rm dx-\\int \\frac\{3m\_2\}\{2\}\\nabla \\psi\_2\\nabla T\_2(T\_2-\\overline\{T\})^+\\mathrm \{dx\} +\\int \\nabla m\_2\\nabla \\overline\{\\psi\_2\}\\nabla T\_2(T\_2-\\overline\{T\})^+\\rm dx\\\\ &=\\int \\frac\{\\mu m\_2\}\{2\}(\\nabla \\psi\_2)^2 T\_2(T\_2-\\overline\{T\})^+ \\rm dx-\\int \\frac\{3m\_2\}\{2\\tau\}(T\_2-T\_\{2L\})(T\_2-\\overline\{T\})^+ \\rm dx\\\\ &=\\int \\frac\{3m\_1\}\{2\}\\nabla \\psi\_1\\nabla T\_1(T\_1-\\overline\{T\})^+ \\rm dx\\leq C\\epsilon\\int |\\nabla(T\_1-\\overline\{T\})^+|^2+|(T\_1-\\overline\{T\})^+|^2 \\rm dx, \\end\{aligned\} \\end\{equation\*\} \\begin\{equation\*\} \\begin\{aligned\} &\\quad-\\int \\nabla m\_1\\nabla \\overline\{\\psi\_1\}\\nabla T\_1(T\_1-\\overline\{T\})^+ \\rm dx +\\int \\frac\{\\varepsilon m\_1\}\{2\\tau\}(\\nabla \\psi\_1)^2 T\_1(T\_1-\\overline\{T\})^+\\rm dx\\\\ &=-\\int \\nabla m\_1\\nabla \\overline\{\\psi\_1\}\\nabla(T\_1-\\overline\{T\}+\\overline\{T\})(T\_1-\\overline\{T\})^+\\rm dx+\\int \\frac\{\\varepsilon m\_1\}\{2\\tau\}(\\nabla \\psi\_1)^2 T\_1(T\_1-\\overline\{T\})^+ \\rm dx \\\\ &\\leq -\\int \\nabla m\_1\\nabla \\overline\{\\psi\_1\}\\nabla (T\_1-\\overline\{T\})(T\_1-\\overline\{T\})^+\\rm dx+\\int \\frac\{\\varepsilon m\_1\}\{2\\tau\}(\\nabla \\psi\_1)^2 T\_1(T\_1-\\overline\{T\})^+ \\rm dx\\\\ &\\leq C\\epsilon \\int \\nabla m\_1\\nabla \\overline\{\\psi\_1\}\\nabla ((T\_1-\\overline\{T\})^+)^2\\rm dx+C(n\_2\\varepsilon)\\epsilon^2\\int T\_1(T\_1-\\overline\{T\})^+ \\rm dx \\\\ &\\quad-\\int \\frac\{3m\_1\}\{2\\tau\}(T\_1-T\_\{1L\})(T\_1-\\overline\{T\})^+ \\rm dx=-\\int \\frac\{3m\_1\}\{2\\tau\}(T\_1-\\overline\{T\}+\\overline\{T\}-T\_\{1L\}(x))(T\_1-\\overline\{T\})^+ \\rm dx\\\\ &=-\\int \\frac \{3m\_1\}\{2\\tau\}(T\_1-\\overline\{T\})(T\_1-\\overline\{T\})^+\\rm dx-\\int \\frac\{3m\_1\}\{2\\tau\}(\\overline\{T\}-T\_\{1L\}(x))(T\_1-\\overline\{T\})^+ \\rm dx \\leq 0. \\end\{aligned\} \\end\{equation\*\}

From Feilin Lan.