Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915748259037184 |
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| author | Amadori, Debora Bressan, Alberto Shen, Wen |
| author_facet | Amadori, Debora Bressan, Alberto Shen, Wen |
| contents | The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions.
For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case Amadori, Debora Bressan, Alberto Shen, Wen Analysis of PDEs 35L65, 35D30, 76A30 The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative, respectively. We study here the unstable case where $f(u)>g(u)$ for all $u\in {\mathbb R}$. Assuming that both $f$ and $g$ are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between $f$ and $g$, remains as small as possible. |
| title | Conservation Laws with Discontinuous Gradient-Dependent Flux: the Unstable Case |
| topic | Analysis of PDEs 35L65, 35D30, 76A30 |
| url | https://arxiv.org/abs/2411.13444 |