ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Constant phase is an energy-stationary state

Example

Let $d \geq 1$ , $T > 0$ , $ε, M, ν > 0$ , $W \in C^{2} (R)$ be bounded below, and $c \in R$ . On $T^{d} = R^{d} / (2 π Z)^{d}$ , consider the periodic equations $\partial_{t} ϕ + u \cdot \nabla ϕ = M Δ μ_{c}$ , $\partial_{t} u + (u \cdot \nabla) u + \nabla π = ν Δ u - ϕ \nabla μ_{c}$ , and $\nabla \cdot u = 0$ , with $μ_{c} = ε^{- 1} W^{'} (ϕ) - ε Δ ϕ$ . The constant fields $u = 0$ , $ϕ = c$ , and $π = 0$ solve this system with constant chemical potential $μ_{c} = ε^{- 1} W^{'} (c)$ . Their total energy is $E (t) = (2 π)^{d} ε^{- 1} W (c)$ for all $t \in [0, T]$ , so this is an energy-stationary state, meaning that its energy is constant in time; no local-minimum or stability claim is made. All quantities are dimensionless mathematical variables.

Example references

Periodic system and total-energy functional

Conditional smooth periodic energy identity

Facts & Assumptions

Given: The parameters, potential, constant $c$ , and periodic equations in the example statement.

[F1]

Cancellation of capillary and transport energy exchange: for a classical periodic solution of these equations, $E^{'} (t) = - ν \int_{T^{d}} ∣ \nabla u ∣^{2} d x - M \int_{T^{d}} ∣ \nabla μ_{c} ∣^{2} d x$ .

Verification

1.1givenalgebra

For $u = 0$ and $ϕ = c$ , all time derivatives and spatial derivatives of $u$ and $ϕ$ vanish, while $μ_{c} = ε^{- 1} W^{'} (c)$ is constant and therefore $\nabla μ_{c} = Δ μ_{c} = 0$ . The phase equation has both sides zero; the momentum equation has every term zero; and $\nabla \cdot u = 0$ . Thus these fields are a classical periodic solution.

2.1givenF1step 1.1algebra∎

Their kinetic and gradient-energy terms vanish, while the potential term integrates to $(2 π)^{d} ε^{- 1} W (c)$ because the fundamental cube has volume $(2 π)^{d}$ . Alternatively, [F1] gives $E^{'} (t) = 0$ because both dissipation integrands vanish. Hence the energy is constant and the state is energy-stationary.

Depends on

Cancellation of capillary and transport energy exchange