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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^n$ be open and let $\\gamma \u003e 0$.\nThe *Cahn–Hilliard equation* is the [[parabolic equation|1CCTH8X2FT]] of fourth order for $c \\colon \\Omega \\times (0, T) \\to \\R$:\n$$c_t = \\Delta \\mu, \\qquad \\mu = c^3 - c - \\gamma\\, \\Delta c,$$\nwhere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] in $x$.\nThe function $\\mu$ is the chemical potential.\nOn a bounded $\\Omega$ one usually imposes $\\partial_\\nu c = \\partial_\\nu \\mu = 0$ on $\\partial \\Omega$, where $\\nu$ is the outer normal.\n\n## Remarks\n\nExample: every constant $c \\equiv c_0$ is a solution, since then $\\mu$ is constant and $\\Delta \\mu = 0$.\nNon-example: the [[Allen–Cahn equation|7S813PEV69]] $c_t = \\gamma \\Delta c - (c^3 - c)$ is of second order and does not conserve mass, so it is not of this form.\nWith the boundary conditions above, $\\frac{d}{dt}\\int_\\Omega c\\, dx = \\int_{\\partial\\Omega} \\partial_\\nu \\mu \\, dS = 0$, so the mass is conserved.\nThe energy $E(c) = \\int_\\Omega \\big(\\frac{\\gamma}{2}\\abs{\\nabla c}^2 + \\frac{1}{4}(1 - c^2)^2\\big)\\, dx$ does not increase along smooth solutions.\n",
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