Solutions of Hamilton–Jacobi Equations and Scalar Conservation Laws with Discontinuous Space–Time Dependence

We establish a unique stable solution to the Hamilton–Jacobi equation ut+H(K(x,t),ux)=0, x (−∞,∞), t [0,∞) with Lipschitz initial condition, where K(x,t) is allowed to be discontinuous in the (x,t) plane along a finite number of (possibly intersecting) curves parameterized by t. We assume that for fixed k, H(k,p) is convex in p and . The solution is determined by showing that if K is made smooth by convolving K in the x direction with the standard mollifier, then the control theory representation of the viscosity solution to the resulting Hamilton–Jacobi equation must converge uniformly as the mollification decreases to a Lipschitz continuous solution with an explicit control theory representation. This also defines the unique stable solution to the corresponding scalar conservation law ut+(f(K(x,t),u))x=0, x (−∞,∞), t [0,∞) with K discontinuous.