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Homogeneous half-derivative local smoothing

PDF p. 7: (2.3); PDF p. 8: Proposition 3.1; (3.6)

Statement

Assume the Axiom of Choice and hypothesis (H): $n \geq 2$ , $Θ \subset R^{n}$ is nonempty and compact with smooth boundary, its exterior $Ω = R^{n} ∖ Θ$ is connected, and $Θ$ is non-trapping. Write $Δ_{D}$ for the Dirichlet Laplacian, $H_{D}^{s} (Ω)$ for the spectral scale of $I - Δ_{D}$ , and $W (t) = e^{i t Δ_{D}}$ for its evolution group. Let $η \in C_{c}^{\infty} (R^{n})$ be a smooth, compactly supported, possibly complex-valued function, restricted to $Ω$ when used as a multiplier.

For every $s \in [0, 1]$ there is a constant $C > 0$ , depending on the obstacle, $η$ and $s$ but independent of $T$ and $v_{0}$ , such that for all $T > 0$ and $v_{0} \in H_{D}^{s} (Ω)$ ,

$\int_{0}^{T} ∥ η W (t) v_{0} ∥_{H_{D}^{s + 1 / 2} (Ω)}^{2} d t \leq C ∥ v_{0} ∥_{H_{D}^{s} (Ω)}^{2} .$

Statement references

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

Continuous and mixed time-space function spaces

Scale isometries and free Schrödinger evolution

The Axiom of Choice

Facts & Assumptions

[F1]

The exterior and smooth-boundary conventions are those of Exterior and truncated obstacle domains. Non-trapping uses complete rays: for every complete unit-speed admissible generalized reflected or glancing ray $x : R \to \overline{Ω}$ and every compact $K \subset R^{n}$ , there is $t \geq 0$ with $x (t) \notin K$ Non-trapping obstacle.

[F2]

Under the Axiom of Choice The Axiom of Choice, the scale is the completion defined by spectral powers of $I - Δ_{D}$ Dirichlet spectral Sobolev scale and duality, and $W (t) = e^{i t Δ_{D}}$ is its strongly continuous unitary evolution Scale isometries and free Schrödinger evolution. The time norm $L^{2} (0, T; H_{D}^{s + 1 / 2})$ is the Bochner norm defined in Continuous and mixed time-space function spaces.

[F3]

The imported homogeneous exterior smoothing estimate Imported homogeneous and retarded exterior smoothing applies on this non-trapping exterior, for every compact smooth cutoff and $0 \leq s \leq 1$ . It bounds $∥ η e^{i t Δ_{D}} v_{0} ∥_{L^{2} (0, T; H_{D}^{s + 1 / 2})}$ by a constant independent of $T$ times $∥ v_{0} ∥_{H_{D}^{s}}$ .

Proof

[F1] The exterior and smooth-boundary conventions are those of Exterior and truncated obstacle domains. Non-trapping uses complete rays: for every complete unit-speed admissible generalized reflected or glancing ray $x : R \to \overline{Ω}$ and every compact $K \subset R^{n}$ , there is $t \geq 0$ with $x (t) \notin K$ Non-trapping obstacle.

[F2] Under the Axiom of Choice The Axiom of Choice, the scale is the completion defined by spectral powers of $I - Δ_{D}$ Dirichlet spectral Sobolev scale and duality, and $W (t) = e^{i t Δ_{D}}$ is its strongly continuous unitary evolution Scale isometries and free Schrödinger evolution. The time norm $L^{2} (0, T; H_{D}^{s + 1 / 2})$ is the Bochner norm defined in Continuous and mixed time-space function spaces.

[F3] The imported homogeneous exterior smoothing estimate Imported homogeneous and retarded exterior smoothing applies on this non-trapping exterior, for every compact smooth cutoff and $0 \leq s \leq 1$ . It bounds $∥ η e^{i t Δ_{D}} v_{0} ∥_{L^{2} (0, T; H_{D}^{s + 1 / 2})}$ by a constant independent of $T$ times $∥ v_{0} ∥_{H_{D}^{s}}$ .

Given: The geometry, Choice assumption, cutoff, order, time and initial datum in the statement.

Goal: Establish the displayed squared integral estimate with a constant independent of $T$ and the datum.

1.1givenF1F2F3

The geometry and non-trapping assumptions in [F1], the Choice assumption, the cutoff and the order satisfy the hypotheses of [F3]. Its spectral spaces and evolution are the spaces and group fixed in [F2]. Therefore there is $C_{0} > 0$ , independent of $T$ and $v_{0}$ , such that $∥ η W (t) v_{0} ∥_{L^{2} (0, T; H_{D}^{s + 1 / 2})} \leq C_{0} ∥ v_{0} ∥_{H_{D}^{s}}$ .

2.1F2step 1.1∎

Square the inequality in step 1.1 and use the definition of the Bochner $L^{2}$ norm in [F2]. Taking $C = C_{0}^{2}$ gives the stated integral inequality, with the same independence from $T$ and $v_{0}$ .

Source notes

The article states the homogeneous estimate at (2.3) and restates it as Proposition 3.1, equation (3.6); its citation here is attribution. The proof above uses the independently sourced smoothing supplier, whose source record states its reading limits.

Depends on

Used by · 6

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2