Manuscript X-D-ST@1 · X-D-ST.tex · lemma · e0cf8fdb

Lemma: Integrated normal-source estimate

The shrinking layer: integrate the normal flux first › An elementary flux estimate · lines 669–726 · proof included or linked

X-D-ST@1#lem:flux, X-D-ST@1#eq:flux-form, X-D-ST@1#eq:flux-bounds

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Suppose

□ u = - \partial_{t} F - H, u (0) = \partial_{t} u (0) = 0, F (0) = 0.

For smooth spatially localized data,

u (t) \partial_{t} u (t) + F (t) = - \int_{0}^{t} C (t - s) F (s) d s - \int_{0}^{t} S (t - s) H (s) d s, = - \int_{0}^{t} C^{'} (t - s) F (s) d s - \int_{0}^{t} C (t - s) H (s) d s .

M_{r} (F) = 0 \leq s \leq ε sup ∥ F (s) ∥_{H^{r}},

and likewise for $H$ , then, for $0 \leq t \leq ε$ ,

∥ u (t) ∥_{H^{r}} ∥ \partial_{t} u (t) + F (t) ∥_{H^{r}} \leq t M_{r} (F) + \frac{1}{2} t^{2} M_{r} (H), \leq \frac{1}{2} t^{2} M_{r + 2} (F) + t M_{r} (H) .

No time derivative of $F$ occurs in these bounds.

Proof. The zero-data wave formula is

u (t) = - \int_{0}^{t} S (t - s) \partial_{s} F (s) d s - \int_{0}^{t} S (t - s) H (s) d s .

Integrating its first integral by parts, using $S (0) = 0$ , $F (0) = 0$ , and $\partial_{s} S (t - s) = - C (t - s)$ , gives the first line of (eq:flux-form). Differentiation gives the second line.

The Fourier multiplier bounds are

∥ C (τ) b ∥_{H^{r}} \leq ∥ b ∥_{H^{r}}, ∥ S (τ) b ∥_{H^{r}} \leq τ ∥ b ∥_{H^{r}},

and

∥ C^{'} (τ) b ∥_{H^{r}} \leq τ ∥ b ∥_{H^{r + 2}} .

Integrating them proves (eq:flux-bounds).

Displayed formulas in this passage

X-D-ST@1#eq:flux-form · line 677
$u (t) \partial_{t} u (t) + F (t) = - \int_{0}^{t} C (t - s) F (s) d s - \int_{0}^{t} S (t - s) H (s) d s, = - \int_{0}^{t} C^{'} (t - s) F (s) d s - \int_{0}^{t} C (t - s) H (s) d s .$
X-D-ST@1#eq:flux-bounds · line 692
$∥ u (t) ∥_{H^{r}} ∥ \partial_{t} u (t) + F (t) ∥_{H^{r}} \leq t M_{r} (F) + \frac{1}{2} t^{2} M_{r} (H), \leq \frac{1}{2} t^{2} M_{r + 2} (F) + t M_{r} (H) .$
unlabeled · line 672
$□ u = - \partial_{t} F - H, u (0) = \partial_{t} u (0) = 0, F (0) = 0.$
unlabeled · line 688
$M_{r} (F) = 0 \leq s \leq ε sup ∥ F (s) ∥_{H^{r}},$
unlabeled · line 705
$u (t) = - \int_{0}^{t} S (t - s) \partial_{s} F (s) d s - \int_{0}^{t} S (t - s) H (s) d s .$
unlabeled · line 716
$∥ C (τ) b ∥_{H^{r}} \leq ∥ b ∥_{H^{r}}, ∥ S (τ) b ∥_{H^{r}} \leq τ ∥ b ∥_{H^{r}},$
unlabeled · line 721
$∥ C^{'} (τ) b ∥_{H^{r}} \leq τ ∥ b ∥_{H^{r + 2}} .$

Lemma: Integrated normal-source estimate

Displayed formulas in this passage

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