Conservation laws

The notion of viscosity solutions was developed first for conservation laws: first-order, partial-differential equations linear, in the derivatives. (Note that the eikonal equation is nonlinear in the derivatives.) Conservation laws arise as simple models of inviscid fluid flow, and their weak solutions are made unique by a viscosity principle as well. These viscosity weak solutions are also called ``entropy'' or ``admissible'' solutions. An excellent survey of this topic is in Lax (1973). We note that the connections between viscosity, entropy, and the uniqueness of weak solutions are completely understood at present only in special situations, notably scalar conservation laws in two variables.

We base our numerical work on the observation that the gradient components of a solution to the eikonal equation satisfy conservation laws. Moreover, the gradient components of viscosity solutions of the eikonal equation are viscosity solutions of the conservation laws. As noted above, recent theoretical work (Crandall and Lions (1984)) and a lot of numerical ediffvidtimeence support the conjecture that the first-arrival condition actually picks out a viscosity solution of the eikonal equation, from amongst all the branches of traveltime (all of which solve the eikonal equation locally). Thus we can characterize first-arrival-time fields by noting that their gradient components are entropy or viscosity solutions to the corresponding conservation laws.