Existence and Smoothness of Elementary Weak Solutions of the Navier–Stokes Equations: Elementary Partial Differential Equations Theory

We construct the solutions depend on external forces, not depend on initial values. Our solutions are unique, smooth, analytic, time global for small initial values. The method is similar to the Banach fixed-point theorem, but we do not assume that the image of domain is contained into the domain itself. Our fixed point theorem can be used to our Navier-Stokes argment even if $${u_n\in S}$$ does not hold. We solve $${\partial_t u-\Delta u=Pf-P((u_n\cdot\nabla)u_n)}$$, $${\partial_t u-\Delta u=Pf-P((u\cdot\nabla)u)}$$ in $${X=\bigcap_{m\ge 5, p=1, 2}W^{m, p}(Ω)}$$ (as a set). We construct $${u=\Phi[u]\in S}$$, where $${S}$$ is a closed subset of $${X}$$. By new fixed point theorem, existence of solutions holds even if $${\Phi[S]\subset S}$$ does not hold. Our solutions are weak solution: $${\langle \partial_t u-\Delta u, \varphi\rangle =\langle f-\nabla p-(u\cdot\nabla)u,\varphi\rangle}$$ for any $${\varphi\in\mathcal D_\sigma(\Omega)}$$ because $${\langle PU, \varphi \rangle =\langle U, \varphi \rangle }$$ for any $${\varphi\in\mathcal D_\sigma(\Omega)}$$, $${\langle\mathrm{div}\,u,\varphi\rangle =0}$$ for any $${\varphi\in\mathcal D(\Omega)}$$. Uniqueness follows without boundary conditions. Smoothness, the energy inequality also follow. Our solutions have extension to $${\R×\R^3}$$, and the motions of fluids occurs by an external force and motions by fluids itselves, so our solutions are physically suitable. Our Elementary PDE Theory also can be applied to non-linear heat equation, non-linear Laplace equation, non-linear wave equation, non-linear Schrödinger equation, compressible Navier-Stokes equations and complex Monge-Ampére equation. We have done mathematics like Hitoshi Kitada: he corrected our book 数理解析学概論 (Introduction to Mathematical Analysis) more than 100 times with our illnesses (ISBN: 4768706622 ).