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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2602.18917 |
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| _version_ | 1866912965539659776 |
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| author | Vorotnikov, Dmitry |
| author_facet | Vorotnikov, Dmitry |
| contents | We investigate a dual variational formulation, in the spirit of Brenier, for several compressible fluid models: the compressible barotropic Euler system, the quantum Euler system, and the Euler-Korteweg system. We identify a unified abstract framework encompassing all three systems, which enables a simultaneous analysis. By introducing time-adaptive weights, we establish the consistency of the duality scheme on large time intervals. We prove the existence of variational dual solutions to the corresponding Cauchy problems for continuous, vacuum-free initial data in spaces of finite Radon measures, and establish the absence of a duality gap. As an application, we formulate and prove a 'Dafermos principle' for these models: no subsolution can dissipate the total entropy earlier or at a faster rate than the corresponding strong solution on its interval of existence. We also discuss connections between our abstract consistency result and Brenier's shock-free substitutes for entropy solutions of Burgers' equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A unified duality framework for barotropic, quantum and Korteweg fluids Vorotnikov, Dmitry Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control 35Q35, 37K58, 49Q99, 76N10 We investigate a dual variational formulation, in the spirit of Brenier, for several compressible fluid models: the compressible barotropic Euler system, the quantum Euler system, and the Euler-Korteweg system. We identify a unified abstract framework encompassing all three systems, which enables a simultaneous analysis. By introducing time-adaptive weights, we establish the consistency of the duality scheme on large time intervals. We prove the existence of variational dual solutions to the corresponding Cauchy problems for continuous, vacuum-free initial data in spaces of finite Radon measures, and establish the absence of a duality gap. As an application, we formulate and prove a 'Dafermos principle' for these models: no subsolution can dissipate the total entropy earlier or at a faster rate than the corresponding strong solution on its interval of existence. We also discuss connections between our abstract consistency result and Brenier's shock-free substitutes for entropy solutions of Burgers' equation. |
| title | A unified duality framework for barotropic, quantum and Korteweg fluids |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis Optimization and Control 35Q35, 37K58, 49Q99, 76N10 |
| url | https://arxiv.org/abs/2602.18917 |