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Lemma 3.2: compactness of the remainder

PDF p. 8: Lemma 3.2; PDF p. 9: proof (a)–(d)

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and it is non-trapping, as specified in Exterior and truncated obstacle domains and Non-trapping obstacle. Let $T > 0$ and $σ \in [- 2, 2)$ . If $v_{j, 0} ⇀ v_{0}$ weakly in the Dirichlet spectral space $H_{D}^{σ} (Ω)$ , then $Q_{σ} (v_{j, 0} - v_{0}) \to 0$ , where the scale, weak convergence and remainder are as in def-se-compact-remainder and Weak convergence notation in normed state spaces.

In the exceptional range $- 1 < σ < 0$ , the indices in the remainder are $τ = σ + 2 \in (1, 2)$ and $ρ = τ - 1 \in (0, 1)$ .

Facts & Assumptions

[F1]

For $k = 0$ when $σ \geq 0$ and $k = 1$ when $σ < 0$ , put $τ = σ + 2 k$ . The remainder is $Q_{σ} (w) = \int_{0}^{T} ∥ ψ R^{k} W (t) w ∥_{H_{D}^{ρ}}^{2} d t$ , with $ρ = σ + 1$ for $- 1 < σ < 0$ and $ρ = 0$ otherwise. Here $R = (I - Δ_{D})^{- 1}$ , $W (t) = e^{i t Δ_{D}}$ , and $ψ$ is the specified smooth compactly supported cutoff Regularized lower-order observation remainder.

[F2]

The map $R^{k} : H_{D}^{σ} \to H_{D}^{τ}$ is an isometry, commutes with $W (t)$ , and $W (t)$ is strongly continuous and unitary on $H_{D}^{τ}$ Scale isometries and free Schrödinger evolution. Weak convergence means convergence under every bounded scalar-linear functional Weak convergence notation in normed state spaces.

[F3]

Under the stated geometry and Choice, a bounded weakly null sequence in $H_{D}^{τ}$ , $0 \leq τ < 2$ , has localized free evolutions converging strongly to zero in $L^{2} (0, T; L^{2} (Ω))$ Localized free-flow bounds and compactness mechanism. This conclusion uses compactness on a bounded localization, not a global compact Sobolev embedding on $Ω$ .

[F4]

Multiplication by $ψ$ is bounded on $H_{D}^{τ}$ for $0 \leq τ < 2$ Endpoint multiplier bounds extended to negative spectral orders. If $0 < ρ < τ$ , a sequence uniformly bounded in $L^{2} (0, T; H_{D}^{τ})$ and converging to zero in $L^{2} (0, T; L^{2})$ also converges to zero in $L^{2} (0, T; H_{D}^{ρ})$ Spectral interpolation and lower-order absorption.

[F5]

A weakly convergent sequence in a real or complex normed space is norm bounded under the real dominated-extension principle HB and Countable Choice Weakly convergent sequences are norm bounded. Under the assumed Axiom of Choice, the real Hahn–Banach dominated-extension theorem supplies precisely this HB premise Hahn-Banach dominated extension theorem for real vector spaces. Countable Choice follows by applying Choice to countable families of nonempty sets The Axiom of Choice.

[F6]

The spaces $L^{2} (0, T; X)$ use strongly measurable Banach-valued maps modulo equality almost everywhere, with norm $(\int_{0}^{T} ∥ f (t) ∥_{X}^{2} d t)^{1 / 2}$ and Lebesgue time measure Continuous and mixed time-space function spaces.

Proof

[F1] For $k = 0$ when $σ \geq 0$ and $k = 1$ when $σ < 0$ , put $τ = σ + 2 k$ . The remainder is $Q_{σ} (w) = \int_{0}^{T} ∥ ψ R^{k} W (t) w ∥_{H_{D}^{ρ}}^{2} d t$ , with $ρ = σ + 1$ for $- 1 < σ < 0$ and $ρ = 0$ otherwise. Here $R = (I - Δ_{D})^{- 1}$ , $W (t) = e^{i t Δ_{D}}$ , and $ψ$ is the specified smooth compactly supported cutoff Regularized lower-order observation remainder.

[F2] The map $R^{k} : H_{D}^{σ} \to H_{D}^{τ}$ is an isometry, commutes with $W (t)$ , and $W (t)$ is strongly continuous and unitary on $H_{D}^{τ}$ Scale isometries and free Schrödinger evolution. Weak convergence means convergence under every bounded scalar-linear functional Weak convergence notation in normed state spaces.

[F3] Under the stated geometry and Choice, a bounded weakly null sequence in $H_{D}^{τ}$ , $0 \leq τ < 2$ , has localized free evolutions converging strongly to zero in $L^{2} (0, T; L^{2} (Ω))$ Localized free-flow bounds and compactness mechanism. This conclusion uses compactness on a bounded localization, not a global compact Sobolev embedding on $Ω$ .

[F4] Multiplication by $ψ$ is bounded on $H_{D}^{τ}$ for $0 \leq τ < 2$ Endpoint multiplier bounds extended to negative spectral orders. If $0 < ρ < τ$ , a sequence uniformly bounded in $L^{2} (0, T; H_{D}^{τ})$ and converging to zero in $L^{2} (0, T; L^{2})$ also converges to zero in $L^{2} (0, T; H_{D}^{ρ})$ Spectral interpolation and lower-order absorption.

[F5] A weakly convergent sequence in a real or complex normed space is norm bounded under the real dominated-extension principle HB and Countable Choice Weakly convergent sequences are norm bounded. Under the assumed Axiom of Choice, the real Hahn–Banach dominated-extension theorem supplies precisely this HB premise Hahn-Banach dominated extension theorem for real vector spaces. Countable Choice follows by applying Choice to countable families of nonempty sets The Axiom of Choice.

[F6] The spaces $L^{2} (0, T; X)$ use strongly measurable Banach-valued maps modulo equality almost everywhere, with norm $(\int_{0}^{T} ∥ f (t) ∥_{X}^{2} d t)^{1 / 2}$ and Lebesgue time measure Continuous and mixed time-space function spaces.

Given: The geometry, Choice, $T$ , $σ$ and weakly convergent initial data in the Statement.

Goal: Prove $Q_{σ} (v_{j, 0} - v_{0}) \to 0$ .

1.1givenF1F2F5

Set $w_{j} = v_{j, 0} - v_{0}$ and $y_{j, 0} = R^{k} w_{j}$ . The sequence $w_{j}$ is weakly null. For every bounded scalar-linear functional $ℓ$ on $H_{D}^{τ}$ , the composition $ℓ R^{k}$ is bounded on $H_{D}^{σ}$ ; hence $y_{j, 0} ⇀ 0$ in $H_{D}^{τ}$ . By weak boundedness and the isometry, there is $M < \infty$ such that $∥ y_{j, 0} ∥_{H_{D}^{τ}} \leq M$ for all $j$ . The case definitions give $0 \leq τ < 2$ .

2.1F2F3F6step 1.1

Put $y_{j} (t) = W (t) y_{j, 0}$ . The localized compactness result applies to this bounded weakly null sequence and yields $ψ y_{j} \to 0$ in $L^{2} (0, T; L^{2} (Ω))$ . By commutation, $ψ y_{j} (t) = ψ R^{k} W (t) w_{j}$ .

3.1F1F6step 2.1

If $ρ = 0$ , the definition of $Q_{σ}$ identifies it with $∥ ψ y_{j} ∥_{L^{2} (0, T; L^{2})}^{2}$ , which tends to zero. This includes $σ = - 2$ , $σ = - 1$ , and all $0 \leq σ < 2$ .

4.1F1F2F4F6step 1.1step 2.1∎

Otherwise $- 1 < σ < 0$ , so $τ = σ + 2 \in (1, 2)$ and $ρ = τ - 1 \in (0, τ)$ . Unitarity and the multiplier bound give $\sup_{j, t} ∥ ψ y_{j} (t) ∥_{H_{D}^{τ}} \leq C M$ . Strong continuity and bounded multiplication give strong measurability, and the finite time interval gives a uniform $L^{2} (0, T; H_{D}^{τ})$ bound. Applying the interpolation convergence statement to step 2.1 gives $ψ y_{j} \to 0$ in $L^{2} (0, T; H_{D}^{ρ})$ . Its squared norm is $Q_{σ} (w_{j})$ , so the desired limit holds in this case as well.

Source notes

The article is cited for the statement and attribution to Lemma 3.2, PDF pages 8–9. The deduction above uses the stated suppliers rather than the article's lemma as a premise.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2