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

=== This is the standard route when you work in a Banach space XXX chosen so the nonlinear term is well behaved (typical choice: X=Hs(R3)X=H^s(\mathbb R^3)X=Hs(R3) with s>3/2s>3/2s>3/2, so HsH^sHs is an algebra and P∇ ⁣⋅(u⊗u)P\nabla\!\cdot(u\otimes u)P∇⋅(u⊗u) maps Hs→Hs−1H^s\to H^{s-1}Hs→Hs−1, etc.). I will present a general abstract template you can instantiate for a concrete XXX. ===

Assume:
# Gt=eνtΔG_t=e^{\nu t\Delta}Gt=eνtΔ is a strongly continuous semigroup on XXX and there exists β∈[0,1)\beta\in[0,1)β∈[0,1) and constant CGC_GCG such that for all t∈(0,T]t\in(0,T]t∈(0,T]

∥Gt∥Y→X≤CG t−β,\|G_t\|_{Y\to X} \le C_G\, t^{-\beta},∥Gt∥Y→X≤CGt−β,
for some Banach space YYY into which P∇ ⁣⋅(⋅⊗⋅)P\nabla\!\cdot(\cdot\otimes\cdot)P∇⋅(⋅⊗⋅) maps (we allow Y=XY=XY=X in many cases).
# The bilinear map B(u,v):=P∇ ⁣⋅(u⊗v)B(u,v):=P\nabla\!\cdot(u\otimes v)B(u,v):=P∇⋅(u⊗v) satisfies a product estimate

∥B(u,v)∥Y≤CB∥u∥X∥v∥X.\|B(u,v)\|_{Y} \le C_B \|u\|_X\|v\|_X.∥B(u,v)∥Y≤CB∥u∥X∥v∥X.
(These two hypotheses are standard heat-kernel + algebraic embedding facts; for X=HsX=H^sX=Hs with s>32s>\tfrac32s>23 one gets β=0\beta=0β=0 and such an estimate with Y=Hs−1Y=H^{s-1}Y=Hs−1.)

Define the Picard map T\mathcal TT on the Banach space C([0,T];X)C([0,T];X)C([0,T];X) with sup norm ∥w∥CTX=sup⁡0≤t≤T∥w(t)∥X\|w\|_{C_TX}=\sup_{0\le t\le T}\|w(t)\|_X∥w∥CTX=sup0≤t≤T∥w(t)∥X by

(Tw)(t):=Gtu0−∫0tGt−sB(w(s),w(s)) ds.(\mathcal T w)(t) := G_t u_0 - \int_0^t G_{t-s} B(w(s),w(s))\,ds.(Tw)(t):=Gtu0−∫0tGt−sB(w(s),w(s))ds.
Estimate T\mathcal TT on a ball. For any w∈C([0,T];X)w\in C([0,T];X)w∈C([0,T];X),

∥∫0tGt−sB(w,w)(s) ds∥X≤∫0t∥Gt−s∥Y→X ∥B(w,w)(s)∥Y ds≤CGCB∫0t(t−s)−β ∥w∥CTX2 ds=CGCB∥w∥CTX2⋅t1−β1−β.\begin{aligned}
\Big\|\int_0^t G_{t-s}B(w,w)(s)\,ds\Big\|_X
&\le \int_0^t \|G_{t-s}\|_{Y\to X}\, \|B(w,w)(s)\|_Y\,ds\\
&\le C_G C_B \int_0^t (t-s)^{-\beta}\, \|w\|_{C_TX}^2\, ds\\
&= C_G C_B \|w\|_{C_TX}^2 \cdot \frac{t^{1-\beta}}{1-\beta}.
\end{aligned}∫0tGt−sB(w,w)(s)dsX≤∫0t∥Gt−s∥Y→X∥B(w,w)(s)∥Yds≤CGCB∫0t(t−s)−β∥w∥CTX2ds=CGCB∥w∥CTX2⋅1−βt1−β.
Taking supremum over t∈[0,T]t\in[0,T]t∈[0,T],

∥Tw∥CTX≤∥u0∥X+CGCB1−β T1−β ∥w∥CTX2.\|\mathcal T w\|_{C_TX} \le \|u_0\|_X + \frac{C_G C_B}{1-\beta}\, T^{1-\beta}\,\|w\|_{C_TX}^2.∥Tw∥CTX≤∥u0∥X+1−βCGCBT1−β∥w∥CTX2.
Now pick R=2∥u0∥XR=2\|u_0\|_XR=2∥u0∥X and choose T>0T>0T>0 small so that

CGCB1−β T1−β R≤12.\frac{C_G C_B}{1-\beta}\, T^{1-\beta}\,R \le \tfrac12.1−βCGCBT1−βR≤21.
Then for every www with ∥w∥CTX≤R\|w\|_{C_TX}\le R∥w∥CTX≤R we have ∥Tw∥CTX≤R\|\mathcal T w\|_{C_TX}\le R∥Tw∥CTX≤R; i.e. T\mathcal TT maps the closed ball BR⊂C([0,T];X)B_R\subset C([0,T];X)BR⊂C([0,T];X) into itself.

Contraction estimate: for w,w~∈BRw,\tilde w\in B_Rw,w~∈BR,

∥Tw−Tw~∥CTX≤sup⁡t≤T∫0t∥Gt−s∥Y→X ∥B(w,w)−B(w~,w~)∥Y ds≤CGT1−β1−β sup⁡s≤T∥B(w+w,  w−w~)∥Y≤CGCBT1−β1−β (∥w∥CTX+∥w~∥CTX)∥w−w~∥CTX≤2CGCBT1−βR1−β ∥w−w~∥CTX.\begin{aligned}
\|\mathcal T w - \mathcal T\tilde w\|_{C_TX}
&\le \sup_{t\le T}\int_0^t \|G_{t-s}\|_{Y\to X}\,\|B(w,w)-B(\tilde w,\tilde w)\|_Y\,ds\\
&\le C_G \frac{T^{1-\beta}}{1-\beta}\, \sup_{s\le T}\|B(w+w,\;w-\tilde w)\|_Y\\
&\le \frac{C_G C_B T^{1-\beta}}{1-\beta}\,( \|w\|_{C_TX}+\|\tilde w\|_{C_TX})\|w-\tilde w\|_{C_TX}\\
&\le \frac{2C_G C_B T^{1-\beta}R}{1-\beta}\,\|w-\tilde w\|_{C_TX}.
\end{aligned}∥Tw−Tw~∥CTX≤t≤Tsup∫0t∥Gt−s∥Y→X∥B(w,w)−B(w~,w~)∥Yds≤CG1−βT1−βs≤Tsup∥B(w+w,w−w~)∥Y≤1−βCGCBT1−β(∥w∥CTX+∥w~∥CTX)∥w−w~∥CTX≤1−β2CGCBT1−βR∥w−w~∥CTX.
By the choice of TTT this Lipschitz factor is <1<1<1, so T\mathcal TT is a contraction on BRB_RBR. Banach fixed point yields a unique fixed point u∈BR⊂C([0,T];X)u\in B_R\subset C([0,T];X)u∈BR⊂C([0,T];X). In particular

u∈C([0,T];X)⊂L∞(0,T;X),u\in C([0,T];X)\subset L^\infty(0,T;X),u∈C([0,T];X)⊂L∞(0,T;X),
and the L∞L^\inftyL∞-bound is ∥u∥L∞(0,T;X)≤R=2∥u0∥X\|u\|_{L^\infty(0,T;X)}\le R=2\|u_0\|_X∥u∥L∞(0,T;X)≤R=2∥u0∥X.

Remarks:
* This shows the L∞(0,T;X)L^\infty(0,T;X)L∞(0,T;X) property is not an independent assumption but a consequence of local existence via Picard.
* The smallness requirement is on TTT (local existence). Global existence requires additional a-priori estimates (energy, dissipation, etc.) to extend the time of existence.