Energy-Variational Solutions

• Energy-variational solutions are weak formulations characterized by an energy envelope and a variational inequality instead of pointwise PDE satisfaction.
• They unify analysis across dissipative and conservative systems, including fluid dynamics, elasticity, and phase-field models by ensuring existence and weak–strong uniqueness.
• Rooted in convex analysis and variational methods, the framework supports robust numerical schemes using minimizing-movement approaches that encode defects in a scalar energy measure.

Energy-variational solutions are a class of weak solutions for nonlinear evolutionary partial differential equations (PDEs) and variational problems, characterized by an energy-based variational inequality or an energy-envelope property rather than direct satisfaction of the PDE in some weak sense. They provide a unified framework for analyzing existence, stability, and selection properties across a wide variety of dissipative, rate-independent, and conservative systems, including hydrodynamics, elasticity, flows of complex fluids, phase-field models, and hyperbolic conservation laws. The methodology underpinning energy-variational solutions is deeply rooted in convex analysis, variational inequalities, and the calculus of variations, with robust connections to measure-valued and dissipative solution frameworks.

1. Mathematical Definition and Core Principles

Energy-variational solutions, sometimes simply referred to as EV solutions, are defined by two essential elements:

• The introduction of an energy envelope or scalar auxiliary function $E(t)\in BV([0,T])$, which majorizes the physical energy along the evolution, i.e., $E(t)\geq \mathcal{E}(u(t))$ for the PDE state $u$ at time $t$.
• A variational (often relative-energy-based) inequality, which combines the energy-dissipation balance and the weak formulation of the PDE, possibly augmented by an error or regularity-weighted term controlling non-attainability of the exact energy or presence of defects.

The archetypical energy-variational inequality for a pair $(u,E)$ reads, for appropriate test functions,

$$E(t) - E(s) + \int_s^t \mathcal{D}(u(\tau), \zeta(\tau))\, d\tau + \int_s^t W(\zeta(\tau))[E(\tau) - \mathcal{E}(u(\tau))]\, d\tau \leq 0,$$

where $\mathcal{D}$ is the dissipation functional and $W$ is a nonnegative weight functional built from the test field $\zeta$ (Lasarzik, 14 Mar 2025, Cheng et al., 13 Jan 2026, Eiter et al., 2022). The solution is not specified by weak satisfaction of the PDE but rather by this inequality, which encodes both energy balance and admissibility with respect to possible lack of strong convergence or oscillation.

2. Abstract Frameworks and Existence Theorems

Several formulations of energy-variational solutions exist. The general construction proceeds as follows:

• State Space and Energy Functional: One specifies a Banach or Hilbert space $V$ for the variable $u$, together with a lower-semicontinuous (usually convex or polyconvex) energy $\mathcal{E}:V\to [0,\infty]$, possibly supplemented by a time-dependent dissipation $\mathcal{Y}(t,u)$.
• Envelope Space: The energy variable $E:[0,T]\rightarrow [0,\infty)$ resides in $BV([0,T])$, i.e., is of bounded variation, and satisfies $E(t)\geq\mathcal{E}(u(t))$ for a.e.\ $t$.
• Energy-Variational Inequality: The pair $(u,E)$ satisfies

$$E(t) - E(s) + \int_s^t \left\langle \mathcal{J}\,\dot{u}(r), \xi(r)\right\rangle dr + \int_s^t D\mathcal{E}(u(r))[\dot{\xi}(r)] dr + \int_s^t \langle \mathcal{A}(r,u(r)), \xi(r)\rangle dr + \int_s^t K(\xi(r)) [\mathcal{E}(u(r)) - E(r)] dr \leq 0,$$

for all admissible test functions $\xi$ (Agosti et al., 2023).

• Existence: Existence is typically obtained constructively by a minimizing-movement or time-discrete scheme (Agosti et al., 2023, Eiter et al., 2022). At each time-step, the solution minimizes an energy-augmented functional, possibly involving a min-max structure due to the presence of dual variables from the variational inequality. Fan's minimax theorem is frequently used to guarantee existence of such minimizers. Interpolating the sequence and passing to the limit yields a pair $(u,E)$ satisfying the continuous variational inequality, with envelope regularity inherited from BV compactness.

3. Relative Energy, Weak–Strong Uniqueness, and Structural Properties

A central feature of energy-variational solutions is the inherent weak–strong uniqueness property: as soon as a classical (strong) solution exists, any energy-variational solution with matching initial data coincides with the strong one (Agosti et al., 2023, Eiter et al., 2022, Il'yasov et al., 2022). This is established via a relative energy (or relative entropy) inequality; for two states $(U,E)$ and a strong solution $\widetilde{U}$, one defines

$$\mathcal{R}(U,E|\widetilde{U}) = E - \mathcal{E}(\widetilde{U}) - \langle D\mathcal{E}(\widetilde{U}), U-\widetilde{U} \rangle,$$

and the relative dissipation $\mathcal{W}$ built from the convexity of energy and dissipation. The key inequality reads

$$\mathcal{R}(U(t), E(t) | \widetilde{U}(t)) + \int_s^t \mathcal{W}(r, U(r) | \widetilde{U}(r)) dr \leq \mathcal{R}(U(s), E(s)|\widetilde{U}(s)),$$

so that $\mathcal{R}$ acts as a Lyapunov function, vanishing identically as soon as the two solutions agree. The selection property, convexity, and the semi-flow (concatenation) property of energy-variational solution sets are further structural consequences (Lasarzik, 2021, Eiter et al., 2022, Lasarzik, 14 Mar 2025).

4. Comparison with Dissipative, Measure-Valued, and Other Weak Solution Concepts

Energy-variational solutions generalize and refine classical dissipative and measure-valued solution notions through two essential mechanisms:

• All defects or lack of strong convergence (e.g., from oscillations, concentration, or turbulence) are absorbed into the single scalar energy envelope $E$, rather than distributed among measure-valued defect fields or Reynolds-level tensors (Lasarzik, 14 Mar 2025).
• The central variational inequality removes the need for multiple defect measures and yields a weak solution concept that is simpler yet stronger in terms of selection properties and compatibility with relative entropy analysis.

Notably, for incompressible Euler and Navier–Stokes equations, energy-variational solutions recover dissipative weak solutions (in the sense of Brenier–Gwiazda–Wiedemann) and coincide with Leray–Hopf solutions when the defect variable vanishes (Eiter et al., 2022, Lasarzik, 2021). For models with complex energies, polyconvexity, or lack of geometric structure (e.g., polyconvex elastodynamics or two-phase Navier–Stokes), it is demonstrated that measure-valued solutions can always be recast as EV solutions, with Young measure information encoded in $E$ (Lasarzik, 14 Mar 2025, Eiter et al., 2022).

5. Applications Across PDE Classes

The energy-variational framework applies to a wide spectrum of both dissipative (parabolic, viscoelastic, viscoplastic) and conservative (hyperbolic, Hamiltonian) models:

Application Area	Example Systems / Models	Representative References
Hydrodynamics	Navier-Stokes, Euler, incompressible/compressible MHD, two-phase flows	(Eiter et al., 2022, Lasarzik, 2021, Lasarzik, 14 Mar 2025)
Nonlinear elasticity	Polyconvex elastodynamics	(Lasarzik, 14 Mar 2025)
Complex/viscoelastic fluids	Oldroyd–B, Symmetric Neo-Hookean, viscoelastoplastic flows with Zaremba–Jaumann rate	(Agosti et al., 2023, Eiter et al., 2021)
Multi-phase/geodynamical flows	Cahn–Hilliard two-phase models with nonlinear mobility and plasticity, obstacle potentials	(Cheng et al., 13 Jan 2026)
Liquid crystals and director models	Ericksen–Leslie equations with Oseen–Frank energy	(Lasarzik et al., 2022, Lasarzik, 14 Mar 2025)
Hyperbolic conservation laws	Entropy solutions for general fluxes, compressible Euler, semilinear wave equations	(Eiter et al., 2022, Lasarzik, 14 Mar 2025)
Rate-independent systems	Visco-energetic solutions (tuning between energetic and BV limits) for nonconvex hysteresis problems	(Minotti et al., 2016, Minotti, 2016)
Biomolecular continuum modeling	Phase field-curvature, membrane, and protein-field interaction models	(Wei et al., 2016)
Linear/nonlinear PDEs via abstract variational resolution	Nonlinear parabolic evolutions, general dissipation/energy structures	(Poliakovsky, 2011, Heid et al., 11 Sep 2025)

In each context, the energy-variational structure enables existence theorems via time-discrete or minimizing-movement schemes, stability under variational convergence (e.g., Gamma or Mosco convergence of non-smooth energies), and can facilitate adaptive and structure-preserving numerical schemes (Heid et al., 11 Sep 2025, Lasarzik et al., 2022).

6. Numerical Schemes and Variational Adaptivity

The energy-variational framework supports the design of numerical approximation strategies that are inherently energy stable. Iterative Galerkin and adaptive finite-element algorithms can be set up such that the discrete solution sequence monotonically reduces the discrete energy at each step, providing convergence guarantees to a critical point governed by the underlying variational principle (Heid et al., 11 Sep 2025). For systems with complex couplings or non-smooth energies, time-discrete envelope constructions (minimizing-movement, Dörfler marking for mesh adaptivity) ensure that the approximate solution satisfies the discrete energy-variational inequality, and hence converges to an EV solution in the continuum limit (Lasarzik et al., 2022).

7. Outlook and Current Research Directions

The generality of energy-variational solution concepts has revealed several recurrent features and suggested ongoing research directions:

• Unified Framework for Weak Solution Concepts: EV solutions subsume dissipative, Young-measure, and measure-valued solutions for a variety of evolutionary PDEs, often providing the minimal closure needed for well-posedness in $L^2$-based topologies (Lasarzik, 14 Mar 2025, Eiter et al., 2022).
• Weak–Strong Uniqueness and Stability: The relative energy machinery provides robust criteria for the selection and stability of physically relevant solutions, critical for multi-scale and highly nonlinear systems (Agosti et al., 2023, Eiter et al., 2021).
• Variational Convergence and Phase Transition Problems: The framework remains stable under variational limits, such as the logarithmic-to-obstacle phase field passage, and is favorable for handling both smooth and non-smooth free energies (Cheng et al., 13 Jan 2026).
• Energy Envelope as a Defect Measure: The contraction of all possible oscillatory or concentration defects into the scalar envelope $E$ enhances compactness and facilitates analysis of solution sets (convexity, semigroup properties, maximal dissipation).
• Flexibility for Both Parabolic and Hyperbolic Regimes: The same structure supports dissipative (parabolic, viscous) as well as conservative (hyperbolic, entropy controlled) systems, allowing a seamless treatment of limiting procedures and singular perturbations (Eiter et al., 2022).

A plausible implication is that energy-variational solution theory forms a foundational layer for future mathematical analysis and numerical simulation of complex evolution problems with coupled energy-dissipation mechanisms, non-standard defect structures, or uncertain model ingredients. Ongoing work is directed toward extending the framework to systems with linear-growth energies, nonlinear constraints, and coarse-graining limits across time and length scales (Agosti et al., 2023, Eiter et al., 2022).