Quarterly of Applied Mathematics· 2026Q2
Energy-variational structure in evolution equations
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- Q2SCImago
- 2026year
Short summary
A novel energy-variational structure simplifies existing measure-valued solvability concepts for evolution equations, demonstrating equivalence between improved measure-valued solutions and energy-variational solutions across diverse examples.
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Key points
- Introduces an energy-variational structure to simplify measure-valued solvability concepts in evolution equations.
- Demonstrates the equivalence between improved measure-valued solutions and energy-variational solutions.
- Applies the concept to four distinct examples: two-phase Navier-Stokes, quasilinear wave, polyconvex elasticity, and Ericksen-Leslie equations.
- Suggests the energy-variational structure is a recurrent feature in evolution equations.
AI-generated from the title and abstract; the full text is not read.
Abstract
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, whereas 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.
The authors' abstract, as published at the source. Quarterly of Applied Mathematics, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics