Scalar conservation law in a bounded domain with strong source at boundary

Abstract

We consider a scalar conservation law with source in a bounded open interval $$\Omega \subseteq \mathbb R$$ Ω ⊆ R . The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function $$\varrho $$ ϱ with an intensity function $$V:\Omega \rightarrow \mathbb R_+$$ V : Ω → R + that grows to infinity at $$\partial \Omega $$ ∂ Ω . We define the entropy solution $$u \in L^\infty $$ u ∈ L ∞ and prove the uniqueness. When V is integrable, u satisfies the boundary conditions introduced by F. Otto (C. R. Acad. Sci. Paris, 322(1):729–734, 1996), which allows the solution to attain values at $$\partial \Omega $$ ∂ Ω different from the given boundary data. When the integral of V blows up, u satisfies an energy estimate and presents essential continuity at $$\partial \Omega $$ ∂ Ω in a weak sense.