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Theorem 1.3: cubic internal control near zero

PDF p. 4: Theorem 1.3; PDF p. 16: Proof of Theorem 1.3

Statement

Assume the Axiom of Choice and hypothesis (H): $n \in {2, 3}$ , the nonempty compact obstacle $Θ \subset R^{n}$ has smooth boundary and connected exterior $Ω = R^{n} ∖ Θ$ , and its generalized reflected or glancing rays leave every compact set in finite forward time. Let $T > 0$ , $s \in {1, 2}$ and $λ \in R ∖ {0}$ . Choose $R > 0$ with $Θ \subset B_{R} (0)$ and a smooth real cutoff $φ \in C^{\infty} (R^{n}; [0, 1])$ that is zero on $∣ x ∣ \leq R$ , positive on $∣ x ∣ > R$ , and one on $∣ x ∣ \geq R + 1$ .

Let $H_{D}^{s} (Ω)$ be the Dirichlet spectral space $D ((I - Δ_{D})^{s / 2})$ , with norm $∥ u ∥_{H_{D}^{s}} = ∥ (I - Δ_{D})^{s / 2} u ∥_{L^{2}}$ . Then there is $δ > 0$ such that, for every $v_{0}, v_{T} \in H_{D}^{s} (Ω)$ with $∥ v_{0} ∥_{H_{D}^{s}}, ∥ v_{T} ∥_{H_{D}^{s}} \leq δ$ , there are a control $h \in C ([0, T]; H_{D}^{s} (Ω))$ and an integral solution $v \in C ([0, T]; H_{D}^{s} (Ω))$ of

$i \partial_{t} v + Δ_{D} v + λ ∣ v ∣^{2} v = φ h, v (0) = v_{0}, v (T) = v_{T} .$

An integral solution here has $λ ∣ v ∣^{2} v, φ h \in L^{1} (0, T; H_{D}^{s} (Ω))$ and, for $W (t) = e^{i t Δ_{D}}$ , satisfies

$v (t) = W (t) v_{0} + i \int_{0}^{t} W (t - r) λ ∣ v (r) ∣^{2} v (r) d r - i \int_{0}^{t} W (t - r) (φ h (r)) d r (0 \leq t \leq T) .$

Both integrals are Bochner integrals in $H_{D}^{s} (Ω)$ . This is the integral-solution convention for the Dirichlet evolution; it does not require $Δ_{D} v (t)$ to belong to $H_{D}^{s} (Ω)$ at each time.

The Dirichlet condition is encoded by $Δ_{D}$ . Here $C ([0, T]; H_{D}^{s})$ denotes norm-continuous paths. The control itself need not be supported in the control region; its force $φ h$ vanishes where $∣ x ∣ \leq R$ .

Statement references

Exterior and truncated obstacle domains

Control region and nested real cutoffs

Linear and nonlinear internally controlled exterior systems

Dirichlet spectral Sobolev scale and duality

Continuous and mixed time-space function spaces

The Axiom of Choice

Mild formulation of the controlled power NLS

Facts & Assumptions

[F1]

The geometry and ray convention are specified by Exterior and truncated obstacle domains, and the cutoff and cubic controlled system by Control region and nested real cutoffs and Linear and nonlinear internally controlled exterior systems. The scale is specified by Dirichlet spectral Sobolev scale and duality, path continuity by Continuous and mixed time-space function spaces, and the assumed choice principle by The Axiom of Choice.

[F2]

Under these geometric and Choice hypotheses, Low-order identifications of the Dirichlet scale identifies $H_{D}^{1}$ with $H_{0}^{1}$ and gives equivalent spectral and $H^{1}$ norms. The latter space is the $H^{1}$ closure of compactly supported smooth functions.

[F3]

With $m = 3$ , Proposition 3.9: energy-space local control supplies a positive radius in the $H^{1}$ norm and a continuous $H_{0}^{1}$ control and integral solution joining every endpoint pair within that radius. With $m = 3$ , Proposition 3.10: order-two local control supplies the corresponding conclusion at order two for endpoints small in the spectral $H_{D}^{2}$ norm.

[F4]

The integral-solution equivalence in Mild formulation of the controlled power NLS applies on $H_{D}^{s}$ to the free group $W (t) = e^{i t Δ_{D}}$ whenever $g (v) = λ ∣ v ∣^{2} v$ and $K h = φ h$ are Bochner integrable there. It identifies the variation-of-constants formula in the Statement with the integrated generator equation. The solutions constructed by the two propositions in F3 use this convention and have both forcings in the required Bochner $L^{1}$ space. The Choice assumption is inherited from these suppliers as well as the scale identification.

Proof

Given: The hypotheses and endpoint data in the Statement.

Goal: Find a positive spectral-norm radius and a control and solution with the stated endpoints.

1.1F2F3given

If $s = 1$ , let $r_{1} > 0$ be the radius supplied by F3. By F2 there is $C > 0$ such that $∥ u ∥_{H^{1}} \leq C ∥ u ∥_{H_{D}^{1}}$ for every $u \in H_{D}^{1}$ . Set $δ = r_{1} / C$ . The endpoint bounds in the Statement then meet the hypotheses of the first proposition in F3.

1.2F1F3given

If $s = 2$ , take $δ > 0$ to be the spectral-norm radius supplied by the second proposition in F3. The endpoint bounds satisfy its assumptions, so it gives an integral cubic controlled solution and a control in $C ([0, T]; H_{D}^{2})$ with the prescribed endpoints. Its equation and cutoff are those of F1.

2.1F1F2F3step 1.1

For $s = 1$ , the first proposition in F3 gives an integral cubic controlled solution and control, both continuous in $H_{0}^{1}$ , with the required endpoints. The norm equivalence in F2 makes both paths continuous in $H_{D}^{1}$ ; F1 identifies the equation and cutoff with those in the Statement. Thus the conclusion holds when $s = 1$ .

3.1F2F3F4step 1.2step 2.1

The solutions supplied by F3 have Bochner-integrable cubic and control forcings and satisfy the variation-of-constants identity of F4. In the case $s = 1$ , the norm equivalence in F2 preserves Bochner integrability and the values of the integrals: convergence of the defining simple-function integrals in either norm is convergence in the other norm. Thus the identity is unchanged on identifying $H_{0}^{1}$ with $H_{D}^{1}$ . For $s = 2$ it is already an identity in $H_{D}^{2}$ . Consequently, in both cases these are integral solutions with exactly the forcing integrability and formula specified in the Statement.

4.1givenstep 2.1step 1.2step 3.1∎

Since $s$ is either $1$ or $2$ , steps 2.1 and 1.2 establish the required conclusion in every stated case. Step 3.1 supplies the stated integral-solution interpretation in both cases.

Source notes

The frozen article is cited for attribution of Theorem 1.3 and its stated reduction. The $[0, 1]$ range of the cutoff is explicit in the setup of (1.8) on PDF page 4 and agrees with (3.1); it restores the original hypothesis omitted from the earlier generated draft. The argument above uses the separately supplied propositions and scale identification, with the independently supplied mild-solution convention, not the article's theorem as a premise.

Depends on

Used by · 3

Dependency tree · 2 levels

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF pages 4 and 16, Theorem 1.3 and its proof