engleski

On a gradient constraint problem for scalar conservation laws

We consider a Dirichlet-Neumann boundary problem in a bounded domain for scalar conservation laws. Using an idea from [1], we propose an informal solution concept by considering an elliptic approximation to the problem (see Chapter 3 of [2]). We are not yet able to prove existence or uniqueness of the solution satisfying the proposed solution concept, but, under appropriate assumptions, we prove that a corresponding weak limit satisfies the considered equation in the interior of the domain. The basic tool is the compensated compactness method. In the case when the flux is continuously differentiable with respect to all the variables, Remark 1 of [3] implies that a weak limit of the elliptic approximation satisfies the Kruzhkov admissibility conditions in the interior of the domain. We also provide numerical examples to justify, in the special situation, one of the limiting assumptions of the theoretical framework.

mixed boundary problem ; scalar conservation laws

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