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Time cutoff bootstrap for invisible adjoint solutions

PDF p. 19: Lemma 4.5 proof

Statement

Assume the Axiom of Choice and hypotheses (H) and (W) of Forward and backward real-linear propagators. Thus $Ω = R^{n} ∖ Θ$ is the connected exterior of a nonempty compact smooth nontrapping obstacle, $T > 0$ , and the reference trajectory $w$ and real $λ \neq 0$ determine $V_{1} = 2 λ ∣ w ∣^{2}$ and $V_{2} = λ w^{2}$ . Let $H_{D}^{s}$ be the Dirichlet spectral scale, with negative indices interpreted by pivot antiduality, and let $W (t) = e^{i t Δ_{D}}$ . Take the real cutoffs $φ, ψ$ from Control region and nested real cutoffs.

Let $θ \in [- 2, 0]$ and let $v \in C ([0, T]; H_{D}^{θ})$ be the mild solution of $i v_{t} + Δ_{D} v + V_{1} v - V_{2} \overline{v} = 0$ with $φ v = 0$ throughout $[0, T]$ , as an equality in $H_{D}^{θ}$ . Then $v = ψ v$ throughout $[0, T]$ . For every nonzero real $ϑ \in C_{c}^{\infty} (0, T)$ , put $u = ϑ v$ and $Φ = ϑ (- V_{1} v + V_{2} \overline{v}) + i ϑ^{'} v$ . The identities $u = ψ u$ and $Φ = ψ Φ$ hold, and $u (t) = - i \int_{0}^{t} W (t - r) ψ Φ (r) d r .$ Moreover $u \in L^{2} (0, T; H_{D}^{θ + 1})$ and $v \in C ([0, T]; H_{D}^{θ + 1})$ . In particular, the gain can be applied successively at $θ = - 2$ and $θ = - 1$ to an invisible solution initially in $C ([0, T]; H_{D}^{- 2})$ .

Statement references

Forward and backward real-linear propagators

Dirichlet spectral Sobolev scale and duality

Control region and nested real cutoffs

Linearized equation and its real-duality adjoint

Continuous and mixed time-space function spaces

The Axiom of Choice

Facts & Assumptions

[F1]

The cutoff construction gives $χ = 1 - φ$ , $ψ χ = χ$ , and real smooth compactly supported $χ, ψ$ . The scale definition gives $A = I - Δ_{D} \geq I$ , consistent spectral spaces and the negative pivot-antidual convention. Smooth cutoff multiplication is bounded and compatible with products on the relevant scale by Lemma 4.1: retarded smoothing down to negative order two. These conventions are in Control region and nested real cutoffs and Dirichlet spectral Sobolev scale and duality.

[F2]

Under the stated hypotheses, $V_{j} (t)$ and conjugation act boundedly on each $H_{D}^{q}$ for $- 2 \leq q \leq 2$ ; the potential multipliers vary continuously in operator norm. The adjoint mild solution is unique on each such scale, forward and backward from any initial time, and satisfies its free-flow Volterra equation. These are Time-dependent potential and conjugation bounds on the scale, Forward and backward real-linear propagators and Linearized equation and its real-duality adjoint.

[F3]

The group $W$ is strongly continuous and unitary on every scale space Scale isometries and free Schrödinger evolution. The retarded smoothing estimate Lemma 4.1: retarded smoothing down to negative order two applies with a real compactly supported cutoff and input $g = ψ f \in L^{2} (0, T; H_{D}^{θ})$ , giving the output localized by $ψ$ in $L^{2} (0, T; H_{D}^{θ + 1})$ . The time spaces and integrals are the Bochner spaces and integrals of Continuous and mixed time-space function spaces. The Axiom of Choice supplies the assumptions of these results The Axiom of Choice.

[F4]

The Volterra formulation and Bochner integration rules for a strongly continuous group with a bounded real-linear perturbation are supplied by Bounded real-linear perturbations preserve solution estimates.

Proof

[F1] The cutoff construction gives $χ = 1 - φ$ , $ψ χ = χ$ , and real smooth compactly supported $χ, ψ$ . The scale definition gives $A = I - Δ_{D} \geq I$ , consistent spectral spaces and the negative pivot-antidual convention. Smooth cutoff multiplication is bounded and compatible with products on the relevant scale by Lemma 4.1: retarded smoothing down to negative order two. These conventions are in Control region and nested real cutoffs and Dirichlet spectral Sobolev scale and duality.

[F2] Under the stated hypotheses, $V_{j} (t)$ and conjugation act boundedly on each $H_{D}^{q}$ for $- 2 \leq q \leq 2$ ; the potential multipliers vary continuously in operator norm. The adjoint mild solution is unique on each such scale, forward and backward from any initial time, and satisfies its free-flow Volterra equation. These are Time-dependent potential and conjugation bounds on the scale, Forward and backward real-linear propagators and Linearized equation and its real-duality adjoint.

[F3] The group $W$ is strongly continuous and unitary on every scale space Scale isometries and free Schrödinger evolution. The retarded smoothing estimate Lemma 4.1: retarded smoothing down to negative order two applies with a real compactly supported cutoff and input $g = ψ f \in L^{2} (0, T; H_{D}^{θ})$ , giving the output localized by $ψ$ in $L^{2} (0, T; H_{D}^{θ + 1})$ . The time spaces and integrals are the Bochner spaces and integrals of Continuous and mixed time-space function spaces. The Axiom of Choice supplies the assumptions of these results The Axiom of Choice.

[F4] The Volterra formulation and Bochner integration rules for a strongly continuous group with a bounded real-linear perturbation are supplied by Bounded real-linear perturbations preserve solution estimates.

Given: The hypotheses and the invisible mild solution $v$ in the Statement.

Goal: Prove the localized Duhamel identity and gain one derivative throughout the time interval.

1.1F1F2F3given

Since $φ v = 0$ and $χ = 1 - φ$ , we have $v = χ v = ψ χ v = ψ v$ in $H_{D}^{θ}$ . Scalar time multiplication gives $u = ψ u$ . Multiplication operators commute with $ψ$ , and conjugation commutes with the real cutoff $ψ$ ; these identities hold on the common scale core and extend by boundedness. Therefore both potential terms and $ϑ^{'} v$ are fixed by $ψ$ , so $Φ = ψ Φ$ . Operator-norm continuity of the potentials and continuity of $v$ show that $Φ \in C ([0, T]; H_{D}^{θ})$ , hence $Φ \in L^{2} (0, T; H_{D}^{θ})$ .

2.1F2F3F4step 1.1

Write $P (r) v = i (V_{1} (r) v - V_{2} (r) \overline{v})$ . The mild equation is $v (t) = W (t) v (0) + \int_{0}^{t} W (t - r) P (r) v (r) d r$ . Multiply it by $ϑ (t)$ . Since $ϑ (0) = 0$ , use $ϑ (t) = \int_{0}^{t} ϑ^{'} (s) d s$ on the free term and $ϑ (t) - ϑ (r) = \int_{r}^{t} ϑ^{'} (s) d s$ on the integral term. The group law and exchange of the resulting triangular Bochner integrals identify the extra terms as $\int_{0}^{t} W (t - s) ϑ^{'} (s) v (s) d s$ . Consequently $u (t) = \int_{0}^{t} W (t - r) (ϑ (r) P (r) v (r) + ϑ^{'} (r) v (r)) d r = - i \int_{0}^{t} W (t - r) Φ (r) d r$ . The exchange is justified on the finite triangle by strong continuity, the uniform group and perturbation bounds, and Bochner integrability. Step 1.1 replaces $Φ$ by $ψ Φ$ .

3.1F3step 1.1step 2.1

Apply the retarded smoothing estimate with cutoff $ψ$ and $g = ψ Φ = Φ$ . It gives $ψ u \in L^{2} (0, T; H_{D}^{θ + 1})$ ; the constant factor $- i$ is harmless. Since $u = ψ u$ in the lower space, $u$ has this higher-regularity representative.

4.1F1F2F3step 3.1∎

Because $ϑ$ is nonzero and continuous, it is nonzero on a nonempty open interval. Choose $t_{0}$ there outside the null exceptional set from step 3.1. Then $v (t_{0}) = u (t_{0}) / ϑ (t_{0}) \in H_{D}^{θ + 1}$ . The spectral inequality $A \geq I$ gives the continuous inclusion $H_{D}^{θ + 1} ↪ H_{D}^{θ}$ by comparison of the two spectral norms on the common core and completion. Solve the adjoint equation forward and backward from $v (t_{0})$ in $H_{D}^{θ + 1}$ . The higher-order solution also satisfies the lower-order Volterra equation, because the scale operations, group and multipliers are compatible. Lower-order uniqueness identifies it with $v$ on all of $[0, T]$ . Thus $v \in C ([0, T]; H_{D}^{θ + 1})$ . Applying the same result first at $- 2$ and then at $- 1$ proves the final assertion.

Source notes

The original article is cited for attribution of the cutoff bootstrap in the proof of Lemma 4.5 on PDF page 19. The argument above uses the stated mathematical suppliers; the article is not used as proof of its own claim. The explicit Choice assumption records the hypotheses of the suppliers and is not attributed to the article.

Depends on

Used by · 1

Lemma 4.5: invisible solutions acquire energy regularity

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2