Hamilton–Jacobi equation

Summary

A Hamilton–Jacobi equation is a first-order nonlinear partial differential equation of the form $u_t + H(\nabla u, x) = 0$, where $H$ is a given function called the Hamiltonian. It comes from classical mechanics, where its solutions generate the motions of a Hamiltonian system. It also appears in optimal control, geometric optics and front propagation. Classical solutions usually stop existing after some time, and the right notion of global solution is the viscosity solution. See the definition.