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

==== Fix a Banach space (X,∥⋅∥X)(X,\|\cdot\|_X)(X,∥⋅∥X) of vector fields on R3\mathbb{R}^3R3 with these properties: ====

(H1) Heat semigroup bounds. There exist constants CG>0C_G>0CG>0 and α∈[0,1)\alpha\in[0,1)α∈[0,1) such that for all τ∈(0,T]\tau\in(0,T]τ∈(0,T]

∥eντΔ∥X←Y≤CG τ−α,\|e^{\nu \tau\Delta}\|_{X\leftarrow Y} \le C_G\,\tau^{-\alpha},∥eντΔ∥X←Y≤CGτ−α,
for some Banach space YYY with a continuous embedding Y↪XY\hookrightarrow XY↪X. (Often Y=X−1Y=X_{-1}Y=X−1 or Hs−1H^{s-1}Hs−1 while X=HsX=H^sX=Hs; typical α=1/2\alpha=1/2α=1/2.)

(H2) Product estimate (algebraic control). There is a constant CBC_BCB such that 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

∥B(u,v)∥Y≤CB∥u∥X∥v∥Xfor all u,v∈X.\|B(u,v)\|_{Y} \le C_B \|u\|_{X}\|v\|_{X}\qquad\text{for all }u,v\in X.∥B(u,v)∥Y≤CB∥u∥X∥v∥Xfor all u,v∈X.
(H3) XXX-continuity in time. We consider functions u∈C([0,T];X)u\in C([0,T];X)u∈C([0,T];X). (You can weaken this, but this is the standard mild-solution class.)

These three hypotheses are exactly the ones used in standard mild/Picard proofs (for example take X=HsX=H^sX=Hs, s>3/2s>3/2s>3/2, Y=Hs−1Y=H^{s-1}Y=Hs−1, then α=1/2\alpha=1/2α=1/2 and (H2) holds).