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

==== Equation (1) is linear in uuu with a known inhomogeneity −P∇ ⁣⋅(u⊗u)-P\nabla\!\cdot(u\otimes u)−P∇⋅(u⊗u). Treat it as ====

∂tu−νΔu=−F(t),F(t):=P∇ ⁣⋅(u⊗u)(t).\partial_t u - \nu\Delta u = -F(t),\qquad F(t):=P\nabla\!\cdot(u\otimes u)(t).∂tu−νΔu=−F(t),F(t):=P∇⋅(u⊗u)(t).
For the linear homogeneous heat equation ∂tv−νΔv=0\partial_t v - \nu\Delta v = 0∂tv−νΔv=0 with initial data v(0)=v0v(0)=v_0v(0)=v0, the solution is v(t)=Gtv0v(t)=G_t v_0v(t)=Gtv0. For the inhomogeneous equation, the Duhamel principle (variation-of-constants formula) gives the unique solution (for sufficiently regular FFF) as

u(t)=Gtu0+∫0tGt−s(−F(s)) ds=Gtu0−∫0tGt−sF(s) ds.u(t) = G_t u_0 + \int_0^t G_{t-s}\big(-F(s)\big)\,ds
= G_t u_0 - \int_0^t G_{t-s} F(s)\,ds.u(t)=Gtu0+∫0tGt−s(−F(s))ds=Gtu0−∫0tGt−sF(s)ds.
Substituting F=P∇ ⁣⋅(u⊗u)F= P\nabla\!\cdot(u\otimes u)F=P∇⋅(u⊗u) yields exactly the mild formula

u(t)=eνtΔu0−∫0teν(t−s)ΔP∇ ⁣⋅(u⊗u)(s) ds.u(t)=e^{\nu t\Delta}u_0 - \int_0^t e^{\nu (t-s)\Delta} P\nabla\!\cdot (u\otimes u)(s)\,ds.u(t)=eνtΔu0−∫0teν(t−s)ΔP∇⋅(u⊗u)(s)ds.
This derivation is valid provided:
'' the convolution Gt−s∗F(s)G_{t-s} '' F(s)Gt−s∗F(s) is defined (e.g. F(s)F(s)F(s) lies in a space where Gt−sG_{t-s}Gt−s acts boundedly), and
* we may interchange integration in time with spatial convolution (standard if FFF is continuous into a Banach space).

Thus any sufficiently regular solution of the PDE satisfies (M). This completes the PDE ⇒\Rightarrow⇒ mild implication.