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Quadratic-cubic estimates near a reference trajectory

PDF p. 20: estimates for F(w,z) and its difference

Statement

Assume the Axiom of Choice. Let $n \in {2, 3}$ , and let $Ω = R^{n} ∖ Θ$ be the connected exterior of a nonempty compact obstacle with smooth boundary. Let $T > 0$ , $λ \in R ∖ {0}$ , and let the reference trajectory $w$ satisfy (W): $w \in C ([0, T]; H_{D}^{2} (Ω) \cap W^{2, \infty} (Ω))$ and $i \partial_{t} w + Δ w + λ ∣ w ∣^{2} w = 0$ in space-time distributions. The intersection has the sum norm. Define $F (w, z) = λ (2 w ∣ z ∣^{2} + \overline{w} z^{2} + ∣ z ∣^{2} z)$ .

Here $H_{D}^{s} = D ((I - Δ_{D})^{s / 2})$ has its spectral norm, with $H_{D}^{1} = H_{0}^{1}$ and $H_{D}^{2} = H^{2} \cap H_{0}^{1}$ as spaces with equivalent norms. The space $H_{0}^{1}$ is the $H^{1}$ closure of compactly supported smooth complex functions. For $s = 1$ put $X_{T} = C ([0, T]; H_{0}^{1} (Ω)) \cap L_{t}^{4} L_{x}^{\infty}$ , with norm $\sup_{t} ∥ z (t) ∥_{H^{1}} + ∥ z ∥_{L_{t}^{4} L_{x}^{\infty}}$ ; its mixed-norm representative is jointly measurable and agrees almost everywhere in time with the continuous path. For $s = 2$ put $X_{T} = C ([0, T]; H_{D}^{2} (Ω))$ with its supremum spectral norm. The mixed norm is the $L^{4} (0, T)$ norm of the spatial essential supremum, and does not require strong measurability into $L^{\infty} (Ω)$ .

For each $s \in {1, 2}$ and $z, \tilde{z} \in X_{T}$ , the functions $F (w, z)$ and $F (w, \tilde{z})$ belong to the Bochner space $L^{1} (0, T; H_{D}^{s} (Ω))$ . There is a constant $C$ depending on the fixed domain, dimension, $T$ , $λ$ and the displayed norms of $w$ , such that

$∥ F (w, z) ∥_{L_{t}^{1} H_{D}^{s}} \leq C (∥ z ∥_{X_{T}}^{2} + ∥ z ∥_{X_{T}}^{3}),$

$∥ F (w, z) - F (w, \tilde{z}) ∥_{L_{t}^{1} H_{D}^{s}} \leq C (∥ z ∥_{X_{T}} + ∥ \tilde{z} ∥_{X_{T}} + ∥ z ∥_{X_{T}}^{2} + ∥ \tilde{z} ∥_{X_{T}}^{2}) ∥ z - \tilde{z} ∥_{X_{T}} .$

Statement references

Exterior and truncated obstacle domains

Reference trajectory hypothesis W

Quadratic-cubic remainder after subtracting a trajectory

Energy control space with a time-integrable supremum norm

Continuous and mixed time-space function spaces

Low-order identifications of the Dirichlet scale

Dirichlet Laplacian and positive shift

Integer-order Sobolev spaces and their norms

The Axiom of Choice

Facts & Assumptions

[F1]

The reference trajectory and polynomial remainder are as defined in Reference trajectory hypothesis W and Quadratic-cubic remainder after subtracting a trajectory. The geometry is that of Exterior and truncated obstacle domains, and the full Choice assumption required by the suppliers is recorded in The Axiom of Choice.

[F2]

The operator $Δ_{D}$ and its positive self-adjoint shift $I - Δ_{D}$ are specified by Dirichlet Laplacian and positive shift. Under these geometry and Choice hypotheses, Low-order identifications of the Dirichlet scale identifies $H_{D}^{1}$ with $H_{0}^{1}$ and $H_{D}^{2}$ with $H^{2} \cap H_{0}^{1}$ , with equivalent norms. The energy control space and compatible mixed representatives are specified by Energy control space with a time-integrable supremum norm and Continuous and mixed time-space function spaces.

[F3]

Tame bounds for the quadratic-cubic trajectory remainder gives strong measurability, Bochner integrability and the quadratic-cubic growth and difference bounds in the energy space when $w \in C_{t} W^{1, \infty}$ . On the stated exterior in dimensions two and three it gives the corresponding bounds in $C_{t} (H^{2} \cap H_{0}^{1})$ when $w \in C_{t} H^{2}$ , using the ordinary $H^{2}$ norm. Its constants depend on the fixed interval, coefficient bounds, coupling and, in the order-two case, the domain.

[F4]

The complex weak Sobolev spaces and norms are defined in Integer-order Sobolev spaces and their norms. For $p = \infty$ the norm is the maximum of the essential-supremum norms of the weak derivatives of order at most $k$ , including the function itself. The Choice assumption in [F1] includes the countable choice required there.

Proof

[F1] The reference trajectory and polynomial remainder are as defined in Reference trajectory hypothesis W and Quadratic-cubic remainder after subtracting a trajectory. The geometry is that of Exterior and truncated obstacle domains, and the full Choice assumption required by the suppliers is recorded in The Axiom of Choice.

[F2] The operator $Δ_{D}$ and its positive self-adjoint shift $I - Δ_{D}$ are specified by Dirichlet Laplacian and positive shift. Under these geometry and Choice hypotheses, Low-order identifications of the Dirichlet scale identifies $H_{D}^{1}$ with $H_{0}^{1}$ and $H_{D}^{2}$ with $H^{2} \cap H_{0}^{1}$ , with equivalent norms. The energy control space and compatible mixed representatives are specified by Energy control space with a time-integrable supremum norm and Continuous and mixed time-space function spaces.

[F3] Tame bounds for the quadratic-cubic trajectory remainder gives strong measurability, Bochner integrability and the quadratic-cubic growth and difference bounds in the energy space when $w \in C_{t} W^{1, \infty}$ . On the stated exterior in dimensions two and three it gives the corresponding bounds in $C_{t} (H^{2} \cap H_{0}^{1})$ when $w \in C_{t} H^{2}$ , using the ordinary $H^{2}$ norm. Its constants depend on the fixed interval, coefficient bounds, coupling and, in the order-two case, the domain.

Given: The domain, trajectory, time interval and functions in the Statement.

Goal: Prove the stated membership and both estimates in each control space.

1.1givenF1F2F3F4

The (W) intersection continuity implies $w \in C ([0, T]; W^{1, \infty})$ : the order-zero and order-one weak derivatives are among those of order at most two, so the maximum norms of [F4] satisfy $∥ u ∥_{W^{1, \infty}} \leq ∥ u ∥_{W^{2, \infty}}$ . The value-plus-first-derivative sum norm used in [F3] is at most $(n + 1) ∥ u ∥_{W^{2, \infty}}$ , which also gives continuity in that norm. It also implies $w \in C ([0, T]; H^{2})$ by the order-two norm equivalence. The corresponding coefficient suprema are finite on the compact time interval. The polynomial in the Statement is exactly the remainder in [F1].

2.1F2F3step 1.1

For $s = 1$ , apply the energy case of [F3] using the compatible representatives in [F2]. It yields the asserted Bochner membership and both inequalities with the $H_{0}^{1}$ norm. The equivalence in [F2] transfers the output norm to $H_{D}^{1}$ , changing only the constant.

3.1F2F3step 1.1step 2.1∎

For $s = 2$ , [F2] identifies the given paths with continuous paths in $H^{2} \cap H_{0}^{1}$ , with equivalent supremum norms. Apply the order-two case of [F3] to $w$ , $z$ and $\tilde{z}$ ; it gives the asserted integrability and inequalities with ordinary $H^{2}$ norms. Use the same equivalence on the input and output norms and enlarge the constant to absorb the finitely many equivalence factors. This proves both estimates for either value of $s$ .

Source notes

The original article is cited for attribution of the two estimates in the proof of Theorem 1.5. The argument above applies the separately authored tame-bound and scale-identification suppliers; the article is not used as proof of those inputs.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF page 20, proof of Theorem 1.5