Duality between Hyperbolic Conservation Laws and Hamilton-Jacobi Equations.

Abstract : The mathematical modelling of vehicular movement (traffic flow) gives rise to an interesting class of partial differential equations, knowns as Hyperbolic conservation laws. An initial value problem for such conservation laws leads to several non-linear wave phenomena, such as shock waves, expansion waves etc. We illustrate these notions using a simple traffic flow model in the context of a traffic junction. Another interesting feature of conservation laws is the non uniqueness of discontinuous weak solutions and an entropy inequality a reminiscent of second law of thermodynamics, is required to characterise the physically relevant weak solution. Hamilton-Jacobi equations, on the other hand arise in geometric optics, control theory etc. We use a duality between Hamilton-Jacobi equations and Hyperbolic conservation laws to establish the existence of a unique entropy satisfying weak solution.