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 $\Omega \subseteq \mathbb{R}$. 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 $\varrho$ with an intensity function $V:\Omega \to {\mathbb{R}}_{+}$ that grows to infinity at $\partial \Omega$. We define the entropy solution $u\in L^{\infty }$ and prove the uniqueness. When $V$ is integrable, $u$ satisfies the boundary conditions introduced in [F. Otto, C. R. Acad. Sci. Paris 1996], which allows the solution to attain values at $\partial \Omega$ different from the given boundary data. When the integral of $V$ blows up, $u$ satisfies an energy estimate and presents essential continuity at $\partial \Omega$ in a weak sense.

References in corpus (2)

• On Traffic Flow with Nonlocal Flux: a Relaxation Representation
• Quasi-static limit for the asymmetric simple exclusion

Cited by in corpus (1)

• Hydrodynamics for asymmetric simple exclusion on a finite segment with Glauber-type source