# Portugaliae Mathematica

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**Volume 72, Issue 2-3, 2015, pp. 161–192**

**DOI: 10.4171/PM/1963**

Published online: 2015-07-23

Solutions for linear conservation laws with gradient constraint

José Francisco Rodrigues^{[1]}and Lisa Santos

^{[2]}(1) FC Universidade de Lisboa, Portugal

(2) Universidade do Minho, Braga, Portugal

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 $t \rightarrow + \infty$ towards the stationary solution. In a 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.

*Keywords: *Linear conservation laws, gradient constraints, transport equation, unilateral constraint, first order variational inequalities

Rodrigues José Francisco, Santos Lisa: Solutions for linear conservation laws with gradient constraint. *Port. Math.* 72 (2015), 161-192. doi: 10.4171/PM/1963