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

=== We consider the incompressible Navier–Stokes equations on R3\mathbb{R}^3R3 (the standard Millennium formulation is equivalent on R3\mathbb{R}^3R3 or the 3-torus). Let u(x,t):R3×[0,∞)→R3u(x,t):\mathbb{R}^3\times[0,\infty)\to\mathbb{R}^3u(x,t):R3×[0,∞)→R3 and pressure p(x,t)p(x,t)p(x,t) solve ===

{∂tu−νΔu+(u⋅∇)u+∇p=0,∇⋅u=0,u(⋅,0)=u0,\begin{cases}
\partial_t u - \nu\Delta u + (u\cdot\nabla)u + \nabla p = 0,\\[4pt]
\nabla\cdot u = 0,\\[4pt]
u(\cdot,0)=u_0,
\end{cases}⎩⎨⎧∂tu−νΔu+(u⋅∇)u+∇p=0,∇⋅u=0,u(⋅,0)=u0,
with viscosity ν>0\nu>0ν>0 and divergence–free smooth initial data u0u_0u0 (rapidly decaying / Schwartz or in the usual function class used in the Millennium problem). We aim to prove: for every smooth, divergence-free initial u0u_0u0 the Navier–Stokes system has a unique, global-in-time smooth solution u(x,t)u(x,t)u(x,t) — subject to the three premises you gave.

I will refer to your premises as:
# (P1) Linear extension of limit (L): the ordinary limit functional lim has been linearly continued to a functional LLL defined on all sequences (or on a much larger space), and hence convergence/limit operations can be applied in a linear way to sequences/series that do not classically converge.
# (P2) Linear extension of derivatives/integrals/series: using LLL one obtains linear extensions of differentiation, integration and summation so that every function has a well-defined derivative, integral and series expansion in this extended sense. In particular one may form termwise derivatives and integrals of series and pass to the extended limit LLL.
# (P3) Preservation of algebraic identities and multiplication: on the extended space the same algebraic identities that held in the classical domain still hold; in particular generalized functions in this space may be multiplied arbitrarily, and multiplication agrees with classical multiplication where both are classically defined.

I will show: under (P1)–(P3), Navier–Stokes admits a (unique) global classical solution for smooth initial data.