Elementary Solutions of the Navier-Stokes Equations: Existence and Smoothness

(October 4, 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:
for the fundamental solution of any linear partial differential operator with constant coefficients $${L}$$ on $${\mathbb{R}^N}$$, that is, for $${E \in \mathcal{D}^{\prime}}$$ that satisfies $${LE=\delta}$$, for $${f \in L^1_{\mathrm{loc}}}$$, one of the solutions of the equation $${Lu=f}$$ on $${\Omega\Subset\mathbb{R}^N}$$ is $${u=E * \chi_\Omega f \in \mathcal{D}^{\prime}(\Omega)}$$ because $${Lu=LE*\chi_\Omega f=\delta*f=f}$$

here, for any $${\varphi\in\mathcal{D}(\Omega)}$$,

$${\langle L(E*\chi_\Omega f), \varphi\rangle}$$
$${=\pm\langle E*\chi_\Omega f, L\varphi\rangle}$$
$${:=\pm\langle E(x), \langle \chi_\Omega(y)f(y), L\varphi(x+y)\rangle\rangle}$$
$${= \pm\langle \chi_\Omega(x)f(x), \langle E(y), L\varphi(x+y)\rangle\rangle}$$
$${= \langle \chi_\Omega(x)f(x), \langle LE(y), \varphi(x+y)\rangle\rangle}$$
$${=\langle LE(x), \langle \chi_\Omega(y)f(y), \varphi(x+y)\rangle\rangle}$$
$${=\langle \chi_\Omega(y)f(y), \varphi(y)\rangle =\langle f, \varphi\rangle}$$.

The policy is, to let $${L}$$ be the heat operator $${\partial_t-\Delta}$$ in the initial value problem of the Navier-Stokes equations
$${\partial_t u -\Delta u=f - \nabla \mathfrak{p}-(u \cdot \nabla)u}$$
$${\mathrm{div}\,u=0}$$
$${u(0, x)=a(x)}$$,
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
$${\lim_{t, |x|\to\infty}\partial^\alpha u(t, x)=0}$$, $${\lim_{t, |x|\to\infty}\partial^\alpha\mathfrak{p}(t, x)=0}$$.

[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 $${B_\delta(0, y)}$$ be the $${\delta}$$-neighborhood of point $${(0, y)}$$. Let $${\Omega}$$ 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 $${|\Omega|}$$ 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 $${V_{\sigma}^{m, p}(\Omega)=\{ u \in C^{\infty}(\Omega) : \|u\|_{W^{m, p}(\Omega)} \lt {\infty},\mathrm{div}\,u=0 \},}$$ $${W_{\sigma}^{m, p}(\Omega)}$$ 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 $${f\in C^\infty(\overline\Omega)}$$. 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,
$${E^{i}(t, x)=\begin{cases} \frac{1}{\sqrt{4 \pi t}^3} e^{-\frac{|x|^2}{4t}} & (t \gt 0) \\ 0 & (t \le 0) \end{cases}}$$.
Let $${A=\{u(0, \cdot):u \in W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega), u(t, x)=\int_{\mathbb{R}\times \mathbb{R}^3} E(s, y) \, \chi_{\Omega}(t-s, x-y)(\,Pf(t-s, x-y) - P((u\cdot \nabla)u)(t-s, x-y))dsdy\}}$$. 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 the norm 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.

Intuitive argment

[Existence of elementary weak solutions]
For any $${f\in C^\infty(\overline\Omega)}$$, if $${a\in A}$$ then there are weak solutions $${u, \mathfrak{p}}$$ of the initial value problem

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

in the sence that, $${u \in W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega),}$$ $${\mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/\{\mathfrak{p}':\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0\}}$$ 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 $${A\neq\{0\}, u \neq 0}$$; if $${f=0}$$ then $${A=\{0\}, u=0}$$. $${\Omega}$$ can be arbitrary large, so $${u, \mathfrak{p}}$$ are time global.

[Intuitive proof]
$${W_\sigma^{m, p}(\Omega)}$$ is completion, so for any $${\{u_n\} \subset V_\sigma^{m, p}(\Omega)}$$,
$${\lim_{n,{n'} \to \infty}\|u_n - u_{n'}\|_{W^{m, p}(\Omega)}=0, \lim_{n,{n'} \to \infty}\|E*\chi_{\Omega}P(u_n\cdot\nabla)u_n - E*\chi_{\Omega}P(u_{n'}\cdot\nabla)u_{n'}\|_{W^{m-1, p}(\Omega)}=0}$$, some $${u \in W_\sigma^{m, p}(\Omega)}$$ exists such that
$${\lim_{n \to \infty}\|u_n - u\|_{W^{m, p}(\Omega)}=0,\lim_{n \to \infty}\|E*\chi_{\Omega}P(u_n\cdot\nabla)u_n- E*\chi_{\Omega}P(u\cdot\nabla)u\|_{W^{m-1, p}(\Omega)}=0}$$.

$${u}$$ 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
$${\langle f, \varphi\rangle = \langle Pf, \varphi\rangle, \langle (u\cdot\nabla)u, \varphi\rangle =\langle P((u\cdot\nabla)u), \varphi\rangle}$$, 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)'' $${\partial_t v_{n} - \Delta v_{n} =Pf-P((u_n \cdot \nabla)u_n)}$$
is
$${v_n=E * \chi_{\Omega}(Pf -P((u_n \cdot \nabla)u_n)) \in V_\sigma^{m-1, p}(\Omega)}$$.

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

We can take the Cauchy sequence $${\{u_n\}}$$ such that the limit of $${\{v_n\}}$$ and the limit of $${\{u_n\}}$$ coincide. Later we can justify it.
We show that $${u=v}$$ is the solution of (N-S)' :
$${v_{n}(t, x)}$$
$${=\int_{\mathbb{R} \times \mathbb{R}^3} E(s, y)\chi_{\Omega}(t-s, x-y) (Pf(t-s, x-y)-P((u_n \cdot \nabla)u_n)(t-s, x-y))dsdy,}$$
$${u_n \to u=v \gets v_n.}$$

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

Therefore, the above calculation and the continuity of the heat operator on $${\mathcal{D}'_\sigma(\Omega)}$$:
$${|\langle \partial_t v_{n} - \Delta v_{n}, \varphi \rangle - \langle \partial_t u - \Delta u, \varphi \rangle|\to 0}$$, 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)' (see [Supplement 3]).

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)

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

[Proof]
$${m}$$ can be arbitrarily large, so the embedding theorem to 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)

Justification and uniqueness

We justify above argment by Banach's fixed point theorem. We put $${X=\bigcap_{m=5}^\infty W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega)}$$. 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, $${d_\alpha=\sup\{|((t, x)-(t', x'))^\alpha|:(t, x), (t', x')\in\Omega\}\gt 1}$$ and
$${\lim_{|\alpha|\to\infty}C_\alpha d_\alpha/\alpha!\le c}$$,
therefore
$${C_\alpha\le c\alpha!}$$ follows. $${X}$$ is a Banach space with norm given by
$${\|u\|_X=\sum_{m=5}^\infty \frac{1}{m!^5}\|u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}}$$.

$${\chi_{\Omega}\in X}$$ so $${X\neq \{0\}}$$.

[Proof]
Let $${\{u_n\}}$$ be a Caucy sequence in $${X}$$. Then, $${\{u_n\}}$$ 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 $${u}$$. If $${u\notin X}$$, for any positive number $${R}$$, there exists a natural number $${m'\ge 5}$$ such that
$${\sum_{m=5}^{m'} \frac{1}{m!^5}\|u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt R}$$.
Then there exists a constant $${C\gt 0}$$,
$${\|u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt CR}$$.
If $${C}$$ does not exists, then, for any $${C\gt 0}$$, $${\|u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\le CR}$$, so $${u=0}$$. This is a contradiction, so there exists a constant $${C\gt 0}$$, $${\|u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt CR}$$. This is a contradiction, so $${u\in X}$$. If
$${\lim_{n\to\infty}\|u_n-u\|_X=0}$$
does not hold, then there exists a positive number $${R'}$$ such that for any natural number $${N}$$, there exists a natural number $${n\gt N}$$, $${\|u_n-u\|_X\gt0}$$. Therefore there exists a natural number $${M'\ge 5}$$ and
$${\sum_{m=5}^{M'} \frac{1}{m!^5}\|u_n-u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt R'}$$. Then there exists a constant $${C'\gt 0}$$,
$${\|u_n-u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt C'R'}$$.
If $${C'}$$ does not exists, then, for any $${C'\gt 0}$$, $${\|u_n-u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\le C'R'}$$. Then $${u_n=u}$$. This is a contradiction, therefore there exists a constant $${C'\gt 0}$$, $${\|u_n-u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)}\gt C'R'}$$. This is a contradiction, so
$${\lim_{n\to\infty}\|u_n-u\|_X=0.}$$
(END)

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)
and
$${\left\|\partial_{x^j}u\right\|_{X}\le C_2\|u\|_X}$$
(absorption of differential)
hold for any $${u, v\in X}$$.

[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 $${\|u_n-u\|_X\to 0}$$
$${\Rightarrow \|u_n-u\|_{W_\sigma^{m, 1}(\Omega)\cap W_\sigma^{m, 2}(\Omega)} \to 0}$$
$${\Rightarrow \|u_n-u\|_{C^{k, \varepsilon}(\overline{\Omega})}\to 0}$$, 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.
(END)

$${X\ni u\mapsto E*(\chi_\Omega u)\in X}$$ 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.

[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
$${|\partial_{t, x}^\alpha(E^i(s, y) \chi_\Omega(t-s, x-y)u^i(t-s, x-y))|\le E^i(s, y)\sup\{|\partial_{t, x}^\alpha u^i(t-s, x-y)|:(t-s, x-y)\in\Omega\}\in L^1_{s, y}(\Omega)}$$, so combine the theorem of differentiation under the integral sign, Hölder's inequality and continues embedding $${X\subset L^\infty(\Omega)}$$, we have
$${\|\partial^\alpha(E*(\chi_{\Omega}u))\|_{L^p(\Omega)}}$$
$${\le\|E*(\partial^\alpha (\chi_{\Omega}u))\|_{L^p(\Omega)}}$$
$${\le \|\|E(s, y)\|_{L_{s, y}^1(-\Omega+(t, x))}\|\partial^\alpha u(t-s, x-y)\|_{L_{s, y}^\infty(-\Omega+(t, x))}\|_{L_{t, x}^p(\Omega)}}$$
$${\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}\|\partial^\alpha u\|_{L^\infty(\Omega)}|\Omega|^{1/p}}$$
$${\le \sup\{\|E\|_{L^1(-\Omega+(t, x))}:(t, x)\in\Omega\}c''C_2^{|\alpha|}\|u\|_X|\Omega|^{1/p}}$$
$${\lt\infty}$$.
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 $${C}$$. For a constant $${M}$$, let $${S}$$ be a subset of $${X}$$:
$${S=\{u\in X:\|u\|_{X}\le M\}}$$. We take $${M}$$ satisfying $${C(1+3C^2)M\le 1}$$. Let the external force $${f\in S}$$ and $${\|f\|_X\le M^2}$$.

We solve
(N-S)'$${\partial_t u -\Delta u=f -(u \cdot \nabla)u}$$,
that is, for any $${a\in A}$$, there exist $${u \in W_{\sigma}^{m, 1}(\Omega)\cap W_{\sigma}^{m, 2}(\Omega),}$$ $${\mathfrak{p}\in L_{\mathrm{loc}}^2(\Omega)/\{\mathfrak{p}':\mathfrak{p}'\sim\mathfrak{q}\iff\nabla\mathfrak{(p'-q)}=0\}}$$, 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 the function sequence $${\{u_n\}\subset S}$$ as $${u_0\in S}$$, 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 a unique weak solution. $${f \neq 0}$$ then $${A\neq\{0\}, u \neq 0}$$. 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 $${f=0\Rightarrow A=\{0\}, u=0}$$. $${\Omega}$$ can be arbitrary large, 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, if

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

holds, the argument is justified.

[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]
$${\partial_t u_{n+1} - \Delta u_{n+1} =Pf-P((u_n \cdot \nabla)u_n)}$$, so similar to the intuitive proof, the limit of $${\{u_n\}}$$ is the solution.
(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
$${\|u_n-u\|_X\to 0,}$$
$${d(\mathfrak{p}_n, \mathfrak{p}):=\|u_n-u\|_X\to 0}$$.

[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.

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

[Proof]
$${\Omega}$$ can be arbitrary large, 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. If there exist $${R, R'\gt 0}$$ such that for any $${\delta'\gt 0}$$, there exists $${(t, x)}$$ such that $${|(t, x)|\gt\delta'}$$ and $${|P\partial^\alpha u(t, x)|\gt R\gt R', |\partial^\alpha\nabla\mathfrak{p}(t, x)|\gt R'}$$ then $${|\partial^\alpha u(t, x)|=|P\partial^\alpha u(t, x)-\partial^\alpha\nabla\mathfrak{p}(t, x)|\ge ({R-R'})/{2}\gt 0}$$ therefore $${\lim_{t, |x|\to\infty}P\partial^\alpha u(t, x)=0}$$ or $${\lim_{t, |x|\to\infty}\partial^\alpha\nabla\mathfrak{p}(t, x)=0}$$. So for any $${\varepsilon'\gt 0}$$, there exists $${\delta'\gt 0}$$ such that $${|(t, x)|\gt\delta'}$$ then $${|\partial^\alpha u(t, x)|\lt\varepsilon', |P\partial^\alpha u(t, x)|\lt\varepsilon'}$$. Hence
$${|\nabla\partial^\alpha\mathfrak{p}(t, x)|\lt 2\varepsilon'}$$
so
$${0\le\limsup_{t, |x|\to\infty}|\nabla\partial^\alpha\mathfrak{p}(t, x)|\le 2\varepsilon'}$$.
$${\partial^\alpha\mathfrak{p}\in L^2(\Omega)}$$, so for suitable respresentative elements,
$${\limsup_{t, |x|\to\infty}|\partial^\alpha\mathfrak{p}(t, x)|=0}$$.
(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
$${\|u\|_X\le \sqrt{\frac{M-C\|f\|_X}{3C^3}}\lt M}$$.

For any $${x, y\in\mathbb{R}^3}$$,
$${\lim_{t\to 0, t\gt s}E^i(t-s, x-y)=\infty}$$.
Thus, for any $${R\gt 0}$$, there exists $${0\lt \tau\lt 1}$$ such that for any $${s\lt\tau}$$,
$${E^i(\tau -s, x-y)\gt R}$$.
Therefore
$${(u^i(\tau, x))^+\gt (\int_{\Omega}((Pf(s, y) - P((u\cdot \nabla)u)(s, y))^i)^+ dsdy)R}$$.

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
$${\|u_n v_n - uv\|_{L^1(\Omega)}\le \|u_n\|_{L^2(\Omega)}\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)}}$$
$${\lt (\|u\|_{L^2(\Omega)}+1)\|v_n-v\|_{L^2(\Omega)}+\|v\|_{L^2(\Omega)}\|u_n-u\|_{L^2(\Omega)} \to 0.}$$

[Supplement 3]
Let $${|\alpha|\le m-1}$$.

$${\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3}E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) - P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
$${=\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3 - B_\delta(0, 0)} E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) - P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
$${+\int_{\Omega}|\int_{B_\delta(0, 0)}E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y)(Pf(t-s, x-y) - P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx.}$$

$${E^i(s, y)}$$ is a locally integrable function, therefore
$${\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3}E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y) Pf(t-s, x-y))dsdy|^pdtdx}$$
is a finite value.

$${\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3}E(s, y)\partial^\alpha( \chi_{\Omega}(t-s, x-y)P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
is also finite.

$${\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3}E(s, y) \partial^\alpha( \chi_{\Omega}(t-s, x-y)P((u \cdot \nabla)u)(t-s, x-y)dsdy|^pdtdx}$$
$${=\int_{\Omega}|\int_{\mathbb{R} \times \mathbb{R}^3 - B_\delta(0, 0)} E(s, y)\partial^\alpha(\chi_{\Omega}(t-s, x-y)P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
$${+\int_{\Omega}|\int_{B_\delta(0, 0)}E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y)P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx.}$$

This first term is a finite value:

$${\int_{\Omega} |\int_{\mathbb{R} \times \mathbb{R}^3 - B_\delta(0, 0)} E(s, y)\partial^\alpha(\chi_{\Omega}(t-s, x-y) P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
$${\le \sup\{E^i(s, y):(s, y) \in \mathbb{R} \times \mathbb{R}^3 - B_\delta(0, 0) \}^p \int_\Omega|\int_{\{(s, y):(t-s, x-y)\in\Omega\}}\partial^\alpha(P((u \cdot \nabla)u)(t-s, x-y))dsdy|^pdtdx}$$
$${\le \sup\{E^i(s, y):(s, y) \in \mathbb{R} \times \mathbb{R}^3 - B_\delta(0, 0) \}^p \sup\{|\partial^\alpha(P((u \cdot \nabla)u))(s, y)|: (s, y) \in \Omega\}^p|\Omega|^{1+p}}$$
$${\lt \infty.}$$

Also, the second term is also a finite value:by Hölder's inequality,
$${\int_{\Omega} |\int_{B_\delta(0, 0)} E(s, y) \partial^\alpha(\chi_{\Omega}(t-s, x-y)P((u \cdot \nabla)u)(t-s, x-y)dsdy|^pdtdx}$$
$${\le \|E\|_{L^1(B_\delta(0, 0))}^p\| \partial^\alpha(P((u \cdot \nabla)u))\|_{L^\infty(B_\delta(0, 0))}^p|\Omega|}$$
$${\lt \infty.}$$
(END)

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