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Zero wave energy means a spatial constant, fixed by the displacement datum

Example

Assume the Axiom of Countable Choice. Let $c > 0$ and let $u$ be a classical solution of the homogeneous equation $□_{c} u = 0$ on $R^{n} \times (0, T)$ with Cauchy data $(u_{0}, u_{1})$ and differentiable displacement $u_{0}$ , so $D u_{0}$ in the initial-energy hypothesis is defined, with $E (t) = E (0)$ for every $t \in (0, T)$ — the sharp form of conservation in the senses of Conservation of total wave energy in three admissible settings — and with $E (0) = \frac{1}{2} \int_{R^{n}} (u_{1}^{2} + c^{2} ∣ D u_{0} ∣^{2}) d x = 0$ . Then $u_{1} = 0$ and $D u_{0} = 0$ , so the displacement datum $u_{0}$ is constant on $R^{n}$ ; the energy seminorm sees only $(u_{t}, D u)$ and cannot fix that constant, and the evolution keeps it: $u (\cdot, t) \equiv u_{0}$ for every $t$ .

In particular every constant displacement with zero initial velocity, $u (x, t) \equiv k$ for a fixed $k \in R$ , is a genuine classical solution of zero energy. Thus "zero energy" is strictly weaker than "zero solution": the displacement datum is what fixes the residual constant (Energy uniqueness for the wave Cauchy problem(ii)).

Facts & Assumptions

Given: ${A C}_{ω}$ ; a classical homogeneous solution $u$ with Cauchy data $(u_{0}, u_{1})$ and differentiable displacement $u_{0}$ , so $D u_{0}$ in the initial-energy hypothesis is defined, conserved total energy in the sharp form $E (t) = E (0)$ for $t \in (0, T)$ and $E (0) = \frac{1}{2} \int (u_{1}^{2} + c^{2} ∣ D u_{0} ∣^{2}) = 0$ ; the density $e = \frac{1}{2} (u_{t}^{2} + c^{2} ∣ D u ∣^{2}) \geq 0$ of Wave energy density, energy flux and total energy.

[F1]

A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere; a continuous nonnegative function with vanishing integral vanishes identically. (A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere)

[F2]

On an open convex set, a $C^{1}$ function with vanishing gradient is constant; $R^{n}$ is convex. (Vanishing gradient and time derivative force constancy on convex sets)

[F3]

In each setting of the conservation theorem the energy is constant on the interval; the hypothesis of this Example records the sharp form $E (t) = E (0)$ for all $t \in (0, T)$ . (Conservation of total wave energy in three admissible settings)

[F4]

A continuous function on an interval with vanishing derivative at every interior point is constant. (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant)

Verification

1.1givenF1F2F3algebra

Vanishing at positive times: sharp conservation gives $E (t) = E (0) = 0$ for each $t \in (0, T)$ . The nonnegative continuous density therefore vanishes everywhere by [F1], so $u_{t} (\cdot, t) = D u (\cdot, t) = 0$ . By [F2], each spatial slice is constant.

2.1givenstep 1.1F4algebra

Time constancy and the data: for each fixed $x$ , the function $t \mapsto u (x, t)$ has derivative zero on $(0, T)$ , so [F4] makes it constant there. Together with step 1.1 this gives one constant $k$ on all space-time. The Cauchy limits then give $u_{0} (x) = k$ and $u_{1} (x) = 0$ at every $x$ , hence $D u_{0} = 0$ and $u (\cdot, t) \equiv u_{0}$ . This derives pointwise data vanishing without inferring it from an almost-everywhere statement at $t = 0$ .

3.1givenstep 2.1algebra∎

The converse check: the constant displacement $u (x, t) \equiv k$ has $u_{t} = D u = 0$ , so $e \equiv 0$ and $E \equiv 0$ , and $□_{c} u = 0$ trivially; hence the zero-energy solutions are exactly the constant displacements, and that constant is precisely the initial displacement datum, which the energy cannot see.

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Sources

Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, AMS Graduate Studies in Mathematics) (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (University of Toronto, 2018, CC BY-SA) (standard reference, not scraped)