Strong and Weak Boundary Conditions for Solutions of Scalar Conservation Laws

Halina Frankowska
Centre National de la Recherche Scientifique (CNRS)

It is well known that the initial and boundary value problem for scalar conservation laws may not have solutions. Several authors proposed appropriate notions of solutions with boundary conditions satisfied in a generalized sense. We consider scalar conservation
laws on the strip (0,infty) x [0,1] with strictly convex smooth flux of a superlinear growth. We associate to it a Hamilton-Jacobi equation with initial and (appropriately defined) boundary conditions and show that it has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. The traces of the gradients V_x satisfy then generalized boundary conditions in a weak sense that was proposed by LeFloch (1988) for conservation
laws with discontinuous initial and boundary conditions. We shall discuss a more precise version of these boundary conditions and show that they are satisfied in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous.

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