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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.11415 |
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Table of Contents:
- We consider a scalar conservation law with source in a bounded open interval $Ω\subseteq\mathbb R$. 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 $V:Ω\to\mathbb R_+$ that grows to infinity at $\partialΩ$. 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 $\partialΩ$ different from the given boundary data. When the integral of $V$ blows up, $u$ satisfies an energy estimate and presents essential continuity at $\partialΩ$ in a weak sense.