An Elementary Solution of the Navier-Stokes Existence and Smoothness

(November 23, 2024 revised)
( MasatoshiOhrui1993@gmail.com )
Abstract

This is an elementary argment in the sense that there are no long or complicated calculations, and the theory of evolution equations is not used at all. Our initial values can be taken arbitrary large, our solutions are time global and physically suitable.

Introduction

The existence of the solutions is actually known. For example, Fujita-Kato Theory, Shibata Theory: Zhang [30], Charve-Danchin [31], Shibata-Miura [32]. The semi-group theory or apriori estimates are in these theories, but these are not elementary. The initial values can be taken arbitrary large in the Leray-Hopf's weak solutions, but the uniqueness and smoothness are unresolved. Semi-group theory or apriori estimates are not used in the proof of existence of the Leray-Hopf's weak solutions (for example, Wasao SIBAGAKI, Hisako RIKIMARU [29]), but it is not elementary, too. We define new weak solutions with uniqueness and smoothness, without semi-group theory or apriori estimates. The initial values mast be small in the Fujita-Kato theory or Shibata theory to show the solutions are time global, but our initial values can be taken arbitrary large and our solutions are time global. We apply locally solvability of the partial differential oparators with constant coefficients. The policy is, to let $${L}$$ be the heat operator $${\partial_t-\Delta}$$ in the initial value problem of the Navier-Stokes equations on $${\Omega}$$
$${\partial_t u -\Delta u=f - \nabla \mathfrak{p}-(u \cdot \nabla)u}$$
$${\mathrm{div}\,u=0}$$
$${u(0, x)=a(x)}$$,
to erase the pressure $${\mathfrak{p}}$$, to approximate the nonlinear term $${(u \cdot \nabla)u}$$ by a sequence of smooth functions, to use the locally solvability for the difference between the external force $${f}$$ and the approximation term, and to show that the limit in the Sobolev space is the solution. Our solutions are physically suitable: $${f\mapsto u, f\mapsto\mathfrak{p};a\mapsto u, a\mapsto\mathfrak{p}}$$ are continuous, and
$${\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0}$$.

[Definition of symbols]
For convenience, we write the index of the component of the vectors in the upper right corner. "Function space" and "space" are abbreviations for "linear topological space" (of functions or distributions), other than pressure $${\mathfrak{p}}$$ are $${\mathbb{R}^3}$$-valued. The absolute value of the functions in the norm of normal function space is interpreted as the length of the number vector (the absolute value of $${\mathbb{R}^3}$$) in the norm of the space of the $${\mathbb{R}^3}$$-value functions. We write the space of the real numeric functions and the space of the $${\mathbb{R}^3}$$-value functions in the same symbol to make symbols simple. For any positive number $${\delta}$$, let $${B_\delta(0, 0)}$$ be the $${\delta}$$-neighborhood of point $${(0, 0)}$$. Let $${|\Omega|}$$ be $${\Omega}$$'s Lebesgue measure. Let $${\chi_{\Omega}}$$ be the characteristic function on $${\Omega}$$. For any natural number $${m \gt \max\{0+4/1, 0+4/2\}=4}$$, $${p=1, 2}$$, let $${V_{\sigma}^{m, p}(\Omega)=\{ u \in C^{\infty}(\R\times\R^3) : \|u\|_{W^{m, p}(\Omega)} \lt {\infty},\mathrm{div}\,u=0 \},}$$ $${W_{\sigma}^{m, p}(\Omega)}$$ be the Sobolev space defined by $${V_{\sigma}^{m, p}(\Omega)}$$'s completion by norm of $${W_{\sigma}^{m, p}(\Omega)=\overline{V_{\sigma}^{m, p}(\Omega)}^{\| \cdot \|_{W^{m, p}(\Omega)}}}$$. Let $${\mathcal{D}(\Omega)}$$ be the space of the test functions ($${C_{0}^{\infty}(\Omega)}$$ as a set), let $${\mathcal{D}_\sigma(\Omega)}$$ be the space of the test functions that the divergence is $${0}$$ for spatial variables (see [Supplement 1]). Let $${P:L^2(\Omega)\to L^2_\sigma(\Omega)=\overline{\mathcal{D}_{\sigma}(\Omega)}^{\| \cdot \|_{L^2(\Omega)}}}$$ be the projection. Let $${C^{k, \varepsilon}(\overline{\Omega})}$$ be the Hölder space. Let
$${\langle w, \varphi \rangle = (w, \varphi)_{L^2(\Omega)}}$$
$${=\int_{\Omega} \sum_{i=1}^{3} w^{i}(t, x)\varphi^{i}(t, x)dtdx}$$
$${=\int_{\Omega} w(t, x) \cdot \varphi(t, x)dtdx}$$
$${(w=(w^1, w^2, w^ 3), \varphi=(\varphi^1, \varphi^2, \varphi^3))}$$.
In general, if for two Banach spaces $${X, Y}$$, there exists a linear Hausdorff space $${Z}$$ such that $${X, Y \subset Z}$$, then $${X\cap Y}$$ is a Banach space with norms given by $${\|u\|_X+\|u\|_Y}$$ or $${\max\{\|u\|_X, \|u\|_Y\}}$$. $${\max\{\|u\|_X, \|u\|_Y\}\le \|u\|_X+\|u\|_Y \le 2\max\{\|u\|_X, \|u\|_Y\}}$$ so these are equivalent. We put
$${X=\bigcap_{m=5}^\infty W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega),}$$
$${X'= \bigcap_{m=5}^\infty W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}$$.
We define for any $${u\in X}$$,
$${\|u\|_X=\sum_{m=5}^\infty \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$,
for any $${u\in X'}$$,
$${\|u\|_{X'}=\sum_{m=5}^\infty \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$.
For a constant $${M\gt 0}$$, let $${S}$$ be a subset of $${X}$$:
$${S=\{u\in X:\|u\|_{X}\le M\}}$$. Let the fundamental solution of $${\partial_t - \Delta}$$ be $${E}$$. That is, in the sence that $${\mathbb{R}^3}$$-valued distribution,
$${(\partial_t - \Delta)E(t, x)=\delta(t, x) = \delta(t) \otimes \delta(x)}$$.
Here,
$${E^{i}(t, x)=\begin{cases} \frac{1}{\sqrt{4 \pi t}^3} e^{-\frac{|x|^2}{4t}} & (t \gt 0) \\ 0 & (t \le 0) \end{cases}}$$.
(END)

Existence of elementary weak solutions

[Assumptions]
Let the domain $${\Omega}$$ be a bounded open set contained in $${\mathbb{R}\times\mathbb{R}^3}$$, have smooth boundary, and satisfy $${(0, 0)\in\Omega}$$. We assume that for any multi-index $${\alpha}$$, $${d_\alpha=\sup\{|((t, x)-(t', x'))^\alpha|:(t, x), (t', x')\in\Omega\}\ge 1}$$. Let the external force $${f:\mathbb{R}\times\mathbb{R}^3\to\R^3}$$ satisfies $${f\in S}$$ and $${\|f\|_X\le M^2}$$. 
Let the set of initial values be
$${A=\{u(0, \cdot):u \in S, u(t, x)=\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf(t-s, x-y) - P((u\cdot \nabla)u)(t-s, x-y))dsdy\}.}$$
(END)

[Locally solvability of linear partial differential operator with constant coefficients]
Let the fundamental solution of linear partial differential operator with constant coefficients $${L}$$ on $${\mathbb{R}^N}$$ be $${E}$$. $${E \in \mathcal{D}^{\prime}}$$ satisfies $${LE=\delta}$$. For $${f \in L^1_{\mathrm{loc}}}$$, one of the weak solutions of the equation $${Lu=f}$$ on $${\Omega\Subset\mathbb{R}^N}$$ is $${u=E * \chi_\Omega f \in \mathcal{D}^{\prime}(\Omega)}$$.
(END)

[Proof]
For any $${\varphi\in\mathcal{D}(\Omega)}$$,
$${\langle L(E*\chi_\Omega f), \varphi\rangle}$$
$${=\pm\langle E*\chi_\Omega f, L\varphi\rangle}$$
$${:=\pm\langle E(x), \langle \chi_\Omega(y)f(y), L\varphi(x+y)\rangle\rangle}$$
$${= \pm\langle \chi_\Omega(x)f(x), \langle E(y), L\varphi(x+y)\rangle\rangle}$$
$${= \langle \chi_\Omega(x)f(x), \langle LE(y), \varphi(x+y)\rangle\rangle}$$
$${=\langle LE(x), \langle \chi_\Omega(y)f(y), \varphi(x+y)\rangle\rangle}$$
$${=\langle \chi_\Omega(y)f(y), \varphi(y)\rangle =\langle f, \varphi\rangle}$$.
(END)

[Proposition]
$${A\neq\empty}$$. Let $${a\in A}$$. Then there are weak solutions $${u, \mathfrak{p}}$$ of the initial value problem on $${\Omega}$$:

$${\partial_t u -\Delta u=f - \nabla \mathfrak{p}-(u \cdot \nabla)u}$$
$${\mathrm{div}\,u=0}$$
$${u(0, x)=a(x)}$$,

in the sence that, $${u \in S,}$$ $${\mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/\{\mathfrak{p}':\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0\}}$$ and $${u, \mathfrak{p}}$$ satisfy for any $${\varphi \in \mathcal{D}_\sigma(\Omega)}$$,
$${\langle \partial_t u + (u \cdot \nabla)u - \Delta u + \nabla \mathfrak{p} - f, \varphi \rangle =0,}$$
for any $${\varphi\in\mathcal{D}(\Omega)}$$,
$${\langle\mathrm{div} \,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0}$$.

If $${f \neq 0}$$ then $${A\notni 0, u \neq 0}$$; if $${f=0}$$ then $${A=\{0\}, u=0}$$. $${u, \mathfrak{p}}$$ are measurable functions on $${\R\times\R^3}$$, so $${u, \mathfrak{p}}$$ are time global and $${\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0}$$.
(END)

f defines A. Our weak solutions do not depend on A.

We proof it later by Banach's fixed point theorem.

[Smoothness of elementary weak solutions]
Solution $${(u, \mathfrak{p})}$$ are $${C^{\infty}}$$-functions.

[Proof of smoothness]
$${m}$$ can be arbitrarily large, so the embedding theorem into Hölder space, 
"if $${\mathbb{N}\ni m-4/p\gt 0}$$ then $${W^{m, p}(\Omega)\subset C^{(m-4/p)-1, \varepsilon}(\overline{\Omega})}$$ for $${\varepsilon\in (0, 1)}$$", in the sence of existence of suitable representative elements, $${u}$$ is $${C^\infty}$$-function.

$${f}$$ is smooth and $${\partial_t u + (u \cdot \nabla)u - \Delta u - f=-\nabla \mathfrak{p}}$$. Because $${-\nabla \mathfrak{p}}$$ is smooth, so $${\mathfrak{p}}$$ is also smooth.
(END)

[$${X, X'}$$ are norm spaces]
$${\|u\|_X}$$ is finity.
[Proof]
By Sobolev embedding theorem, $${\partial^\alpha u}$$ is bounded. From Taylor's formula, $${|\partial^\alpha u(t, x)|\le C_\alpha \Rightarrow C_\alpha =O(\alpha!)\,(|\alpha|\to\infty)}$$. In fact, $${d_\alpha=\sup\{|((t, x)-(t', x'))^\alpha|:(t, x), (t', x')\in\Omega\}\ge 1}$$ and
$${\lim_{|\alpha|\to\infty}C_\alpha d_\alpha/\alpha!\le c}$$,
therefore
$${C_\alpha\le c\alpha!}$$ follows.
(END)

[Completeness]
$${X, X'}$$ are Banach spaces. $${\chi_{\Omega}\in X, \chi_{\Omega}\in X'}$$ so $${X, X'\neq \{0\}}$$.
(END)

[Proof]
Let $${\{u_n\}}$$ be a Cauchy sequence in $${X}$$. Then, for any $${m\ge 5}$$, $${\{u_n\}}$$ is a Caucy sequence of $${W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega)}$$. $${W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega)}$$ is a Banach space, so $${\{u_n\}}$$ converges. Let the limit be $${u}$$. $${u\in X}$$. For any positive number $${\varepsilon}$$, there exists a natural number $${N}$$ such that if $${\ell, n\ge N}$$ then
$${\|u_\ell-u_n\|_X \lt \varepsilon}$$.
From using Fatou's lemma for counting measure,
$${\|u-u_n\|_X}$$
$${=\sum_{m=5}^{\infty} \frac{1}{m!^5}\|u-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$
$${= \sum_{m=5}^{\infty} \frac{1}{m!^5}\liminf_{\ell\to\infty}\|u_\ell-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$
$${\le\liminf_{\ell\to\infty} \sum_{m=5}^{\infty} \frac{1}{m!^5}\|u_\ell-u_n\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$
$${\le \varepsilon}$$.
(END)

[Separation of product and absorption of differential]
Constants $${C_1, C_2 \gt 0}$$ exist such that
$${\left\|u^i v^i\right\|_{X'}\le C_1\|u^i\|_{X'}\|v^i\|_{X'}}$$
(separation of product)
holds for any $${u, v\in X'}$$, and
$${\left\|\partial_{x^j}u\right\|_{X}\le C_2\|u\|_{X}}$$
(absorption of differential)
holds for any $${u\in X}$$.
(END)

[Proof]
For the binomial coefficients $${c_{\alpha, \beta}}$$, let
$${c_{\alpha}=\sum_{\beta\le\alpha}c_{\alpha, \beta}}$$.
There is a continuous embedding $${X'\subset C^{k, \varepsilon}(\overline{\Omega})}$$ for any natural number $${k}$$, because $${\|u_n-u\|_X\to 0}$$
$${\Rightarrow \|u_n-u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)} \to 0}$$
$${\Rightarrow \|u_n-u\|_{C^{k, \varepsilon}(\overline{\Omega})}\to 0}$$, so there exists a constant $${c'\gt 0}$$ such that $${\|u\|_{C^{k, \varepsilon}(\overline{\Omega})}\le c'\|u\|_{X'}}$$. If $${|\alpha|\le k}$$, by Leibniz' formula,
$${\|\partial^\alpha (u^i v^i)\|_{L^p(\Omega)}}$$
$${\le c_{\alpha}\|u^i\|_{C^{k, \varepsilon}(\overline{\Omega})}\|v^i\|_{C^{k, \varepsilon}(\overline{\Omega})}|\Omega|^{1/p}}$$
$${\le c_{\alpha}c' |\Omega|^{1/p}\|u^i\|_{X'} c'\|v^i\|_{X'}}$$
$${\le c_{\alpha}c'^2 |\Omega|^{1/p}\|u^i\|_{X'}\|v^i\|_{X'}}$$. Therefore,
$${\|\partial^\alpha (u^i v^i)\|_{L^p(\Omega)}\le c_{\alpha}c'^2 |\Omega|^{1/p}\|u^i\|_{X'}\|v^i\|_{X'}}$$, so there exists a constant $${C_1\gt 0}$$ such that
$${\|u^i v^i\|_{X'}\le C_1\|u^i\|_{X'}\|v^i\|_{X'}}$$.

Let $${\{u_n\}\subset X}$$ satisfies $${u_n\to u, \partial_{x^j}u_n\to v}$$. From Hölder's inequality, we have
$${|\langle \partial_{x^j}u_n - v, \varphi\rangle|}$$
$${\le \|\partial_{x^j}u_n - v\|_{L^p(\Omega)}\|\varphi\|_{L^q(\Omega)}}$$
$${\to 0\,(p=1\Rightarrow q=\infty, p=2\Rightarrow q=2)}$$ and the weak differentiation is continuous in $${\mathcal{D}'_\sigma(\Omega)}$$, so $${\partial_{x^j}u_n\to \partial_{x^j}u\, \mathrm{in}\,\mathcal{D}'_\sigma(\Omega)}$$. From
$${v=\partial_{x^j}u\in X}$$, $${\{u\in X:\partial_{x^j}u\in X\}=X}$$, the absorption of differential is true by the closed graph theorem.

Or
$${\|\partial_{x^j}u\|_{X}= \sum_{m=5}^{\infty} \frac{1}{m!^5}\|\partial_{x^j}u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$
$${\le\sum_{m=4}^{\infty} \frac{1}{m!^5}\|u\|_{W^{m+1, 1}(\Omega)\cap W^{m+1, 2}(\Omega)}}$$
$${\le C_2\sum_{m=5}^{\infty} \frac{1}{m!^5}\|u\|_{W^{m, 1}(\Omega)\cap W^{m, 2}(\Omega)}}$$.
(END)

[Boundness of $${X\ni u\mapsto E*(\chi_\Omega u)\in X}$$]
$${X\ni u\mapsto E*(\chi_\Omega u)\in X}$$ is a bounded operator, so constant $${C_3\gt 0}$$ exists such that for any $${u\in X}$$,
$${\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y)\chi_\Omega(t-s, x-y)u(t-s, x-y)dsdy\|_{X}}$$
$${\le C_3\|u\|_X}$$
holds.
(END)

[Proof]
As a function of $${(s, y)}$$, for any $${(t, x)\in\Omega}$$,
$${\mathrm{supp}(E^i(s, y) \chi_\Omega(t-s, x-y)u^i(t-s, x-y))}$$
$${\subseteq -\overline{\Omega}+(t, x)}$$
$${=\overline{\{(s, y)\in \mathbb{R}\times\mathbb{R}^3:(t-s, x-y)\in\Omega\}}}$$
is the translation of reverse of $${\overline{\Omega}}$$, so it is compact, and
$${|\partial_{t, x}^\alpha(E^i(s, y) \chi_\Omega(t-s, x-y)u^i(t-s, x-y))|\le E^i(s, y)\sup\{|\partial_{t, x}^\alpha u^i(t-s, x-y)|:(t, x)\in\Omega\}\le C_\alpha E^i(s, y)\in L^1_{s, y}(\Omega)}$$, so combine the theorem of differentiation under the integral sign, Hölder's inequality and continues embedding $${X\subset L^\infty(\Omega)}$$,
$${\|\partial^\alpha(E*(\chi_{\Omega}u))\|_{L^p(\Omega)}}$$
$${\le\|E*(\partial^\alpha (\chi_{\Omega}u))\|_{L^p(\Omega)}}$$
$${\le \|\|E(s, y)\|_{L_{s, y}^1(-\Omega+(t, x))}\|\partial^\alpha u(t-s, x-y)\|_{L_{s, y}^\infty(-\Omega+(t, x))}\|_{L_{t, x}^p(\Omega)}}$$
$${\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}\|\partial^\alpha u\|_{L^\infty(\Omega)}|\Omega|^{1/p}}$$
$${\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}c''C_2^{|\alpha|}\|u\|_X|\Omega|^{1/p}}$$
$${\lt\infty}$$.
So we have
$${\|E*(\chi_\Omega u)\|_X\le C_3\|u\|_X.}$$
(END)

We take $${C=\max\{C_1, C_2, C_3\}}$$. The separation of product, the absorption of differential, and the boundness of $${X\ni u\mapsto E*(\chi_\Omega u)\in X}$$ hold for $${C}$$.

We solve
(N-S)'$${\partial_t u -\Delta u=f -(u \cdot \nabla)u}$$,
that is, for any $${a\in A}$$, there exist $${u \in S,}$$ $${\mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/\{\mathfrak{p}':\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0\}}$$, such that for any $${\varphi \in \mathcal{D}_\sigma(\Omega)}$$,
$${\langle \partial_t u + (u \cdot \nabla)u - \Delta u + \nabla \mathfrak{p} - f, \varphi \rangle =0,}$$
for any $${\varphi\in\mathcal{D}(\Omega)}$$,
$${\langle\mathrm{div} \,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0}$$,
$${u(0, x)=a(x)}$$.

$${\varPhi:S\to S}$$ can be defined as
$${\varPhi[u](t, x)}$$
$${=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u\cdot\nabla)u)(t-s, x-y))dsdy}$$. We take a function sequence $${\{u_n\}\subset S}$$ as $${u_0:\Omega\to\R^3}$$ and $${u_0\in S}$$, if $${n\ge 0}$$ then $${u_{n+1}(t, x)=\varPhi[u_n](t, x)}$$
$${=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y) \chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u_n\cdot\nabla)u_n)(t-s, x-y))dsdy}$$. If $${X}$$ is a complete metric space, then $${S}$$ is complete because it is a closed subset that is not empty, and if $${\varPhi}$$ is a contraction mapping, according to the Banach's fixed point theorem, the uniqueness and the existence of a fixed point of $${\varPhi}$$ follows:

Some $${u \in S}$$ exists uniquely and $${\varPhi[u]=u}$$.

Then, due to the uniqueness of the fixed point in Banach's fixed point theorem, $${u}$$ is an unique weak solution. If $${f \neq 0}$$ then $${A\notni 0, u \neq 0}$$. Let $${f=0}$$. From the properties of $${X}$$, if $${u\in X}$$ and $${u(t, x)=\int_{\mathbb{R}\times\mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) -P((u\cdot\nabla)u)(t-s, x-y))dsdy}$$ then $${\|u\|_X\le 3C^3\|u\|_X^2}$$. So, if $${u\neq 0}$$ then $${1\le 3C^3\|u\|_X}$$. By $${C=O(|\Omega|)\,(|\Omega|\to 0)}$$ and the absolute continuity of Lebesgue integral, $${|\Omega|\to 0\Rightarrow C\to 0, \|u\|_X\to 0}$$, therefore $${f=0\Rightarrow A=\{0\}, u=0}$$. $${u, \mathfrak{p}}$$ are measurable functions on $${\R\times\R^3}$$, so $${u, \mathfrak{p}}$$ are time global.

[Proof of the possibility that $${\varPhi}$$ can be defined as a contraction mapping]
$${u\in S\Rightarrow \|E*(\chi_{\Omega}(Pf-P((u\cdot \nabla)u)))\|_X\lt\infty}$$
holds. Therefore
$${\|\varPhi[u]\|_X\le M.}$$

$${\|P\|=1}$$, so
$${\|\chi_{\Omega}(Pf-P((u\cdot\nabla)u))\|_X}$$
$${\le\|f\|_X+\|u^1 \partial_{x^1}u+u^2 \partial_{x^2}u+u^3 \partial_{x^3}u\|_X}$$
$${\le M^2+3C^2M^2\lt\infty}$$.

If
$${\|\varPhi[u]\|_X}$$
$${\le CM^2+3C^3M^2}$$
$${\le M}$$, $${M}$$ must be $${C(1+3C^2)M\le 1}$$.
(END)

$${\varPhi:S\to S}$$ may be Lipschitz continuous: there may be a constant $${L\gt 0}$$ such that $${\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y) \chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla)v)(t-s, x-y)-P((u\cdot\nabla)u)(t-s, x-y))dsdy\|_X}$$
$${\le L \|u- v\|_X}$$.

If the Lipschitz continuity established,

$${\|\varPhi[u]-\varPhi[v]\|_X}$$
$${\le\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y)\chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla)v)(t-s, x-y)-P((u\cdot\nabla)u)(t-s, x-y))dsdy\|_X}$$
$${\le L\|u-v\| _X}$$
follows. Here,

[$${\varPhi}$$ may be a contraction mapping]
$${L \lt 1}$$

holds.

[Proof of Lipschitz continuity]
$${(v \cdot \nabla)v(t-s, x-y)-(u \cdot \nabla)u(t-s, x-y)}$$
$${=\sum_{j=1}^3 (v^j (\partial_{x^j}v(t-s, x-y) - \partial_{x^j}u(t-s, x-y)) + (v^j \partial_{x^j}u(t-s, x-y)) - (u^j \partial_{x^j}u(t-s, x-y)))}$$, so we have

$${\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y) \chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla)v)(t-s, x-y)-P((u\cdot\nabla)u)(t-s, x-y))dsdy\|_X}$$
$${\le C^2\|v\|_X\max_j(\|\partial_{x^j}v - \partial_{x^j}u\|_X)+C^2\|v-u\|_X\max_j(\|\partial_{x^j}u\|_X)}$$
$${\le C^3M\|v-u\|_X+C^3M\|v-u\|_X}$$
$${= 2C^3M\|u- v\|_X.}$$

Therefore, Lipschitz continuity follows for $${L=2C^3M}$$.
(END)

[Proof of the possibility that $${\varPhi}$$ is a contraction mapping]
From the above argment
$${\|\int_{\mathbb{R}\times\mathbb{R}^3}E(s, y)\chi_{\Omega}(t-s, x-y)(P((v \cdot \nabla) v(t-s, x-y))-P((u \cdot \nabla)u)(t-s, x-y))dsdy\|_X}$$
$${\le 2C^3M\|u- v\|_X}$$
and
$${2C^3M\lt 1.}$$
(END)

[Solvability of the Navier-Stokes equations]
The fixed point $${u}$$ of $${\varPhi:S\to S}$$ is the solution of (N-S)'.

[proof]
$${u}$$ satisfies $${\mathrm{div}\,u=0}$$ in the sense of a distribution belonging to $${\mathcal{D}'(\Omega)}$$. That is, for any $${\varphi\in\mathcal{D}(\Omega)}$$, $${\langle\mathrm{div}\,u, \varphi\rangle=-\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle=0}$$.
In fact, for any $${u\in W_\sigma^{m, p}(\Omega)}$$ there exists a Cauchy sequence $${\{u_n\}\subset V_\sigma^{m, p}(\Omega)}$$, by the integration by parts and Hölder's inequality, we have
$${0=-\sum_{j=1}^3\langle u_n^j, \partial_{x^j}\varphi\rangle}$$
$${\to -\sum_{j=1}^3\langle u^j, \partial_{x^j}\varphi\rangle}$$.

For any $${\varphi\in\mathcal{D}_\sigma(\Omega)}$$,
$${\mathrm{div}(\varphi)=0}$$, so by the integration by parts
$${\langle \nabla\mathfrak{p}, \varphi\rangle}$$
$${=\int_{\Omega} \sum_{i=1}^{3} (\nabla\mathfrak{p})^i(t, x)\varphi^i(t, x)dtdx}$$
$${=-\int_{\Omega}\mathfrak{p}(t, x)\mathrm{div}(\varphi)(t, x)dtdx=0.}$$

Therefore, boundness of $${u, \partial_{x^j}u}$$ by Sobolev embedding theorem and $${|\Omega|\lt\infty}$$, we have $${(u\cdot\nabla)u\in L^2(\Omega)}$$, so by Helmholtz decomposition,
if we let $${f=Pf+\nabla\mathfrak{f}, (u\cdot\nabla)u=P((u\cdot\nabla)u)+\nabla\mathfrak{u}}$$
then for any $${\varphi\in\mathcal{D}_\sigma(\Omega)}$$,
$${\langle f, \varphi\rangle = \langle Pf, \varphi\rangle, \langle (u\cdot\nabla)u, \varphi\rangle =\langle P((u\cdot\nabla)u), \varphi\rangle}$$, hence we solve

(N-S)' $${\partial_t u - \Delta u= f -(u \cdot \nabla)u\,\mathrm{in}\, \mathcal{D}'_\sigma(\Omega)}$$.

By the locally solvability, the solution of the approximate equation on $${\Omega}$$

(N-S)'' $${\partial_t v_{n} - \Delta v_{n} =Pf-P((u_n \cdot \nabla)u_n)}$$
satisfies
$${v_n=u_{n+1}=E * \chi_{\Omega}(Pf -P((u_n \cdot \nabla)u_n))}$$.

Therefore, the solution of (N-S)'' satisfies
$${u_{n+1}(t, x)=\int_{\mathbb{R} \times \mathbb{R}^3} E(s, y) \chi_{\Omega}(t-s , x-y)(Pf(t-s, x-y) -P((u_n \cdot \nabla)u_n)(t-s, x-y))dsdy.}$$

$${\partial_t u_{n+1} (t, x)- \Delta u_{n+1} (t, x)}$$
$${=\langle(\partial_t E(t-s, x-y) - \Delta E(t-s, x-y)),\chi_{\Omega}(s, y)(Pf(s, y)-P((u_n \cdot \nabla)u_n)(s, y))\rangle}$$
$${=\langle\delta(\tau) \otimes \delta(z),\chi_{\Omega}(t-\tau, x-z)(Pf(t-\tau, x-z)-P((u_n \cdot \nabla)u_n)(t-\tau, x-z)) \rangle}$$
$${=Pf(t, x)-P((u_n \cdot \nabla)u_n)(t, x).}$$

Therefore, the above calculation and the continuity of the heat operator on $${\mathcal{D}'_\sigma(\Omega)}$$:
$${|\langle \partial_t u_{n+1} - \Delta u_{n+1}, \varphi \rangle - \langle \partial_t u - \Delta u, \varphi \rangle|\to 0}$$, and from Hölder's inequality, $${\|P\|=1}$$, and product of the functions $${L^2(\Omega)\times L^2(\Omega) \ni (u, v) \mapsto uv \in L^1(\Omega)}$$ is continuous (see [Supplement 2]), so
$${| \int_{\Omega} (P((u_n \cdot \nabla)u_n)(t, x)}$$
$${-P((u \cdot \nabla)u)(t, x))) \cdot \varphi(t, x) dtdx |}$$
$${\le \|((u_n \cdot \nabla)u_n)(t, x)-((u \cdot \nabla)u)(t, x)\|_{L^1(\Omega)}\| \varphi(t, x) \|_{L^\infty(\Omega)}\to 0\,(n \to \infty)}$$, hence
$${\partial_t u - \Delta u =Pf-P((u \cdot \nabla)u)}$$ holds, so we have
$${u(t, x)=\int_{\mathbb{R} \times \mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y) (Pf(t-s, x-y)-P((u \cdot \nabla)u)(t-s, x-y))dsdy}$$.

$${u}$$ is a solution in the sence of distribution in $${\mathcal{D}_\sigma'(\Omega)}$$ of (N-S)' .

For any $${U\in\mathcal{D}_\sigma'(\Omega)}$$,
" $${\varphi\in\mathcal{D}_\sigma(\Omega)\Rightarrow\langle U, \varphi\rangle =0}$$ "
$${\iff}$$ " there exists a distribution $${\mathfrak{p}}$$ such that $${U=\nabla\mathfrak{p}}$$ "
by Helmholtz decomposition, therefore there exists $${\mathfrak{p}}$$ such that $${\partial_t u + (u \cdot \nabla)u - \Delta u - f=-\nabla \mathfrak{p}}$$ holds.
(END)

Properties of the solutions

[Continuity of $${f\mapsto u, f\mapsto\mathfrak{p}}$$]
Let $${f_n, f\in S, \|f_n\|_X, \|f\|\le M^2, \|f_n-f\|_X\to 0}$$. Let the solutions be $${u_n,\mathfrak{p}_n}$$ for $${f_n}$$ and $${a_n\in A}$$, let the solutions be $${u,\mathfrak{p}}$$ for $${f}$$ and $${a\in A}$$. Then
$${\|u_n-u\|_X\to 0,}$$
$${d(\mathfrak{p}_n, \mathfrak{p}):=\|u_n-u\|_X\to 0}$$.
(END)

[Proof]
$${\|u_n-u\|_X=\|\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf_n(t-s, x-y) - P((u_n\cdot \nabla)u_n)(t-s, x-y))dsdy-\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf(t-s, x-y) - P((u\cdot \nabla)u)(t-s, x-y))dsdy\|_X}$$
$${\le C\|f_n-f\|_X+2C^3M\|u_n-u\|_X}$$.
So
$${\limsup_{n\to\infty}\|u_n-u\|_X}$$
$${\le 2C^3M\limsup_{n\to\infty}\|u_n-u\|_X}$$.

$${\limsup_{n\to\infty}\|u_n-u\|_X\le 2M}$$,
therefore
$${0\le (1-2C^3M)\limsup_{n\to\infty}\|u_n-u\|_X}$$
$${\le 0}$$.
Hence
$${\limsup_{n\to\infty}\|u_n-u\|_X}$$
$${=\lim_{n\to\infty}\|u_n-u\|_X=0}$$.
There exist the maps $${f\mapsto u, u\mapsto\mathfrak{p}}$$ so $${d(\mathfrak{p}_n, \mathfrak{p}):=\|u_n-u\|_X\to 0}$$.
(END)

[Continuity of $${a\mapsto u, a\mapsto\mathfrak{p}}$$]
Let the solutions be $${u_a, v_b, \mathfrak{p}_a, \mathfrak{q}_b}$$ for $${a, b}$$. If we define the metrics given by
$${d_A(a, b)=\|u_a-v_b\|_X, D(\mathfrak{p}, \mathfrak{q})=\|u_a-v_b\|_X}$$, then $${a\mapsto u, a\mapsto\mathfrak{p}}$$ are continuous.
(END)

[Vanishing]
$${\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0}$$.
(END)

[Proof]
$${u}$$ is a measurable function on $${\R\times\R^3}$$, so we can take the limits as $${t, |x|\to\infty}$$.

$${\partial^\alpha u(t, x)=\int_{\Omega} E(t-s, x-y) \, \partial^\alpha(Pf(s, y) - P((u\cdot \nabla)u)(s, y))dsdy}$$, for any $${t_0\gt 0}$$, if $${t-s\gt t_0}$$ then
$${|E^i|\le 1/t_0^{3/2},}$$
$${\partial^\alpha(Pf- P((u\cdot \nabla)u))\in X\subset C^{0, \varepsilon}(\overline{\Omega})}$$
so $${\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0}$$ follows from the bounded convergence theorem.
(END)

From the properties of $${X}$$ and $${u=\varPhi[u]}$$,
$${\|u\|_X\le C\|f\|_X +3C^3\|u\|_X^2\le M}$$.
$${CM\le C(1+3C^2)M\le 1}$$
so
$${C\|f\|_X\le CM^2\le M}$$.
Therefore, from
$${C\|f\|_X +3C^3\|u\|_X^2\le M}$$, we have
$${\|u\|_X\le \sqrt{\frac{M-C\|f\|_X}{3C^3}}\lt M}$$.

Supplements

[Supplement 1]
As functions $${\varphi}$$ that $${\mathrm{div} \varphi = \nabla \cdot \varphi=0}$$, it is sufficient to take any $${\psi \in \mathcal{D}(\Omega)}$$ and set to $${\varphi = \mathrm{curl} \psi}$$.

[Supplement 2]
Let $${\|u_n-u\|_{L^2(\Omega)}\to 0, \|v_n-v\|_{L^2(\Omega)}\to 0}$$. By the triangle inequality, we have
$${| \|u_n\|_{L^2(\Omega)}-\|u\|_{L^2(\Omega)}|\le \|u_n-u\|_{L^2(\Omega)}}$$ for any sufficientaly large $${n}$$. On the other hand, $${\|u_n\|_{L^2(\Omega)}\lt \|u\|_{L^2(\Omega)}+1}$$. Therefore
$${\|u_n v_n - uv\|_{L^1(\Omega)}\le \|u_n\|_{L^2(\Omega)}\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)}}$$
$${\lt (\|u\|_{L^2(\Omega)}+1)\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)} \to 0.}$$

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