Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2603.10214 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908877966016512 |
|---|---|
| author | Bressan, Alberto Shen, Wen |
| author_facet | 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. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10214 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux 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. In the stable case where $f(u)<g(u)$ for all $u\in R$, it was proved in [1] that the limits of vanishing viscosity approximations form a contractive semigroup w.r.t. the $L^1$ distance. Further, they coincide with the limits of a suitable family of front tracking approximations. In the present paper we introduce a simple condition that guarantees that every weak, entropy admissible solution of a Cauchy problem coincides with the corresponding semigroup trajectory, and hence is unique. |
| title | A Uniqueness Condition for Conservation Laws with Discontinuous Gradient-Dependent Flux |
| topic | Analysis of PDEs 35L65, 35D30, 76A30 |
| url | https://arxiv.org/abs/2603.10214 |