$L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911130727743488 |
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| author | Cheng, Jeffrey |
| author_facet | Cheng, Jeffrey |
| contents | In this paper, we study stability properties of solutions to scalar conservation laws with a class of non-convex fluxes. Using the theory of $a$-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient $a$ in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2 \times 2$ system setting by Chen, Golding, Krupa, $\&$ Vasseur. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes Cheng, Jeffrey Analysis of PDEs 35L65, 35B35, 35L45 In this paper, we study stability properties of solutions to scalar conservation laws with a class of non-convex fluxes. Using the theory of $a$-contraction with shifts, we show $L^2$-stability for shocks among a class of large perturbations, and give estimates on the weight coefficient $a$ in regimes where the shock amplitude is both large and small. Then, we use these estimates as a building block to show a uniqueness theorem under minimal entropy conditions for weak solutions to the conservation law via a modified front tracking algorithm. The proof is inspired by an analogous program carried out in the $2 \times 2$ system setting by Chen, Golding, Krupa, $\&$ Vasseur. |
| title | $L^2$-stability $\&$ Minimal Entropy Conditions for Scalar Conservation Laws with Concave-Convex Fluxes |
| topic | Analysis of PDEs 35L65, 35B35, 35L45 |
| url | https://arxiv.org/abs/2411.03578 |