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An explicit conditional nonlinear energy closure

Statement

Let $I = [0, T)$ with $0 < T \leq \infty$ , and let $E : I \to [0, \infty)$ be the energy functional of a wave system of the kind specified in the preceding positive matrix-potential energy theorem, restricted to its existence interval. Here absolute continuity on compact subintervals has the endpoint-increment meaning in the cited definition. Suppose $E$ is absolutely continuous on every compact subinterval of $I$ and, for constants $C \geq 0$ , $δ > 0$ , satisfies for almost every $t \in I$ $E^{'} (t) \leq C (1 + t)^{- 1 - δ} E (t)^{3 / 2} .$ If $E (0) > 0$ and $C \sqrt{E (0)} < 2 δ$ , then $E (t) \leq E (0) {[1 - \frac{C \sqrt{E (0)}}{2 δ}]}^{- 2} (t \in I) .$ This is a bound only on the given existence interval; a separate continuation theorem is needed to infer global existence. For comparison, if $C > 0$ and the coefficient is changed to $(1 + t)^{- 1}$ , the equality model $E^{'} = C (1 + t)^{- 1} E^{3 / 2}$ with the same positive initial value blows up at the finite time $\exp (2 / (C \sqrt{E (0)})) - 1$ . This comparison is an ODE example, not a claim that an RN or Kerr–Newman perturbation satisfies either differential inequality. Time and energy are dimensionless in this scalar comparison.

Statement references

Positive matrix-potential wave system energy

Facts & Assumptions

Given: The nonnegative energy from the coupled-potential estimate, the stated interval, absolute continuity, constants $C, δ$ , the differential inequality, ${A C}_{ω}$ , and $D C$ .

[F1]

Positive matrix-potential wave system energy: its matrix wave energy is nonnegative and absolutely continuous for the stated smooth solutions, with its exact energy functional and finite-time derivative identity.

[F2]

Absolute continuity on a compact interval: $F$ is absolutely continuous on $[a, b]$ when for every $ε > 0$ there is $η > 0$ such that every finite family of pairwise interior-disjoint subintervals of total length below $η$ has total endpoint increments below $ε$ ; taking a single interval also gives continuity.

[F3]

Absolutely continuous functions have integrable derivatives: under ${A C}_{ω}$ , an absolutely continuous real function has a derivative almost everywhere and that derivative is integrable.

[F4]

Fundamental theorem of calculus for absolutely continuous functions: under ${A C}_{ω}$ and $D C$ , an absolutely continuous $F$ satisfies $F (b) - F (a) = \int_{a}^{b} F^{'} (t) d t$ .

[F5]

A Lipschitz function after an absolutely continuous function is absolutely continuous: composing an absolutely continuous function with a Lipschitz function on its image preserves absolute continuity.

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : for a continuous function on $[a, b]$ differentiable on $(a, b)$ , its endpoint increment is $f^{'} (c) (b - a)$ for some $c \in (a, b)$ .

[F7]

Continuity and derivatives of positive-base real powers: for every real $α$ , $x \mapsto x^{α}$ is differentiable on $(0, \infty)$ with derivative $α x^{α - 1}$ ; in particular $(x^{- 1 / 2})^{'} = - \frac{1}{2} x^{- 3 / 2}$ .

[F8]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : at every point where $E$ is differentiable and $h$ is differentiable at $E (t)$ , $(h \circ E)^{'} (t) = h^{'} (E (t)) E^{'} (t)$ .

[F9]

Every continuous function on an interval has a primitive; two primitives differ by a constant; and $\int_{a}^{b} f = G (b) - G (a)$ for any primitive $G$ Continuity and derivatives of positive-base real powers: the primitive of a continuous scalar function evaluates by the oriented integral; applying the power derivative in [F7] to $- (1 + s)^{- δ} / δ$ gives $\int_{0}^{t} (1 + s)^{- 1 - δ} d s = [1 - (1 + t)^{- δ}] / δ$ .

[F10]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $(\log (1 + t))^{'} = 1 / (1 + t)$ for $t > - 1$ .

[A1]

$E (0) > 0$ , $C \geq 0$ , $δ > 0$ , and $C \sqrt{E (0)} < 2 δ$ .

[A2]

$E^{'} (t) \leq C (1 + t)^{- 1 - δ} E (t)^{3 / 2}$ for almost every $t$ in the existence interval.

[A3]

The real-analysis suppliers use countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and dependent choice $D C$ (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain); no full axiom of choice is assumed.

[A4]

In the comparison example only, $C > 0$ .

Proof

1.1givenF1F2F3F4F5F6F7F8F9A1A2A3algebra

Fix $t > 0$ in the positive component of ${s : E (s) > 0}$ containing $0$ . Since $E$ is absolutely continuous on $[0, t]$ , [F2]'s definition applied to a single subinterval gives continuity there; positivity on this component and compactness then give a positive minimum and a finite maximum for $E$ on $[0, t]$ . The derivative of $h (x) : = x^{- 1 / 2}$ is bounded on this range by [F7]. If the range is a singleton then $h$ is constant on it; otherwise [F6] makes $h$ Lipschitz there by the mean value theorem. In either case [F5] makes $z : = h \circ E$ absolutely continuous. By [F3], $E^{'}$ exists almost everywhere, and [F8] gives $z^{'} (t) = - \frac{1}{2} E (t)^{- 3 / 2} E^{'} (t) \geq - \frac{C}{2} (1 + t)^{- 1 - δ} .$ The absolutely continuous fundamental theorem [F4] and the primitive evaluation [F9] now give $z (t) \geq E (0)^{- 1 / 2} - \frac{C}{2 δ} (1 - (1 + t)^{- δ}) \geq E (0)^{- 1 / 2} - \frac{C}{2 δ} > 0.$ Taking inverse squares gives the claimed bound wherever $E (t) > 0$ in that component.

2.1step 1.1F2F3F4F5F6F7F8F9A2A3algebra

If $E$ reaches $0$ , the claimed inequality is immediate at that time. It cannot later become positive: otherwise some connected component $(a, b)$ of ${t : E (t) > 0}$ has $a$ at or after a zero and $E (a) = 0$ , by continuity from [F2]. For any fixed $t \in (a, b)$ and $s \in (a, t)$ , the same compact-interval argument as in step 1.1 makes $z$ absolutely continuous on $[s, t]$ . Integrating with [F4] gives $z (t) \geq z (s) - C / (2 δ)$ . As $s ↓ a$ , continuity gives $E (s) \to 0$ and $z (s) \to \infty$ , contradicting finite $z (t)$ . Thus after a zero $E$ remains zero, and the bound holds throughout $I$ .

3.1givenF7F8F10A1A4algebra∎

For the comparison kernel $(1 + t)^{- 1}$ , set $t_{*} : = \exp ⁣ (\frac{2}{C \sqrt{E (0)}}) - 1, E (t) : = {(E (0)^{- 1 / 2} - \frac{C}{2} \log (1 + t))}^{- 2} (0 \leq t < t_{*}) .$ The denominator is positive on this interval and tends to $0$ as $t ↑ t_{*}$ . By [F10], [F7], and [F8], differentiation gives $E^{'} = C (1 + t)^{- 1} E^{3 / 2}$ , so this solution is unbounded on its existence interval. Therefore the nonintegrable coefficient does not give the uniform bound proved above.

Remarks

No continuation criterion or PDE estimate establishing [A2] is included. The proposition cannot by itself be used as a global stability theorem.

An explicit conditional nonlinear energy closure

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources