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  "body": "\n## Definition\n\nLet $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, let $(N, h)$ be a Riemannian manifold, and let $u_0 \\colon M \\to N$ be a smooth map.\nLet $0 \u003c T \\le \\infty$.\nA solution of the *harmonic map heat flow* with initial map $u_0$ on $[0, T)$ is a map $u \\colon M \\times [0, T) \\to N$ with these properties.\nIt is smooth on $M \\times (0, T)$, continuous up to $t = 0$, and\n$$\\frac{\\partial u}{\\partial t}(x, t) = \\tau\\big(u(\\cdot, t)\\big)(x) \\quad \\text{on } M \\times (0, T), \\qquad u(x, 0) = u_0(x),$$\nwhere $\\tau$ is the [[tension field|W8F8S8DX5X]].\n\n## Remarks\n\nThe [[energy|E6HDE1DFXD]] decreases along a solution:\n$$\\frac{d}{dt} E\\big(u(\\cdot, t)\\big) = -\\int_M \\abs{\\tau(u(\\cdot, t))}^2 \\, dV_g \\le 0.$$\nIf $N$ is compact, a solution exists on some interval $[0, T)$ with $T \u003e 0$, since the equation is a semilinear parabolic system.\nA map $u_0$ is a fixed point of the flow exactly when it is a harmonic map.\n\nExample: for $N = \\R$ the Christoffel symbols vanish, and the flow is the [[heat equation|VP2HN86VAG]] $\\partial_t u = \\Delta_g u$ on $M$.\nIts solutions exist for all time and converge to the constant function equal to the mean of $u_0$.\n\nNon-example: a family with $\\partial_t u = -\\tau(u)$ is not a solution of the harmonic map heat flow but of the backward flow, along which the energy increases.\nFor $N = \\R$ this is the backward heat equation, which is ill-posed.\n",
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