Unfortunately, the class of weak solutions is too big: the usual (initial) data for these problems does not determine weak solutions uniquely. (The class of classical solutions is too small, since there usually are no classical solutions.) Therefore an additional criterion must be supplied to fix an unique solution. In the fluid-mechanics problems that motivated much of the work on these equations, it was recognized early on that the effect of viscosity had been neglected. It was found that when viscosity is included properly, no actual gradient discontinuities develop, and the solutions are generally classical, i.e. smooth. Because smooth solutions are uniquely determined by the appropriate initial data, the addition of viscosity fixes the uniqueness problem.
A very important property of viscosity solutions, first understood in the context of conservation laws (Lax, 1973), is that these are the unique, weak solutions, stable with respect to perturbations in the equation. Because round-off and discretization errors always perturb a numerical solution, viscosity solutions are a natural target for numerical methods.