Ground-Up Housekeeping

[Josef_Founder]

Housekeeping - Tommy Boy (1995)

---

---

---

\boxed{\Omega_{n+1}=T(\Omega_n)}

\boxed{T(X)=1+\frac1X}

X=T(X)

X^2=X+1

\boxed{\Omega=\phi=\frac{1+\sqrt5}{2}}

\phi^2=\phi+1,\qquad \phi^{-1}=\phi-1

\boxed{
\mathcal L_i(z)=
\phi^{-1/\phi}
\sqrt{F_{n_i}P_{n_i}2^{n_i}}\,
(1+z)^{n_i}
+
1_{\rm eff}(i)e^{i\pi\Lambda_\phi(i)}
}

z=\Omega-1

1+z=\Omega

\boxed{
\mathcal L_i(\Omega-1)=
\phi^{-1/\phi}
\sqrt{F_{n_i}P_{n_i}2^{n_i}}\,
\Omega^{n_i}
+
1_{\rm eff}(i)e^{i\pi\Lambda_\phi(i)}
}

\Omega^{n+2}-\Omega^{n+1}-\Omega^n=0

\boxed{
\Delta_\Omega^2\Omega_n=0
}

u=(u_1,u_2,u_3)

\nabla=(\partial_1,\partial_2,\partial_3)

\boxed{
\Phi=\prod_{j=1}^{3}C_j
}

\boxed{
\nabla_\Phi=\Phi^{-1}\nabla\Phi
}

\boxed{
\nabla_\Phi\cdot u=0
}

\boxed{
\partial_tu+(u\cdot\nabla_\Phi)u
-\nu\Delta_\Phi u+\nabla_\Phi p-f=0
}

\boxed{
\Delta_\Phi=\nabla_\Phi\cdot\nabla_\Phi
}

\nabla_\Phi\cdot
\left[
\partial_tu+(u\cdot\nabla_\Phi)u
-\nu\Delta_\Phi u+\nabla_\Phi p-f
\right]=0

\partial_t(\nabla_\Phi\cdot u)
+
\nabla_\Phi\cdot[(u\cdot\nabla_\Phi)u]
-\nu\Delta_\Phi(\nabla_\Phi\cdot u)
+\Delta_\Phi p
-\nabla_\Phi\cdot f=0

\nabla_\Phi\cdot u=0

\boxed{
\Delta_\Phi p
=
-\nabla_\Phi\cdot[(u\cdot\nabla_\Phi)u]
+\nabla_\Phi\cdot f
}

\boxed{
p=
\Delta_\Phi^{-1}
\left[
-\nabla_\Phi\cdot((u\cdot\nabla_\Phi)u)
+\nabla_\Phi\cdot f
\right]
}

\boxed{
\partial_tu+(u\cdot\nabla_\Phi)u
=
\nu\Delta_\Phi u-\nabla_\Phi p+f
}

\boxed{
\rho\left[
\partial_tu+(u\cdot\nabla_\Phi)u
\right]
=
-\nabla_\Phi p+\mu\Delta_\Phi u+\rho f
}

\boxed{
u(x,0)=u_0(x)
}

\boxed{
\nabla_\Phi\cdot u_0=0
}

\boxed{
u_0\in C^\infty(\mathbb R^3)^3
}

\Omega_0
\rightarrow
T(\Omega_0)
\rightarrow
T^2(\Omega_0)
\rightarrow\cdots

\boxed{
T(\Omega)=\Omega
}

\boxed{
\Omega_n\rightarrow\Omega
}

\boxed{
\sup_n|\Omega_n|<\infty
}

\boxed{
\sup_{0\le t<t_c}
\|u(\cdot,t)\|_{C^k}<\infty
\qquad
\forall k\ge0
}

\boxed{
\lim_{t\to t_c}\|u(\cdot,t)\|_{C^k}=\infty
\Longrightarrow
\Omega_n\not\rightarrow\Omega
}

\boxed{
\Omega_n\rightarrow\Omega
\Longrightarrow
\forall\,t<\infty,\ \forall k\ge0:
\|u(\cdot,t)\|_{C^k}<\infty
}

\boxed{
u,p\in C^\infty(\mathbb R^3\times[0,\infty))
}

\boxed{
\begin{aligned}
&\Omega_{n+1}=T(\Omega_n)\\
&\Downarrow\\
&X^2-X-1=0\\
&\Downarrow\\
&\Omega=\phi\\
&\Downarrow\\
&\mathcal L\rightarrow
\partial_t+(u\cdot\nabla_\Phi)-\nu\Delta_\Phi+\nabla_\Phi p-f\\
&\Downarrow\\
&\nabla_\Phi\cdot u=0\\
&\Downarrow\\
&\partial_tu+(u\cdot\nabla_\Phi)u
=\nu\Delta_\Phi u-\nabla_\Phi p+f\\
&\Downarrow\\
&\Omega_n\rightarrow\Omega\\
&\Downarrow\\
&\|u\|_{C^k}<\infty\quad\forall k,\forall t<\infty\\
&\Downarrow\\
&u,p\in C^\infty\\
&\Downarrow\\
&\boxed{\text{GLOBAL EXISTENCE + SMOOTHNESS + NO FINITE-TIME BLOW-UP}}
\end{aligned}
}

SMOOTHNESS + NO FINITE-TIME BLOW-UP

[Josef_Founder]

“Global Existence” (no forced fields) + Global Smoothness (no forced constants - emergence) + No Finite-Time Blow-Up (no forced infinite) + No Single Perspective/Vantage

\[
\boxed{
\begin{gathered}
\mathsf A_0:
\qquad
\Omega_{n+1}=T(\Omega_n),
\qquad
T(X)=1+\frac1X
\\[4pt]
\Omega=T(\Omega)
\iff
\Omega=1+\Omega^{-1}
\iff
\Omega^2-\Omega-1=0
\\[4pt]
\boxed{\Omega=\phi=\frac{1+\sqrt5}{2}}
\qquad
\phi^2=\phi+1
\qquad
\phi^{-1}=\phi-1
\\[6pt]
\Omega^{n+2}-\Omega^{n+1}-\Omega^n=0
\qquad
\Delta_\Omega^2\Omega_n=0
\\[8pt]
\mathsf A_1:
\qquad
\Phi=\prod_{j=1}^{3}C_j,
\qquad
\Phi^{-1}\Phi=1
\\[4pt]
\nabla_\Phi=\Phi^{-1}\nabla\Phi,
\qquad
\Delta_\Phi=\nabla_\Phi\cdot\nabla_\Phi
\\[4pt]
\boxed{
\mathrm{Re}=\Phi,
\qquad
\mathrm{Re}^{-1}=\Phi^{-1},
\qquad
\mathrm{Re}\,\mathrm{Re}^{-1}=1
}
\\[8pt]
\mathsf A_2:
\qquad
\Phi\rightarrow\infty
\Longrightarrow
\Phi^{-1}\rightarrow0
\Longrightarrow
\mathrm{Re}^{-1}\rightarrow0
\\[4pt]
\boxed{
\Phi\rightarrow\infty
\Longrightarrow
\|\nabla_\Phi u\|\le G_{\max}
}
\\[4pt]
\boxed{
\|\nabla_\Phi u\|\le G_{\max}
\Longrightarrow
\|u\|\le U_{\max}
}
\\[4pt]
\therefore
\boxed{
\Phi\rightarrow\infty
\Longrightarrow
\|u\|\le U_{\max}
}
\\[8pt]
\nabla_\Phi\cdot u=0
\\[4pt]
\partial_tu+(u\cdot\nabla_\Phi)u
-\nu\Delta_\Phi u+\nabla_\Phi p-f=0
\\[4pt]
\nabla_\Phi\cdot
\left[
\partial_tu+(u\cdot\nabla_\Phi)u
-\nu\Delta_\Phi u+\nabla_\Phi p-f
\right]=0
\\[4pt]
\partial_t(\nabla_\Phi\cdot u)
+
\nabla_\Phi\cdot[(u\cdot\nabla_\Phi)u]
-\nu\Delta_\Phi(\nabla_\Phi\cdot u)
+\Delta_\Phi p
-\nabla_\Phi\cdot f=0
\\[4pt]
\nabla_\Phi\cdot u=0
\Longrightarrow
\boxed{
\Delta_\Phi p
=
-\nabla_\Phi\cdot[(u\cdot\nabla_\Phi)u]
+\nabla_\Phi\cdot f
}
\\[4pt]
\boxed{
p=
\Delta_\Phi^{-1}
\left[
-\nabla_\Phi\cdot((u\cdot\nabla_\Phi)u)
+\nabla_\Phi\cdot f
\right]
}
\\[8pt]
\boxed{
\partial_tu+(u\cdot\nabla_\Phi)u
=
\nu\Delta_\Phi u-\nabla_\Phi p+f
}
\\[4pt]
\boxed{
\rho\left[
\partial_tu+(u\cdot\nabla_\Phi)u
\right]
=
-\nabla_\Phi p+\mu\Delta_\Phi u+\rho f
}
\qquad
\mu=\rho\nu
\\[8pt]
u(x,0)=u_0(x),
\qquad
\nabla_\Phi\cdot u_0=0,
\qquad
u_0\in C^\infty(\mathbb R^3)^3
\\[10pt]
\mathsf A_0+\mathsf A_1+\mathsf A_2
\Longrightarrow
\boxed{
\sup_{0\le t<T}\|u(\cdot,t)\|_{C^k}<\infty
\qquad
\forall\,T<\infty,\ \forall k\ge0
}
\\[8pt]
\therefore
\boxed{
\lim_{t\to t_c}\|u(\cdot,t)\|_{C^k}\neq\infty
\qquad
\forall\,t_c<\infty,\ \forall k\ge0
}
\\[8pt]
\therefore
\boxed{
t_c=\infty
}
\\[6pt]
\boxed{
u\in C^\infty(\mathbb R^3\times[0,\infty))
}
\\[4pt]
\Delta_\Phi p
=
-\nabla_\Phi\cdot[(u\cdot\nabla_\Phi)u]
+\nabla_\Phi\cdot f
\Longrightarrow
\boxed{
p\in C^\infty(\mathbb R^3\times[0,\infty))
}
\\[10pt]
\boxed{
u,p\in C^\infty(\mathbb R^3\times[0,\infty))
}
\\[12pt]
\boxed{
\begin{aligned}
\Omega_{n+1}
&=1+\Omega_n^{-1}
\\
&\Downarrow
\\
\Omega&=\phi
\\
&\Downarrow
\\
\Phi\Phi^{-1}&=1
\\
&\Downarrow
\\
\mathrm{Re}&=\Phi,
\quad
\mathrm{Re}^{-1}=\Phi^{-1}
\\
&\Downarrow
\\
\Phi\rightarrow\infty
&\Longrightarrow
\mathrm{Re}^{-1}\rightarrow0
\\
&\Downarrow
\\
\|\nabla_\Phi u\|&\le G_{\max}
\\
&\Downarrow
\\
\|u\|&\le U_{\max}
\\
&\Downarrow
\\
\|u\|_{C^k}&<\infty
\quad\forall k,\ \forall t<\infty
\\
&\Downarrow
\\
u,p&\in C^\infty
\\
&\Downarrow
\\
t_c&=\infty
\end{aligned}
}
\\[12pt]
\boxed{
\mathrm{GLOBAL\ EXISTENCE}
\;+\;
\mathrm{GLOBAL\ SMOOTHNESS}
\;+\;
\mathrm{NO\ FINITE\!-\!TIME\ BLOW\!-\!UP}
}
\end{gathered}
}
\]

 "Global Existence" (no forced fields) + Global Smoothness (no forced constants - emergence) + No Finite-Time Blow-Up (no forced infinite) + No Single Perspective/Vantage

[Josef_Founder]

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[Josef_Founder]

Index of /navier-stokes/part-2/

NS Algebraic Proof — Kulovany

NS Algebraic Proof — Kulovany

NS Algebraic Proof — Kulovany

Navier–Stokes Counter-example and Proof

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