One-Dimensional Carrollian Fluids III: Global Existence and Weak Continuity in $L^\infty$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915322715439104 |
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| author | Petropoulos, P. Marios Schulz, Simon Taujanskas, Grigalius |
| author_facet | Petropoulos, P. Marios Schulz, Simon Taujanskas, Grigalius |
| contents | The Carrollian fluid equations arise as the $c \to 0$ limit of the relativistic fluid equations and have recently experienced a surge of activity in the flat-space holography community. However, the rigorous mathematical well-posedness theory for these equations does not appear to have been previously studied. This paper is the third in a series in which we initiate the systematic analysis of the Carrollian fluid equations. In the present work we prove the global-in-time existence of bounded entropy solutions to the isentropic Carrollian fluid equations in one spatial dimension for a particular constitutive law ($γ= 3$). Our method is to use a vanishing viscosity approximation for which we establish a compensated compactness framework. Using this framework we also prove the compactness of entropy solutions in $L^\infty$, and establish a kinetic formulation of the problem. This global existence result in $L^\infty$ extends the $C^1$ theory presented in our companion paper ``One-Dimensional Carrollian Fluids II: $C^1$ Blow-up Criteria''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05972 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | One-Dimensional Carrollian Fluids III: Global Existence and Weak Continuity in $L^\infty$ Petropoulos, P. Marios Schulz, Simon Taujanskas, Grigalius Analysis of PDEs General Relativity and Quantum Cosmology High Energy Physics - Theory 35L65, 35Q35, 35Q75, 85A30 The Carrollian fluid equations arise as the $c \to 0$ limit of the relativistic fluid equations and have recently experienced a surge of activity in the flat-space holography community. However, the rigorous mathematical well-posedness theory for these equations does not appear to have been previously studied. This paper is the third in a series in which we initiate the systematic analysis of the Carrollian fluid equations. In the present work we prove the global-in-time existence of bounded entropy solutions to the isentropic Carrollian fluid equations in one spatial dimension for a particular constitutive law ($γ= 3$). Our method is to use a vanishing viscosity approximation for which we establish a compensated compactness framework. Using this framework we also prove the compactness of entropy solutions in $L^\infty$, and establish a kinetic formulation of the problem. This global existence result in $L^\infty$ extends the $C^1$ theory presented in our companion paper ``One-Dimensional Carrollian Fluids II: $C^1$ Blow-up Criteria''. |
| title | One-Dimensional Carrollian Fluids III: Global Existence and Weak Continuity in $L^\infty$ |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology High Energy Physics - Theory 35L65, 35Q35, 35Q75, 85A30 |
| url | https://arxiv.org/abs/2407.05972 |