01 · The idea
In everyday language
A value function from an optimisation problem can have corners, so the derivatives demanded by a classical Hamilton–Jacobi equation may not exist. Crandall and Lions’ viscosity framework compares the unknown function with smooth test functions touching it from above or below. Inequalities for those test functions express what it means to satisfy the equation at a nonsmooth point. Under suitable assumptions, a comparison principle orders subsolutions and supersolutions, often yielding uniqueness and stability. The word viscosity comes from a regularisation viewpoint, but the definition is mathematical and need not describe a physically viscous fluid.
Why it matters
Viscosity solutions made many nonlinear equations from control and optimisation mathematically well posed, connecting nonsmooth value functions with reliable principles for uniqueness, approximation and stability.
