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Free and retarded bounds in the energy control space

PDF p. 15: Proposition 3.9 proof (i)–(ii)

Statement

Assume the Axiom of Choice and hypothesis (H): for $n \in {2, 3}$ , $Ω = R^{n} ∖ Θ$ is the connected exterior of a nonempty compact obstacle with smooth boundary, and the obstacle is non-trapping, meaning that every complete admissible generalized ray $x : R \to \overline{Ω}$ , parametrized at unit speed and including reflected and glancing behavior, leaves every compact set in finite forward time. Let $T > 0$ , let $Δ_{D}$ be the Dirichlet Laplacian on $Ω$ , and write $W (t) = e^{i t Δ_{D}}$ . Let $H_{0}^{1} (Ω)$ be the closure of $C_{c}^{\infty} (Ω)$ in $H^{1} (Ω)$ , with its inherited $H^{1}$ norm. Let $X_{T}$ be the space of continuous $H_{0}^{1}$ -valued paths with a compatible jointly measurable representative of finite scalar mixed norm $L^{4} (0, T; L^{\infty} (Ω))$ , equipped with the sum of that mixed norm and the supremum $H_{0}^{1}$ norm.

There is a constant $C$ , depending on $T$ and the fixed domain, such that, for every $u_{0} \in H_{0}^{1} (Ω)$ and every strongly measurable $f \in L^{1} (0, T; H_{0}^{1} (Ω))$ , $∥ W (\cdot) u_{0} ∥_{X_{T}} \leq C ∥ u_{0} ∥_{H_{0}^{1} (Ω)}, {∥ \int_{0}^{t} W (t - r) f (r) d r ∥}_{X_{T}} \leq C ∥ f ∥_{L^{1} (0, T; H_{0}^{1} (Ω))} .$ The integral is the $H_{0}^{1}$ -valued Bochner integral. The $L_{t}^{4} L_{x}^{\infty}$ term is the scalar sectionwise mixed norm; the second bound does not assert strong measurability into $L^{\infty} (Ω)$ .

Facts & Assumptions

[F1]

The domain and non-trapping convention are those of Exterior and truncated obstacle domains and Non-trapping obstacle. The Dirichlet Laplacian and its spectral scale are specified by Dirichlet Laplacian and positive shift and Dirichlet spectral Sobolev scale and duality. The stated Sobolev closure, time spaces and scalar mixed norm are specified by Complex integer-order Sobolev notation and Continuous and mixed time-space function spaces; Energy control space with a time-integrable supremum norm defines $X_{T}$ and its norm. We assume The Axiom of Choice.

[F2]

Low-order identifications of the Dirichlet scale identifies $H_{D}^{1} (Ω)$ with $H_{0}^{1} (Ω)$ with equivalent norms. Scale isometries and free Schrödinger evolution makes $W$ a strongly continuous unitary group on the Hilbert space $H_{D}^{1}$ .

[F3]

With the stated geometry and $p = 4$ , Lemma 3.8: energy-data time-integrable supremum bound bounds the homogeneous orbit in $L^{4} (0, T; L^{\infty} (Ω))$ by a constant times its initial $H_{0}^{1}$ norm, and supplies a strongly measurable $L^{\infty}$ -valued representative.

[F4]

Homogeneous bounds transfer to L1 retarded forcing transfers a homogeneous scalar mixed-norm bound for a strongly continuous unitary group on a Hilbert space $H$ into a continuous $H$ -valued retarded integral and the corresponding scalar $L_{t}^{p} L_{x}^{\infty}$ bound. Its scalar convention applies when $H$ embeds continuously in $L^{2}$ on the underlying sigma-finite spatial space; its proof uses the homogeneous bound on intervals of length at most $T$ .

Proof

[F1] The domain and non-trapping convention are those of Exterior and truncated obstacle domains and Non-trapping obstacle. The Dirichlet Laplacian and its spectral scale are specified by Dirichlet Laplacian and positive shift and Dirichlet spectral Sobolev scale and duality. The stated Sobolev closure, time spaces and scalar mixed norm are specified by Complex integer-order Sobolev notation and Continuous and mixed time-space function spaces; Energy control space with a time-integrable supremum norm defines $X_{T}$ and its norm. We assume The Axiom of Choice.

[F2] Low-order identifications of the Dirichlet scale identifies $H_{D}^{1} (Ω)$ with $H_{0}^{1} (Ω)$ with equivalent norms. Scale isometries and free Schrödinger evolution makes $W$ a strongly continuous unitary group on the Hilbert space $H_{D}^{1}$ .

[F3] With the stated geometry and $p = 4$ , Lemma 3.8: energy-data time-integrable supremum bound bounds the homogeneous orbit in $L^{4} (0, T; L^{\infty} (Ω))$ by a constant times its initial $H_{0}^{1}$ norm, and supplies a strongly measurable $L^{\infty}$ -valued representative.

[F4] Homogeneous bounds transfer to L1 retarded forcing transfers a homogeneous scalar mixed-norm bound for a strongly continuous unitary group on a Hilbert space $H$ into a continuous $H$ -valued retarded integral and the corresponding scalar $L_{t}^{p} L_{x}^{\infty}$ bound. Its scalar convention applies when $H$ embeds continuously in $L^{2}$ on the underlying sigma-finite spatial space; its proof uses the homogeneous bound on intervals of length at most $T$ .

Given: The domain, time, initial datum and forcing in the statement.

Goal: Establish both bounds in $X_{T}$ .

1.1givenF1F2F3

By the equivalent norms in F2, unitarity on $H_{D}^{1}$ gives $\sup_{0 \leq t \leq T} ∥ W (t) u_{0} ∥_{H_{0}^{1}} \leq C_{1} ∥ u_{0} ∥_{H_{0}^{1}}$ , and the orbit is continuous in $H_{0}^{1}$ . F3 supplies its $L_{t}^{4} L_{x}^{\infty}$ bound. Its representative is compatible with the continuous orbit, so the definition in F1 gives the first inequality after adding the two bounds.

1.2givenF2F3F4

For the forcing estimate take $H = H_{D}^{1}$ with its spectral norm and $p = 4$ in F4. By F2, $H$ embeds continuously in $L^{2} (Ω)$ , and $f$ is strongly measurable and integrable as an $H$ -valued function; equivalent norms preserve both properties. By F3 and norm equivalence, the required homogeneous estimate holds with the $H$ norm. F4 therefore gives a continuous $H$ -valued retarded integral with a compatible jointly measurable representative and a scalar mixed-norm estimate, bounded together by a constant times $∥ f ∥_{L^{1} (0, T; H)}$ . The restriction from $(0, T)$ to $(0, T - r)$ provides the same homogeneous constant for each shifted interval.

2.1F1F2step 1.1step 1.2∎

Convert the $H$ norms in step 1.2 to the equivalent inherited $H_{0}^{1}$ norms. The resulting path and representative satisfy the defining conditions of $X_{T}$ , and the sum of its continuous-path and mixed norms is bounded by a constant times $∥ f ∥_{L^{1} (0, T; H_{0}^{1})}$ . Together with step 1.1 this proves both inequalities, with the larger of the constants.

Source notes

The frozen article is cited for the two estimates in the proof of Proposition 3.9, parts (i) and (ii). The deduction above uses the declared completed suppliers; the article citation is attribution rather than a premise proving its own claim.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2