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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $H \\colon \\R^n \\times \\R^n \\to \\R$ be a continuous function, called the *Hamiltonian*.\nLet $T \\in (0, \\infty]$.\nThe *Hamilton–Jacobi equation* with Hamiltonian $H$ is the [[partial differential equation|GC5BDZ6QBC]]\n$$\\partial_t u + H(\\nabla u, x) = 0$$\nfor $u \\colon \\R^n \\times (0, T) \\to \\R$.\nHere $\\nabla u$ is the [[gradient|0ABK9V9Y64]] of $u$ in $x$.\nThe *stationary* Hamilton–Jacobi equation is $H(\\nabla u, x) = 0$ for $u$ defined on an open subset of $\\R^n$.\n\n## Remarks\n\nOften $H$ does not depend on $x$, and the equation reads $u_t + H(\\nabla u) = 0$, usually with an initial condition $u(x, 0) = g(x)$.\nIn mechanics the Hamiltonian also depends on time, and the equation is $\\partial_t S + H(\\nabla S, x, t) = 0$ for Hamilton's principal function $S$.\nThe [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] of optimal control is a Hamilton–Jacobi equation whose Hamiltonian is an infimum or a supremum over controls.\n\nExample: $H(p, x) = \\tfrac12 \\abs{p}^2$ and $u(x, t) = \\dfrac{\\abs{x}^2}{2(t + 1)}$ for $t \u003e 0$.\nCheck: $\\partial_t u = -\\dfrac{\\abs{x}^2}{2(t + 1)^2}$ and $\\nabla u = \\dfrac{x}{t + 1}$, so $\\tfrac12 \\abs{\\nabla u}^2 = \\dfrac{\\abs{x}^2}{2(t + 1)^2}$ and the sum is $0$.\n\nNon-example: the heat equation $u_t - \\Delta u = 0$ is not a Hamilton–Jacobi equation, since it involves second derivatives of $u$.\nAdding a term $-\\varepsilon \\Delta u$ to a Hamilton–Jacobi equation gives a viscous approximation, whose limits as $\\varepsilon \\to 0$ lead to viscosity solutions.\n",
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