Nonexistence of a class of $T_N$ configurations for a hyperbolic system with one entropy
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912111548956672 |
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| author | Peng, Guanying Vuolo, Anthony |
| author_facet | Peng, Guanying Vuolo, Anthony |
| contents | In Kirchheim, Müller and Šverák [Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, 2003], the authors proposed the program to use the differential inclusion approach to study entropy solutions for systems of conservation laws. In particular, they raised questions concerning the local structure of the rank-one convex hull of a set $K_a\subset\mathbb{R}^{3\times 2}$, which arises from the differential inclusion formulation of a classical $2\times 2$ system of conservation laws (the $p$-system) coupled with one entropy. Recently, this question has been studied extensively by showing that the set $K_a$ does not contain the so-called $T_N$ configurations for $N=4$ and $N=5$. In this paper, we continue this program by showing that the set $K_a$ does not contain a class of three-dimensional $T_N$ configurations, as well as two-dimensional $T_N$ configurations for general $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05637 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonexistence of a class of $T_N$ configurations for a hyperbolic system with one entropy Peng, Guanying Vuolo, Anthony Analysis of PDEs In Kirchheim, Müller and Šverák [Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, 2003], the authors proposed the program to use the differential inclusion approach to study entropy solutions for systems of conservation laws. In particular, they raised questions concerning the local structure of the rank-one convex hull of a set $K_a\subset\mathbb{R}^{3\times 2}$, which arises from the differential inclusion formulation of a classical $2\times 2$ system of conservation laws (the $p$-system) coupled with one entropy. Recently, this question has been studied extensively by showing that the set $K_a$ does not contain the so-called $T_N$ configurations for $N=4$ and $N=5$. In this paper, we continue this program by showing that the set $K_a$ does not contain a class of three-dimensional $T_N$ configurations, as well as two-dimensional $T_N$ configurations for general $N$. |
| title | Nonexistence of a class of $T_N$ configurations for a hyperbolic system with one entropy |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.05637 |