We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L^1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.
Asymptotic behavior of solutions to conservation laws with periodic forcing
AMADORI, DEBORA;
2006-01-01
Abstract
We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L^1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.Pubblicazioni consigliate
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