Hamilton–Jacobi equation

Remarks

Often $H$ does not depend on $x$, and the equation reads $u_t + H(\nabla u) = 0$, usually with an initial condition $u(x, 0) = g(x)$. In 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 $S$. The Hamilton–Jacobi–Bellman equation of optimal control is a Hamilton–Jacobi equation whose Hamiltonian is an infimum or a supremum over controls.

Example: $H(p, x) = \tfrac12 \abs{p}^2$ and $u(x, t) = \dfrac{\abs{x}^2}{2(t + 1)}$ for $t > 0$. Check: $\partial_t u = -\dfrac{\abs{x}^2}{2(t + 1)^2}$ 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 $0$.

Non-example: the heat equation $u_t - \Delta u = 0$ is not a Hamilton–Jacobi equation, since it involves second derivatives of $u$. Adding a term $-\varepsilon \Delta u$ to a Hamilton–Jacobi equation gives a viscous approximation, whose limits as $\varepsilon \to 0$ lead to viscosity solutions.