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Existence and selection of solutions in the energy-variational framework with applications in fluid dynamics

Thomas Eiter, Robert Lasarzik, Marcel Śliwiński

TL;DR

The paper develops a general energy-variational framework for abstract evolution equations, allowing energies with linear growth and two distinct regularity weights to ensure lower semicontinuity and compactness. Existence of global energy-variational solutions is established via a minimizing-movements scheme, with robust selection criteria to choose among nonunique solutions. The framework is then applied to the Euler–Korteweg system and to binormal curvature flow, yielding global solvability and, in the geometric setting, a weak-strong uniqueness principle. The results unify fluid-mechanical and geometric PDEs under a common variational structure and provide tools for selecting physically relevant solutions through energy defect control and functional minimization. This has potential implications for analyzing singular limits, turbulence, and numerical schemes that preserve energy-dissipation structure.

Abstract

We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation of the assumptions on the regularity weight and the admissibility of energies with merely linear growth. We apply the abstract theory to the Euler--Korteweg system and to the equation for binormal curvature flow, which serve as examples that require the first and second generalization, respectively. Moreover, we discuss criteria that are suitable for the selection of particular energy-variational solutions in the possibly multi-valued solution set.

Existence and selection of solutions in the energy-variational framework with applications in fluid dynamics

TL;DR

The paper develops a general energy-variational framework for abstract evolution equations, allowing energies with linear growth and two distinct regularity weights to ensure lower semicontinuity and compactness. Existence of global energy-variational solutions is established via a minimizing-movements scheme, with robust selection criteria to choose among nonunique solutions. The framework is then applied to the Euler–Korteweg system and to binormal curvature flow, yielding global solvability and, in the geometric setting, a weak-strong uniqueness principle. The results unify fluid-mechanical and geometric PDEs under a common variational structure and provide tools for selecting physically relevant solutions through energy defect control and functional minimization. This has potential implications for analyzing singular limits, turbulence, and numerical schemes that preserve energy-dissipation structure.

Abstract

We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation of the assumptions on the regularity weight and the admissibility of energies with merely linear growth. We apply the abstract theory to the Euler--Korteweg system and to the equation for binormal curvature flow, which serve as examples that require the first and second generalization, respectively. Moreover, we discuss criteria that are suitable for the selection of particular energy-variational solutions in the possibly multi-valued solution set.
Paper Structure (18 sections, 32 theorems, 159 equations)

This paper contains 18 sections, 32 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Energy-variational solutions in an abstract setting
  5. Assumptions
  6. Definition of energy-variational solutions
  7. General properties of energy-variational solutions
  8. Existence of energy-variational solutions
  9. Construction of approximate solutions
  10. Convergence of approximate solutions
  11. Selection criteria
  12. The Euler--Korteweg system
  13. Energy-variational framework
  14. Existence of energy-variational solutions
  15. Binormal curvature flow
  16. ...and 3 more sections

Key Result

Proposition 2.6

The convex conjugate ${\mathcal{E}}^*$ of ${\mathcal{E}}$ satisfies ${\mathcal{E}}^*\leq \iota_{\overline B_{\alpha}}+\beta$, where $\alpha,\beta\geq 0$ are the constants from energy as and $B_{\alpha}=\{\eta\in \mathbbm{V}^{**}\mid \|\eta\|_{\mathbbm{V}^{**}}< \alpha\}$. In particular, it holds $\o then $\mathop{\mathrm{dom}}\nolimits{\mathcal{E}}^*=\mathop{\mathrm{dom}}\nolimits\partial{\mathcal

Theorems & Definitions (73)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 63 more