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Two-dimensional exponents removing the Strichartz loss

PDF p. 15: Lemma 3.8 proof, n=2

Statement

Assume the Axiom of Choice. Let $Θ \subset R^{2}$ be a nonempty compact obstacle with smooth boundary and connected exterior $Ω = R^{2} ∖ Θ$ . Assume it is non-trapping: each generalized reflected or glancing ray leaves each compact set in finite forward time. Let $Δ_{D}$ be the Dirichlet Laplacian and $W (t) = e^{i t Δ_{D}}$ . Let $H_{0}^{1} (Ω)$ be the closure of $C_{c}^{\infty} (Ω)$ in $H^{1} (Ω)$ .

For every $T > 0$ and finite $p > 2$ , put $s^{'} = 1 - 1 / p$ and $q = 2 p / (p - 2)$ . Then $2 / p + 2 / q = 1$ , $s^{'} + 1 / p = 1$ , and $s^{'} q = 2 (p - 1) / (p - 2) > 2$ . There is a constant $C$ depending on $Ω$ , $T$ and $p$ such that, for every $u_{0} \in H_{0}^{1} (Ω)$ , $∥ W (\cdot) u_{0} ∥_{L^{p} (0, T; L^{\infty} (Ω))} \leq C ∥ u_{0} ∥_{H^{1} (Ω)} .$ Here the time-space norm is the Bochner norm. The fractional space used in the deduction is $W^{s^{'}, q} (Ω) = [L^{q} (Ω), W^{1, q} (Ω)]_{s^{'}}$ , defined by complex interpolation.

Statement references

Exterior and truncated obstacle domains

Non-trapping obstacle

Dirichlet spectral Sobolev scale and duality

Complex integer-order Sobolev notation

Continuous and mixed time-space function spaces

An explicit common interpolation model for the fractional Strichartz targets

The Axiom of Choice

Facts & Assumptions

Given: The hypotheses and data in the Statement.

[F1]

The geometry and generalized-ray convention are those of Exterior and truncated obstacle domains and Non-trapping obstacle. The Dirichlet scale supplier Dirichlet spectral Sobolev scale and duality defines $H_{D}^{r}$ and $W (t)$ under the Axiom of Choice The Axiom of Choice.

[F2]

The lossy Strichartz estimate Global Strichartz estimate with derivative loss applies on this exterior for $2 < p < \infty$ , $2 \leq q < \infty$ , $2 / p + 2 / q = 1$ and $0 \leq s \leq 1$ : it bounds the Bochner $L_{t}^{p} W_{x}^{s, q}$ norm of $W (\cdot) u_{0}$ by a constant times $∥ u_{0} ∥_{H_{D}^{s + 1 / p}}$ .

[F3]

The interpolation model An explicit common interpolation model for the fractional Strichartz targets defines $W^{s, q} = [L^{q}, W^{1, q}]_{s}$ for $0 < s < 1$ and $1 < q < \infty$ . The exterior embedding Independent exterior fractional Sobolev embedding gives a bounded linear inclusion $W^{s, q} (Ω) \to L^{\infty} (Ω)$ when $s q > 2$ .

[F4]

The scale identification Low-order identifications of the Dirichlet scale gives $H_{D}^{1} (Ω) = H_{0}^{1} (Ω)$ with equivalent norms. The integer Sobolev definition Complex integer-order Sobolev notation specifies the closure and $H^{1}$ norm, and Continuous and mixed time-space function spaces specifies strong measurability and Bochner $L^{p}$ norms.

Proof

Given: The hypotheses and data in the Statement.

[F1] The geometry and generalized-ray convention are those of Exterior and truncated obstacle domains and Non-trapping obstacle. The Dirichlet scale supplier Dirichlet spectral Sobolev scale and duality defines $H_{D}^{r}$ and $W (t)$ under the Axiom of Choice The Axiom of Choice.

[F2] The lossy Strichartz estimate Global Strichartz estimate with derivative loss applies on this exterior for $2 < p < \infty$ , $2 \leq q < \infty$ , $2 / p + 2 / q = 1$ and $0 \leq s \leq 1$ : it bounds the Bochner $L_{t}^{p} W_{x}^{s, q}$ norm of $W (\cdot) u_{0}$ by a constant times $∥ u_{0} ∥_{H_{D}^{s + 1 / p}}$ .

[F3] The interpolation model An explicit common interpolation model for the fractional Strichartz targets defines $W^{s, q} = [L^{q}, W^{1, q}]_{s}$ for $0 < s < 1$ and $1 < q < \infty$ . The exterior embedding Independent exterior fractional Sobolev embedding gives a bounded linear inclusion $W^{s, q} (Ω) \to L^{\infty} (Ω)$ when $s q > 2$ .

[F4] The scale identification Low-order identifications of the Dirichlet scale gives $H_{D}^{1} (Ω) = H_{0}^{1} (Ω)$ with equivalent norms. The integer Sobolev definition Complex integer-order Sobolev notation specifies the closure and $H^{1}$ norm, and Continuous and mixed time-space function spaces specifies strong measurability and Bochner $L^{p}$ norms.

Given: The obstacle, $T$ , $p$ and $u_{0}$ in the Statement.

Goal: Prove the parameter identities and the stated energy-data estimate.

1.1F1F2F3givenalgebra

Since $2 < p < \infty$ , the numbers $s^{'} = 1 - 1 / p$ and $q = 2 p / (p - 2)$ satisfy $0 < s^{'} < 1$ and $2 < q < \infty$ . Direct calculation gives $2 / p + 2 / q = 1$ , $s^{'} + 1 / p = 1$ , and $s^{'} q = 2 (p - 1) / (p - 2) = 2 + 2 / (p - 2) > 2$ . Thus the hypotheses of both the lossy estimate and the fractional embedding hold.

2.1F2F4step 1.1given

The scale identification puts $u_{0}$ in $H_{D}^{1}$ and bounds its spectral norm by a constant times its $H^{1}$ norm. Applying the lossy estimate with $s = s^{'}$ and the exponents from step 1.1 yields $∥ W (\cdot) u_{0} ∥_{L^{p} (0, T; W^{s^{'}, q})} \leq C_{1} ∥ u_{0} ∥_{H^{1}}$ .

3.1F3F4step 1.1step 2.1∎

The bounded linear inclusion in F3 maps the strongly measurable $W^{s^{'}, q}$ -valued representative in step 2.1 to a strongly measurable $L^{\infty}$ -valued representative: apply the inclusion to its finite-range simple approximants. Its pointwise norm is at most the inclusion norm times the $W^{s^{'}, q}$ norm. Integrating the $p$ th power and using step 2.1 gives the asserted Bochner estimate, with $C$ the product of the two constants.

Source notes

The original article is cited for attribution to the two-dimensional argument in the proof of Lemma 3.8. The proof above uses the separately supplied lossy estimate, scale identification and fractional embedding. The interpolation convention is explicit; no identification with a Slobodeckij space for $q \neq 2$ is asserted.

Depends on

Used by · 1

Lemma 3.8: energy-data time-integrable supremum bound

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Nonlinear Schrödinger equation in an exterior domain, v2