Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

===== Let s>3/2s>3/2s>3/2. Suppose =====

u∈C([0,T];Hs(R3)).u\in C([0,T];H^{s}(\mathbb{R}^3)).u∈C([0,T];Hs(R3)).
Define the bilinear map B(u,v)=P∇ ⁣⋅(u⊗v)B(u,v)=P\nabla\!\cdot(u\otimes v)B(u,v)=P∇⋅(u⊗v). Then for every fixed t∈(0,T]t\in(0,T]t∈(0,T] the HsH^sHs-valued function

Φ(s):=Gt−s B(u(s),u(s)),s∈[0,t],\Phi(s):=G_{t-s}\,B\big(u(s),u(s)\big),\qquad s\in[0,t],Φ(s):=Gt−sB(u(s),u(s)),s∈[0,t],
is strongly measurable on [0,t][0,t][0,t] and belongs to L1([0,t];Hs(R3))L^1([0,t];H^s(\mathbb{R}^3))L1([0,t];Hs(R3)). Moreover there is a constant CCC (depending on ν,s\nu,sν,s) such that for a.e. s∈[0,t]s\in[0,t]s∈[0,t]

∥Φ(s)∥Hs≤C (t−s)−1/2 ∥u(s)∥Hs2.\|\Phi(s)\|_{H^s}
\le C\,(t-s)^{-1/2}\,\|u(s)\|_{H^s}^2.∥Φ(s)∥Hs≤C(t−s)−1/2∥u(s)∥Hs2.
In particular Φ\PhiΦ is Bochner-integrable on [0,t][0,t][0,t] since ∫0t(t−s)−1/2 ds<∞\int_0^t (t-s)^{-1/2}\,ds<\infty∫0t(t−s)−1/2ds<∞.

====== 1. Product estimate (algebra property). For s>3/2s>3/2s>3/2 we have the continuous product embedding Hs×Hs→HsH^s\times H^s\to H^sHs×Hs→Hs. Hence ======

∥u(s)⊗u(s)∥Hs≤C1∥u(s)∥Hs2.\|u(s)\otimes u(s)\|_{H^s} \le C_1 \|u(s)\|_{H^s}^2.∥u(s)⊗u(s)∥Hs≤C1∥u(s)∥Hs2.
Applying ∇ ⁣⋅\nabla\!\cdot∇⋅ reduces regularity by one:

∥B(u,u)(s)∥Hs−1=∥P∇ ⁣⋅(u⊗u)(s)∥Hs−1≤C2∥u(s)∥Hs2,\|B(u,u)(s)\|_{H^{s-1}} = \|P\nabla\!\cdot(u\otimes u)(s)\|_{H^{s-1}}
\le C_2 \|u(s)\|_{H^s}^2,∥B(u,u)(s)∥Hs−1=∥P∇⋅(u⊗u)(s)∥Hs−1≤C2∥u(s)∥Hs2,
because PPP is bounded on Sobolev spaces.
# Heat-semigroup gain estimate. The heat semigroup satisfies the smoothing bound

∥Gτf∥Hs≤C3 τ−1/2 ∥f∥Hs−1,τ>0,\|G_\tau f\|_{H^{s}} \le C_3\,\tau^{-1/2}\,\|f\|_{H^{s-1}},\qquad \tau>0,∥Gτf∥Hs≤C3τ−1/2∥f∥Hs−1,τ>0,
which is the r=1r=1r=1 case of the general estimate ∥Gτf∥Hs+r≤Cτ−r/2∥f∥Hs\|G_\tau f\|_{H^{s+r}}\le C\tau^{-r/2}\|f\|_{H^s}∥Gτf∥Hs+r≤Cτ−r/2∥f∥Hs.
# Combine the two with τ=t−s\tau=t-sτ=t−s:

∥Φ(s)∥Hs=∥Gt−sB(u,u)(s)∥Hs≤C3(t−s)−1/2∥B(u,u)(s)∥Hs−1≤C (t−s)−1/2∥u(s)∥Hs2.\|\Phi(s)\|_{H^s} = \|G_{t-s} B(u,u)(s)\|_{H^s}
\le C_3 (t-s)^{-1/2} \|B(u,u)(s)\|_{H^{s-1}}
\le C\,(t-s)^{-1/2}\|u(s)\|_{H^s}^2.∥Φ(s)∥Hs=∥Gt−sB(u,u)(s)∥Hs≤C3(t−s)−1/2∥B(u,u)(s)∥Hs−1≤C(t−s)−1/2∥u(s)∥Hs2.
# Since u∈C([0,T];Hs)u\in C([0,T];H^s)u∈C([0,T];Hs), the function s↦∥u(s)∥Hs2s\mapsto \|u(s)\|_{H^s}^2s↦∥u(s)∥Hs2 is bounded on [0,t][0,t][0,t]. Therefore the right-hand side is dominated by a constant times (t−s)−1/2(t-s)^{-1/2}(t−s)−1/2, which is integrable on [0,t][0,t][0,t]. This proves Bochner integrability.

Strong measurability follows from continuity of the composition s↦B(u,u)(s)s\mapsto B(u,u)(s)s↦B(u,u)(s) into Hs−1H^{s-1}Hs−1 and continuity of the map s↦Gt−ss\mapsto G_{t-s}s↦Gt−s into bounded operators on Hs−1H^{s-1}Hs−1 for s<ts<ts<t; standard measurability results then imply Φ\PhiΦ is strongly measurable on [0,t][0,t][0,t].

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