Scalar conservation law in a bounded domain with strong source at boundary
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911890668519424 |
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| author | Xu, Lu |
| author_facet | Xu, Lu |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11415 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Scalar conservation law in a bounded domain with strong source at boundary Xu, Lu Analysis of PDEs 35L04, 35D40 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. |
| title | Scalar conservation law in a bounded domain with strong source at boundary |
| topic | Analysis of PDEs 35L04, 35D40 |
| url | https://arxiv.org/abs/2310.11415 |