Solutions for linear conservation laws with gradient constraints

TL;DR

This paper studies solutions to linear boundary value problems with gradient constraints, establishing existence, uniqueness, and stability, and explores their relation to obstacle problems and finite-time stabilization.

Contribution

It introduces a method combining regularization and penalization to prove existence and uniqueness for problems with low-regularity coefficients, and links gradient constraints to obstacle problems.

Findings

Existence and uniqueness of solutions with $L^2$ coefficients.

Continuous dependence of solutions on data.

Conditions for finite-time stabilization and equivalence to obstacle problems.

Abstract

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.