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Cubic and quintic energy-space nonlinear time estimates

PDF p. 15: time-integrated estimate in Proposition 3.9; PDF p. 16: opening difference estimate

Statement

Assume the Axiom of Choice. Let $n \geq 2$ , let $Θ \subset R^{n}$ be nonempty and compact with smooth boundary, and suppose that $Ω = R^{n} ∖ Θ$ is connected. Let $T > 0$ , $m \in {3, 5}$ and $λ \in R ∖ {0}$ , and set $g (z) = λ ∣ z ∣^{m - 1} z$ .

Let $H_{0}^{1} (Ω)$ be the closure of $C_{c}^{\infty} (Ω; C)$ in the norm $∥ w ∥_{H^{1}} = (∥ w ∥_{2}^{2} + \sum_{j} ∥ D_{j} w ∥_{2}^{2})^{1 / 2}$ . Define $X_{T} = C ([0, T]; H_{0}^{1} (Ω)) \cap L_{t}^{4} L_{x}^{\infty}$ using compatible jointly measurable representatives and the norm $∥ w ∥_{X_{T}} = \sup_{t \in [0, T]} ∥ w (t) ∥_{H^{1}} + ∥ w ∥_{L_{t}^{4} L_{x}^{\infty}}$ . The mixed norm is the $L^{4} (0, T)$ norm of the spatial essential supremum; it does not require strong measurability into $L^{\infty} (Ω)$ .

For $u, v \in X_{T}$ , both $g (u)$ and $g (v)$ belong to the Bochner space $L^{1} (0, T; H_{0}^{1} (Ω))$ , and

$∥ g (v) ∥_{L_{t}^{1} H_{0}^{1}} \leq m ∣ λ ∣ T^{(5 - m) / 4} ∥ v ∥_{X_{T}}^{m} .$

There is a constant $C$ depending only on $m$ and $∣ λ ∣$ such that

$∥ g (v) - g (u) ∥_{L_{t}^{1} H_{0}^{1}} \leq C T^{(5 - m) / 4} (∥ v ∥_{X_{T}}^{m - 1} + ∥ u ∥_{X_{T}}^{m - 1}) ∥ v - u ∥_{X_{T}} .$

For $m = 5$ the factor involving $T$ is $1$ .

Statement references

Energy control space with a time-integrable supremum norm

Complex integer-order Sobolev notation

Continuous and mixed time-space function spaces

The Axiom of Choice

Facts & Assumptions

[F1]

The energy-space definition supplies the compatible representative convention and the displayed $X_{T}$ norm; the integer Sobolev and time-space definitions supply the weak-derivative norm, closure and Bochner and mixed-norm conventions. Energy control space with a time-integrable supremum norm Complex integer-order Sobolev notation Continuous and mixed time-space function spaces

[F2]

For bounded $w \in H_{0}^{1} (Ω)$ , the power chain rule gives $g (w) \in H_{0}^{1} (Ω)$ and $∥ g (w) ∥_{H^{1}} \leq m ∣ λ ∣ ∥ w ∥_{\infty}^{m - 1} ∥ w ∥_{H^{1}}$ . The pointwise-differences proposition gives the corresponding energy difference bound with one term containing $∥ v - u ∥_{H^{1}}$ and one containing $∥ v - u ∥_{\infty}$ . Power nonlinearity preserves the zero-boundary energy space Pointwise and energy difference estimates for odd powers

[F3]

The time-integrability lemma applies to the stated compatible mixed-norm convention and establishes strong $H_{0}^{1}$ measurability and $L^{1}$ membership for the nonlinear terms, without assuming Bochner measurability into $L^{\infty}$ . Cubic and quintic time bounds from L4 Linfinity control

[F4]

Hölder's integral inequality applies to nonnegative measurable time functions, including its $1, \infty$ endpoint. Holder's inequality for integrals, including the endpoint cases

[F5]

The assumed Axiom of Choice supplies the choice hypotheses carried by these inputs. The Axiom of Choice

Proof

[F1] The energy-space definition supplies the compatible representative convention and the displayed $X_{T}$ norm; the integer Sobolev and time-space definitions supply the weak-derivative norm, closure and Bochner and mixed-norm conventions. Energy control space with a time-integrable supremum norm Complex integer-order Sobolev notation Continuous and mixed time-space function spaces

[F2] For bounded $w \in H_{0}^{1} (Ω)$ , the power chain rule gives $g (w) \in H_{0}^{1} (Ω)$ and $∥ g (w) ∥_{H^{1}} \leq m ∣ λ ∣ ∥ w ∥_{\infty}^{m - 1} ∥ w ∥_{H^{1}}$ . The pointwise-differences proposition gives the corresponding energy difference bound with one term containing $∥ v - u ∥_{H^{1}}$ and one containing $∥ v - u ∥_{\infty}$ . Power nonlinearity preserves the zero-boundary energy space Pointwise and energy difference estimates for odd powers

[F3] The time-integrability lemma applies to the stated compatible mixed-norm convention and establishes strong $H_{0}^{1}$ measurability and $L^{1}$ membership for the nonlinear terms, without assuming Bochner measurability into $L^{\infty}$ . Cubic and quintic time bounds from L4 Linfinity control

[F4] Hölder's integral inequality applies to nonnegative measurable time functions, including its $1, \infty$ endpoint. Holder's inequality for integrals, including the endpoint cases

[F5] The assumed Axiom of Choice supplies the choice hypotheses carried by these inputs. The Axiom of Choice

Given: The domain, parameters, functions and norms in the Statement.

Goal: Establish Bochner integrability and both displayed estimates.

1.1givenF1F3F5

For $w \in {u, v, v - u}$ put $A_{w} (t) = ∥ w (t) ∥_{L^{\infty} (Ω)}$ , $a_{w} = ∥ A_{w} ∥_{L^{4} (0, T)}$ and $B_{w} = \sup_{t \in [0, T]} ∥ w (t) ∥_{H^{1}}$ . The compatible representative convention makes $A_{w}$ measurable and finite almost everywhere. The domain and norm hypotheses match [F3], so $g (u)$ and $g (v)$ are strongly measurable and integrable as $H_{0}^{1}$ -valued functions; their difference is integrable as well.

2.1F2F4step 1.1algebra

Apply [F4] to $A_{v}^{m - 1}$ and $1$ with reciprocal exponents $(m - 1) / 4$ and $(5 - m) / 4$ . At $m = 5$ the second exponent is $\infty$ . Thus $\int_{0}^{T} A_{v}^{m - 1} d t \leq T^{(5 - m) / 4} a_{v}^{m - 1}$ . Using the sharp pointwise-in-time growth estimate in [F2] and $a_{v}, B_{v} \leq ∥ v ∥_{X_{T}}$ gives the first displayed inequality.

3.1givenF2F4step 1.1step 2.1

Let $h = v - u$ . The energy difference bound in [F2], applied at almost every time, is bounded by $C [(A_{v}^{m - 1} + A_{u}^{m - 1}) B_{h} + (A_{v}^{m - 2} + A_{u}^{m - 2}) (B_{v} + B_{u}) A_{h}]$ . The first part integrates by step 2.1. Repeated Hölder applied to the second part gives $\int_{0}^{T} A_{w}^{m - 2} A_{h} d t \leq T^{(5 - m) / 4} a_{w}^{m - 2} a_{h}$ for $w = u, v$ : the three reciprocal exponents are $(m - 2) / 4$ , $1 / 4$ and $(5 - m) / 4$ , with the last factor in $L^{\infty}$ when $m = 5$ .

4.1step 3.1F1algebra∎

Bound each $a_{w}$ and $B_{w}$ by $∥ w ∥_{X_{T}}$ . For nonnegative $x, y$ , $x^{m - 2} y + x y^{m - 2} \leq x^{m - 1} + y^{m - 1}$ , so the mixed terms from step 3.1 are absorbed into the displayed difference bound after enlarging $C$ . The constant depends only on $m$ and $∣ λ ∣$ , and $(5 - m) / 4 = 0$ for $m = 5$ .

Source notes

The original article is cited for attribution of the estimates in Proposition 3.9. The proof uses the separately supplied local polynomial and time-integrability results and Hölder's inequality.

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Sources

Nonlinear Schrödinger equation in an exterior domain, v2 · PDF pages 15–16, proof of Proposition 3.9