The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations

We present a closed form expression for the viscosity solution to the Riemann problem for any scalar nonconvex conservation law. We then define an analogous problem for any scalar nonconvex Hamilton-Jacobi equation and obtain its (even simpler) solution. Extensions to two (and by inference, higher) space dimensional problems, when the initial discontinuity lies on a hyperplane, are also given. We shall be concerned with solutions to the following nonlinear scalar partial differential equations: (H-J) (Pt + (wT) + g(cPy) = 0 and (CL) ut + (u)X + g(u)y = 0, for t > 0, with special initial data. This restriction to two space dimensions is made for simplicity only. A rather complete existence and uniqueness theory for the Cauchy problem for (CL) has been known for some time (see, e.g. [2]). Recently a parallel theory was developed for (H-J) [1]. (More general equations have also been treated by both of these theories.) Solutions of (CL) are well known to generally develop discontinuities, even if the initial data are smooth. Solutions of (H-J) stay continuous, but generally develop jumps in derivatives. To obtain uniqueness and the rest of the theory mentioned above, it suffices to consider only viscosity solutions. A solution to (1) is a viscosity solution if it is the local uniform limit of qE(x, y, t) as E t 0, where TE solves () +f () + g( g ) ( (XX + cPyp) + G( x, y, t, , W, x E Ty' Here, GE converges uniformly to zero on compact subsets of R2 X R+ X R4. Received by the editors March 21, 1983. 1980 Mathematics Subject Classification. Primary 35L60, 35L65, 35L99. Kev words and phrases. Conservation law, Hamilton-Jacobi equations, Riemann problem. ' Research supported by NSF Grant MCS 82-007788, NASA University Consortium Agreement NCA2-OR390-202, and ARO Grant DAAG29-82-KO090. ?, 983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page 641 This content downloaded from 207.46.13.118 on Sun, 11 Sep 2016 04:59:02 UTC All use subject to http://about.jstor.org/terms