Solutions for linear conservation laws with gradient constraint

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 → +∞ 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.Dedicado a João Paulo Dias, no seu ativo septuagésimo aniversário!

Variational and Quasi-Variational Inequalities with Gradient Type Constraints

“…x ∈ B r (y), for all y ∈ Σ . Similar results were obtained in [79] for the transported sandpile problem, for u(t) ∈ K k , such that…”

“…⊓ ⊔ However, in this case, for the corresponding variational inequality, i.e. when G ≡ g(x,t), in [79] it was shown that the problem is well-posed and has similar stability properties, as in Section 3.1, with coefficients only in L 2 .…”

“…for all v ∈ K k , with b b b ∈ R 2 , ∂ Ω ∈ C 2 , f = f (t) ≥ 0 nondecreasing and f ∈ L ∞ (0, ∞), which also satisfies (100). Moreover, it was also shown in [79] that u(t) equivalently solves (102) where u ε ∈ C 1 (Ω ) ∩ W 1,∞ 0 (Ω ) is an approximation of the initial condition u 0 ∈ W 1,∞ 0 (Ω ). We observe that the existence of a quasi-variational solution of (102) with this G ε is also guaranteed by Theorem 15 or Corollary 1.…”

“…we may prove the following result Theorem 16. For any δ ≥ 0 and 1 < p < ∞, under the assumptions (70)-( 72) and (75)- (79), problem (74…”

“…The first section treats weak and strong solutions of variational inequalities with time dependent convex sets, following [66] and giving explicit estimates on the continuous dependence results. The next two sections are, respectively, dedicated to the scalar problems with gradient constraint, relating the original works [83] and [84] to the more recent inequality for the transport equation of [79] for the variational case, and to the scalar quasi-variational strong solutions presenting a synthesis of [77] with [78] and an extension to the linear first order problem as a new corollary. The following section, based on [66], briefly describes the regularisation penalisation method to obtain the existence of weak solutions by compactness and monotonicity.…”