Energy-variational structure in evolution equations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913736192688128 |
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| author | Lasarzik, Robert |
| author_facet | Lasarzik, Robert |
| contents | We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered examples, we prove that a certain class of improved measure-valued solutions can be equivalently expressed as an energy-variational solution. The first concept represents the solution as a high-dimensional Young measure, whether for the second concept, only a scalar auxiliary variable is introduced and the formulation is relaxed to an energy-variational inequality. We investigate four examples: the two-phase Navier--Stokes equations, a quasilinear wave equation, a system stemming from polyconvex elasticity, and the Ericksen--Leslie equations equipped with the Oseen--Frank energy. The wide range of examples suggests that this is a recurrent feature in evolution equations in general. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_11438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy-variational structure in evolution equations Lasarzik, Robert Analysis of PDEs 35D99, 35R35, 35R45, 35Q35, 35Q74, 76T06 We consider different measure-valued solvability concepts from the literature and show that they could be simplified by using the energy-variational structure of the underlying system of partial differential equations. In the considered examples, we prove that a certain class of improved measure-valued solutions can be equivalently expressed as an energy-variational solution. The first concept represents the solution as a high-dimensional Young measure, whether for the second concept, only a scalar auxiliary variable is introduced and the formulation is relaxed to an energy-variational inequality. We investigate four examples: the two-phase Navier--Stokes equations, a quasilinear wave equation, a system stemming from polyconvex elasticity, and the Ericksen--Leslie equations equipped with the Oseen--Frank energy. The wide range of examples suggests that this is a recurrent feature in evolution equations in general. |
| title | Energy-variational structure in evolution equations |
| topic | Analysis of PDEs 35D99, 35R35, 35R45, 35Q35, 35Q74, 76T06 |
| url | https://arxiv.org/abs/2503.11438 |