Manuscript X-D@3 · X-D-v3.tex · lemma · b7774529

Lemma: Normal-flux form

D2: normal-flux estimate and the profile lemma › Integrating before estimating · lines 658–705 · proof included or linked

X-D@3#lem:flux, X-D@3#eq:flux-form

Suppose

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

Then

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 .

If $F, H$ are uniformly bounded in every spatial smooth seminorm on $0 \leq t \leq ε$ , then

u = O (ε), \partial_{t} u + F = O (ε)

in every local spatial smooth seminorm, uniformly on that layer. No time derivative of $F$ is needed for these bounds.

Proof. Integrate the zero-data Duhamel formula once by parts in time. The initial term vanishes because $F (0) = 0$ .

Fourier multipliers give

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

Consequently, if $M_{r} (F)$ and $M_{r} (H)$ denote the corresponding suprema on the layer,

∥ u (t) ∥_{r} ∥ \partial_{t} u (t) + F (t) ∥_{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) .

Spatial Sobolev embedding proves the stated smooth estimates. The bounds integrate the flux rather than differentiate the ramp before estimating.

Displayed formulas in this passage

X-D@3#eq:flux-form · line 666
$u (t) \partial_{t} u (t) + F (t) = - \int_{0}^{t} C (t - s) F (s) d s - \int_{0}^{t} S (t - s) H (s) d s, = - \int_{0}^{t} C^{'} (t - s) F (s) d s - \int_{0}^{t} C (t - s) H (s) d s .$
unlabeled · line 696
$∥ u (t) ∥_{r} ∥ \partial_{t} u (t) + F (t) ∥_{r} \leq t M_{r} (F) + \frac{1}{2} t^{2} M_{r} (H), \leq \frac{1}{2} t^{2} M_{r + 2} (F) + t M_{r} (H) .$
unlabeled · line 661
$□ u = - \partial_{t} F - H, u (0) = \partial_{t} u (0) = 0, F (0) = 0.$
unlabeled · line 677
$u = O (ε), \partial_{t} u + F = O (ε)$
unlabeled · line 689
$∥ C (τ) b ∥_{r} \leq ∥ b ∥_{r}, ∥ S (τ) b ∥_{r} \leq τ ∥ b ∥_{r}, ∥ C^{'} (τ) b ∥_{r} \leq τ ∥ b ∥_{r + 2} .$