Solutions for linear conservation laws with gradient constraint

We consider variational inequality solutions with prescribed gradient constraints for first order linear boundary value problems. For operators with coefficients only in L^2, we show the existence and uniqueness of the solution by using a combination of parabolic regularization with a penalization in the nonlinear diffusion coefficient. We also prove the continuous dependence of the solution with respect to the data, as well as, in a coercive case, the asymptotic stabilization as time t tends to infinity towards the stationary solution. In particular situation, motivated by the transported sandpile problem, we give sufficient conditions for the equivalence of the first order problem with gradient constraint with a two obstacles problem, the obstacles being the signed distances to the boundary. This equivalence, in special conditions, illustrates also the possible stabilization of the solution in finite time.This research was partially supported by CMAT-"Centro de Matematica da Universidade do Minho'', financed by FEDER Funds through "Programa Operacional Factores de Competitividade-COMPETE'' and by Portuguese Funds through FCT, "Fundacao para a Ciencia e a Tecnologia'', within the Project PEst-OE/MAT/UI0013/2014