Variational Principles for Shock Dynamics in Compressible Euler Flows
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914499061088256 |
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| author | Gay-Balmaz, François Yang, Cheng |
| author_facet | Gay-Balmaz, François Yang, Cheng |
| contents | Hamilton's principle plays a central role in fluid mechanics as a fundamental tool for deriving governing equations, analyzing conservation laws, and designing structure-preserving numerical schemes. However, its classical formulation is restricted to smooth solutions and does not directly accommodate shock discontinuities. Addressing this limitation within a variational framework has long been a challenging issue. In this paper, we develop a variational framework that extends Hamilton's principle to shock solutions in compressible fluid dynamics. For the barotropic Euler equations, we introduce a modified action principle that incorporates additional contributions localized at discontinuities. This allows the Rankine--Hugoniot conditions for mass and momentum to emerge directly from unrestricted variations, without imposing continuity across shocks. The additional term admits a natural interpretation as a dissipation potential, linking the variational structure to energy loss at shocks. We then extend the approach to the full compressible Euler equations. Using a variational formulation of nonequilibrium thermodynamics together with suitable variational and phenomenological constraints, we recover the Rankine--Hugoniot relations for mass, momentum, and energy. This yields a unified variational description of shock dynamics in compressible fluids and highlights a fundamental distinction between barotropic and full compressible models in the treatment of energy and entropy at discontinuities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20635 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variational Principles for Shock Dynamics in Compressible Euler Flows Gay-Balmaz, François Yang, Cheng Mathematical Physics Analysis of PDEs Dynamical Systems Hamilton's principle plays a central role in fluid mechanics as a fundamental tool for deriving governing equations, analyzing conservation laws, and designing structure-preserving numerical schemes. However, its classical formulation is restricted to smooth solutions and does not directly accommodate shock discontinuities. Addressing this limitation within a variational framework has long been a challenging issue. In this paper, we develop a variational framework that extends Hamilton's principle to shock solutions in compressible fluid dynamics. For the barotropic Euler equations, we introduce a modified action principle that incorporates additional contributions localized at discontinuities. This allows the Rankine--Hugoniot conditions for mass and momentum to emerge directly from unrestricted variations, without imposing continuity across shocks. The additional term admits a natural interpretation as a dissipation potential, linking the variational structure to energy loss at shocks. We then extend the approach to the full compressible Euler equations. Using a variational formulation of nonequilibrium thermodynamics together with suitable variational and phenomenological constraints, we recover the Rankine--Hugoniot relations for mass, momentum, and energy. This yields a unified variational description of shock dynamics in compressible fluids and highlights a fundamental distinction between barotropic and full compressible models in the treatment of energy and entropy at discontinuities. |
| title | Variational Principles for Shock Dynamics in Compressible Euler Flows |
| topic | Mathematical Physics Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2604.20635 |