The Least Action Admissibility Principle
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866916646635962368 |
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| author | Gimperlein, Heiko Grinfeld, Michael Knops, Robin J. Slemrod, Marshall |
| author_facet | Gimperlein, Heiko Grinfeld, Michael Knops, Robin J. Slemrod, Marshall |
| contents | This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided to Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos's entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by $p(ρ)=ρ^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07191 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Least Action Admissibility Principle Gimperlein, Heiko Grinfeld, Michael Knops, Robin J. Slemrod, Marshall Analysis of PDEs This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided to Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos's entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by $p(ρ)=ρ^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data. |
| title | The Least Action Admissibility Principle |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.07191 |