Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Energy-variational structure in evolution equations

Robert Lasarzik

TL;DR

The article develops a unifying energy-variational framework that recasts multiple generalized solvability concepts for nonlinear evolution PDEs as energy-variational solutions. By analyzing four representative systems—the two-phase Navier–Stokes flow, a quasilinear wave equation, polyconvex elastodynamics, and the Ericksen–Leslie equations with Oseen–Frank energy—the authors demonstrate equivalence between energy-variational solutions and existing measure-valued or dissipative weak formulations, often reducing the amount of auxiliary data required (e.g., replacing varifolds with a BV energy function). The work relies on an abstract existence theory and Young-measure techniques to connect variational energy inequalities with measure-valued limits, offering a time-discrete minimizing-movement interpretation and potential selection criteria. Overall, the energy-variational approach provides a broad, computationally friendly framework for a wide class of evolution equations and may guide the selection of physically relevant solutions via dissipation-based criteria.

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, 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.

Energy-variational structure in evolution equations

TL;DR

The article develops a unifying energy-variational framework that recasts multiple generalized solvability concepts for nonlinear evolution PDEs as energy-variational solutions. By analyzing four representative systems—the two-phase Navier–Stokes flow, a quasilinear wave equation, polyconvex elastodynamics, and the Ericksen–Leslie equations with Oseen–Frank energy—the authors demonstrate equivalence between energy-variational solutions and existing measure-valued or dissipative weak formulations, often reducing the amount of auxiliary data required (e.g., replacing varifolds with a BV energy function). The work relies on an abstract existence theory and Young-measure techniques to connect variational energy inequalities with measure-valued limits, offering a time-discrete minimizing-movement interpretation and potential selection criteria. Overall, the energy-variational approach provides a broad, computationally friendly framework for a wide class of evolution equations and may guide the selection of physically relevant solutions via dissipation-based criteria.

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, 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.

Paper Structure

This paper contains 6 sections, 8 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Interface evolution in two phase Navier--Stokes equations
  4. Quasilinear wave equation with convex energy
  5. Polyconvex elastodynamics
  6. Ericksen--Leslie equations equipped with the Oseen--Frank energy

Key Result

Lemma 2.1

Let ${\mathbf{ l}} : \mathcal{V} \mathop{\mathrm{\rightarrow}}\nolimits \mathop{\mathrm{\mathbb{R}}}\nolimits$ be a linear continuous functional, where $\mathcal{V}$ is a closed subspace of with $d$, $m\in \mathop{\mathrm{\mathbb{N}}}\nolimits$ and let $\mathcal{W}$ be a Banach space such that $\nabla \mathcal{V}\subset \mathcal{W}$, where Let $\mathfrak p : L^1(0,T;\mathcal{W}) \mathop{\mathr

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Construction of probability measure
  • proof
  • Lemma 2.3: Factorization of linear maps
  • proof
  • Definition 3.1: Varifold solution for the two-phase Navier--Stokes equation
  • Remark 3.1
  • Definition 3.2: Energy-variational solution
  • Theorem 3.3
  • ...and 20 more