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Energy dispersive bounds under bounded trajectory perturbations

Statement

Assume the Axiom of Choice. Let $Ω = R^{n} ∖ Θ$ , $n \geq 2$ , where $Θ$ is compact with smooth boundary and $Ω$ is open. Let $A = I - Δ_{D} \geq I$ be the self-adjoint shifted Dirichlet Laplacian and let $H_{D}^{s} = D (A^{s / 2})$ with norm $∥ A^{s / 2} \cdot ∥_{2}$ for $s \in {1, 2}$ . In particular $H_{D}^{1} = H_{0}^{1} (Ω)$ and $H_{D}^{2} = H^{2} (Ω) \cap H_{0}^{1} (Ω)$ , with equivalent standard Sobolev norms. Write $W (t) = e^{i t Δ_{D}}$ , a strongly continuous unitary group on each of these spaces.

Fix $0 < T < \infty$ . Suppose that, for some $C_{T}$ , the scalar mixed-norm estimate $∥ W (\cdot) h ∥_{L^{4} (0, T; L^{\infty} (Ω))} \leq C_{T} ∥ h ∥_{H_{D}^{1}}$ holds for every $h \in H_{D}^{1}$ . Here the mixed norm is the $L^{4}$ norm in time of the spatial essential supremum; it does not require Bochner measurability into $L^{\infty}$ .

Let $B : [0, T] \to B_{R} (H_{D}^{1})$ be norm-continuous, where $B_{R}$ denotes bounded real-linear operators, and let $K = \sup_{t \in [0, T]} ∥ B (t) ∥$ . Let $S (t, r)$ be the real-linear evolution determined by $S (t, r) h = W (t - r) h + \int_{r}^{t} W (t - ρ) B (ρ) S (ρ, r) h d ρ$ . For $0 \leq r \leq b \leq T$ , put $I = [r, b]$ and $∥ u ∥_{X_{I}} = ∥ u ∥_{C (I; H_{D}^{1})} + ∥ u ∥_{L^{4} (I; L^{\infty})}$ . Then, for every $h \in H_{D}^{1}$ and strongly measurable $f \in L^{1} (I; H_{D}^{1})$ , the function $u (t) = S (t, r) h + \int_{r}^{t} S (t, ρ) f (ρ) d ρ$ belongs to $X_{I}$ and satisfies $∥ u ∥_{X_{I}} \leq C_{T, K} (∥ h ∥_{H_{D}^{1}} + ∥ f ∥_{L^{1} (I; H_{D}^{1})})$ , where one may take $C_{T, K} = e^{K T} + C_{T} (1 + T K e^{K T})$ . In particular the homogeneous perturbed orbit and the retarded perturbed forcing operator both obey this bound.

The analogous energy conclusion holds on $H_{D}^{2}$ : if $B$ is also a norm-continuous bounded real-linear family on $H_{D}^{2}$ , consistent with its action on $H_{D}^{1}$ , and $K_{2}$ is its supremum norm there, then the same evolution restricted to $H_{D}^{2}$ satisfies $∥ u ∥_{C (I; H_{D}^{2})} \leq e^{K_{2} T} (∥ h ∥_{H_{D}^{2}} + ∥ f ∥_{L^{1} (I; H_{D}^{2})})$ . This order-two assertion does not require an $L^{4} L^{\infty}$ estimate at order two.

For the trajectory form $B (t) z = i (V_{1} (t) z + V_{2} (t) \overline{z})$ , it is sufficient that each $V_{j} : [0, T] \to W^{2, \infty} (Ω)$ be norm-continuous. Here $W^{2, \infty}$ consists of functions with essentially bounded weak derivatives through order two. The resulting multipliers preserve $H_{0}^{1}$ and hence the Dirichlet boundary condition. No global-in-time smoothing or unique-continuation estimate is asserted.

Facts & Assumptions

[F1]

The exterior realization identifies the self-adjoint operator, its form domain $H_{0}^{1}$ and its operator domain $H^{2} \cap H_{0}^{1}$ Exterior Dirichlet realization, H² domain and form domain. The positive spectral-scale contract supplies the scale and the consistent unitary group Positive spectral Hilbert scale: duality and interpolation.

[F2]

For a strongly continuous bounded free group and a norm-continuous bounded real-linear perturbation, the real evolution supplier constructs the unique evolution, its variation-of-constants formula and its energy bound Bounded real-linear perturbations preserve solution estimates.

[F3]

The shifted retarded-transfer lemma converts the assumed homogeneous bound into the causal $L_{t}^{1} H_{D}^{1}$ to $L_{t}^{4} L_{x}^{\infty}$ bound on every translated interval of length at most $T$ , using its scalar $L^{\infty}$ convention Homogeneous bounds transfer to L1 retarded forcing.

[F4]

The spectral multiplier lemma bounds multiplication by $a \in W^{2, \infty}$ on both $H_{D}^{1}$ and $H_{D}^{2}$ , preserves the zero-boundary spaces, and identifies conjugation as an isometry Endpoint multiplier bounds extended to negative spectral orders.

[F5]

The Axiom of Choice supplies the choice hypotheses inherited by these scale, multiplier and evolution inputs The Axiom of Choice.

[F6]

The complex integer Sobolev spaces use $H^{k} (Ω) = W^{k, 2} (Ω; C)$ with $∥ u ∥_{H^{k}}^{2} = \sum_{∣ α ∣ \leq k} ∥ D^{α} u ∥_{2}^{2}$ for $k = 1, 2$ . The coefficient norm is $∥ a ∥_{W^{2, \infty}} = \max_{∣ α ∣ \leq 2} ∥ D^{α} a ∥_{\infty}$ , with $D^{0} a = a$ , as in Integer-order Sobolev spaces and their norms. Its open-set, dimensional and scalar hypotheses hold here, and its Countable Choice hypothesis follows from [F5].

[F7]

Strong measurability on $I$ means almost-everywhere convergence in the relevant $H_{D}^{j}$ norm, $j = 1, 2$ , of measurable simple functions, with respect to Lebesgue measure on $I$ Strongly measurable Banach-valued function. These Hilbert spaces are Banach spaces, so the definition applies to the forcing and integrands in the energy spaces; it imposes no strong measurability into spatial $L^{\infty}$ .

Proof

[F1] The exterior realization identifies the self-adjoint operator, its form domain $H_{0}^{1}$ and its operator domain $H^{2} \cap H_{0}^{1}$ Exterior Dirichlet realization, H² domain and form domain. The positive spectral-scale contract supplies the scale and the consistent unitary group Positive spectral Hilbert scale: duality and interpolation.

[F2] For a strongly continuous bounded free group and a norm-continuous bounded real-linear perturbation, the real evolution supplier constructs the unique evolution, its variation-of-constants formula and its energy bound Bounded real-linear perturbations preserve solution estimates.

[F3] The shifted retarded-transfer lemma converts the assumed homogeneous bound into the causal $L_{t}^{1} H_{D}^{1}$ to $L_{t}^{4} L_{x}^{\infty}$ bound on every translated interval of length at most $T$ , using its scalar $L^{\infty}$ convention Homogeneous bounds transfer to L1 retarded forcing.

[F4] The spectral multiplier lemma bounds multiplication by $a \in W^{2, \infty}$ on both $H_{D}^{1}$ and $H_{D}^{2}$ , preserves the zero-boundary spaces, and identifies conjugation as an isometry Endpoint multiplier bounds extended to negative spectral orders.

[F5] The Axiom of Choice supplies the choice hypotheses inherited by these scale, multiplier and evolution inputs The Axiom of Choice.

Given: The domain, free homogeneous estimate and real-linear perturbation in the Statement.

Goal: Establish the finite-interval perturbed homogeneous and forcing estimates and the stated multiplier application.

1.1F1F2F5F7given

By the exterior realization and spectral-scale contract, $H_{D}^{1}$ is a complex Hilbert space, $W$ is a strongly continuous unitary group on it, and its scalar realization embeds continuously into $L^{2} (Ω)$ . The real evolution result therefore gives the stated $S$ and, for the prescribed data and forcing, the Volterra identity $u (t) = W (t - r) h + \int_{r}^{t} W (t - ρ) (B (ρ) u (ρ) + f (ρ)) d ρ$ . It also gives $∥ u ∥_{C (I; H_{D}^{1})} \leq e^{K (b - r)} (∥ h ∥_{H_{D}^{1}} + ∥ f ∥_{L^{1} (I; H_{D}^{1})})$ . The integrand is strongly measurable because $B$ and $u$ are continuous and $f$ is strongly measurable.

1.2F1F2F5given

On $H_{D}^{2}$ , the same real evolution theorem applies with the consistent unitary free group and bound $K_{2}$ , giving the stated energy estimate. The $H_{D}^{2}$ solution also satisfies the $H_{D}^{1}$ Volterra equation because the embeddings, free groups and perturbation actions are consistent; uniqueness in $H_{D}^{1}$ identifies the evolutions.

2.1F3step 1.1given

Apply the shifted retarded-transfer lemma in its scalar convention to $g = B u + f$ . Its $L^{\infty}$ endpoint applies to the scalar mixed norm, and translation and restriction give the same $C_{T}$ on $I$ . Since $∥ g ∥_{L^{1} (I; H_{D}^{1})} \leq T K ∥ u ∥_{C (I; H_{D}^{1})} + ∥ f ∥_{L^{1} (I; H_{D}^{1})}$ , the free homogeneous and retarded estimates yield $∥ u ∥_{L^{4} (I; L^{\infty})} \leq C_{T} (1 + T K e^{K T}) (∥ h ∥_{H_{D}^{1}} + ∥ f ∥_{L^{1} (I; H_{D}^{1})})$ . Combining this with step 1.1 gives the claimed $X_{I}$ estimate. Setting $f = 0$ or $h = 0$ gives the two separate operator estimates.

3.1F4F6step 2.1step 1.2given∎

For the trajectory coefficients, the multiplier estimate gives $∥ B (t) - B (s) ∥_{B_{R} (H_{D}^{j})} \leq C \sum_{k = 1}^{2} ∥ V_{k} (t) - V_{k} (s) ∥_{W^{2, \infty}}$ for $j = 1, 2$ , using the conjugation isometry and multiplication by $i$ . Thus $B$ is norm-continuous and bounded on both levels, and its actions are consistent. The same multiplier result preserves $H_{0}^{1}$ and $H^{2} \cap H_{0}^{1}$ , so the Dirichlet condition is retained. Applying the previous steps completes the proof.

Depends on

Used by · 1

Linearized propagator bounds in the nonlinear control spaces

Dependency tree · 2 levels

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