Signed section flux in logarithmic wall coordinates

For real variables $p,s,a,b,c$, set

$$K=a^2+b^2+c^2-2(ab+bc+ca),\qquad S=p^2+s^2,\qquad C=S+K.$$

The shear components and wall rates are

$$P=Kp-2a^2+b^2+c^2+ab-2bc+ac,$$ $$Q=Ks+\sqrt{3}(-b^2+c^2+ab-ac),$$ $$r_1=-S+2p,\qquad r_2=-S-p+\sqrt{3}s,\qquad r_3=-S-p-\sqrt{3}s.$$

These functions define the normalized vacuum Bianchi IX vector field $(P,Q,r_1a,r_2b,r_3c)$.

Let $X=\mathbb R^5$ with coordinates $x=(p,s,a,b,c)$ and the sup norm. Write

$$F(x)=(P,Q,r_1a,r_2b,r_3c),\qquad \mathcal C(x)=C(p,s,a,b,c),$$

and let $W=\{(p,s,a,b,c)\in X:a>0,\ b>0,\ c>0\}$.

Write $N_1(x)=a$, $N_2(x)=b$, and $N_3(x)=c$. For initial data $(x_0,t_0)$, let $I_{\max}(x_0,t_0)$ be the union of all open convex domains of solutions with these initial data, and let $\alpha_{x_0,t_0}$ be their compatible curve. A solution $S$ has such a domain $I_S$, a curve $\alpha_S$, and $\alpha_S(t_0)=x_0$.

Let $\mathcal L(X,X)$ be the continuous linear operators on $X$ and set

$$A(x)=dF_x,\qquad \widetilde F(x,J)=(F(x),A(x)\circ J).$$

For a fixed initial time $t_0$, put

$$\mathcal D_{t_0}=\{(x,t):t\in I_{\max}(x,t_0)\},\qquad \Phi_{t_0}(x,t)=\alpha_{x,t_0}(t),\qquad J_{x,t_0}(t)=d_x\Phi_{t_0}(x,t).$$

Write $\lambda_x=d\mathcal C_x:X\to\mathbb R$ for the constraint differential.

For $x=(p,s,a,b,c)$, define

$$H(x)=(abc)^{2/3},\quad w(x)=abc(1+4H(x)),\quad \delta(x)=2K-3S,\quad \bar\delta(x)=\frac{2(K+4H(x))}{1+4H(x)},$$ $$V(x,t_0,t)=\frac{w(x)\det J_{x,t_0}(t)}{w(\alpha_{x,t_0}(t))}.$$

Write $[L]$ for the matrix of a continuous linear operator $L:X\to X$ in the standard basis. Set $\Omega(v_0,\ldots,v_4)=\det(v_0,\ldots,v_4)$,

$$\omega_x(v_0,\ldots,v_3)=\frac{\Omega(x/2,v_0,\ldots,v_3)}{w(x)},\qquad \varphi_x(v_0,v_1,v_2)=\omega_x(F(x),v_0,v_1,v_2).$$

A normalized symmetry is a continuous linear equivalence $L:X\to X$ satisfying

$$\mathcal C(Lx)=\mathcal C(x),\quad Lx\in W\ \Longleftrightarrow\ x\in W, \quad F(Lx)=LF(x),\quad w(Lx)=w(x),\quad\det L=1.$$

These symmetries include the identity, composition, inverse, and the two maps

$$R(p,s,a,b,c)=((-p+\sqrt3s)/2,(-\sqrt3p-s)/2,b,c,a),\qquad S_0(p,s,a,b,c)=(p,-s,a,c,b).$$

Transforming a solution replaces its curve $\alpha$ by $L\alpha$ and keeps its time domain.

Write $\Phi^s_t(x)=\alpha_{x,s}(t)$, with domain $I(x,s)$, and $\mathcal X=\{x:\mathcal C(x)=1,\ N_i(x)>0\ (i=1,2,3)\}$. For $y=(v,l,r)\in\mathbb R^3$ and a wall height $h$, put

$$b=e^{-l},\quad c=e^{-r},\quad d=1-v^2-K(h,b,c),\quad U_h=\{y:d>0\},\quad c_h(y)=(-\sqrt d,v,h,b,c).$$

For a normalized axis symmetry $A$, the chart is $c_{A,h}=A^{-1}c_h$, with coordinates $\pi_A(x)=((Ax)_1,-\log (Ax)_3,-\log (Ax)_4)$. At $p=-\sqrt d$ its flux density and measure are

$$\rho_h(y)=\frac{p^2+v^2-2p}{2|p|(1+4(hbc)^{2/3})}, \qquad d\mu_h={\bf1}_{U_h}\rho_h^+\,dy.$$

Write $\varphi$ for the signed section flux and $\bar\delta$ for the corrected divergence. The three primary frames interchange axis zero with the chosen dominant axis. For $h>0$ and $y\in U_h$, the three coordinate tangents, taken in their given order, satisfy

$$\varphi_{c_h(y)}(Dc_h(y)e_0,Dc_h(y)e_1,Dc_h(y)e_2)=-\rho_h(y).$$

import Definitions.Def_BianchiIX_Dynamics_MaximalFlow
import Definitions.Def_BianchiIX_Dynamics_FlowTransport
import Definitions.Def_BianchiIX_Dynamics_CorrectedVolume
import Definitions.Def_BianchiIX_Dynamics_SectionFlux
import Definitions.Def_BianchiIX_AxisAndOrbitData
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Real.Sqrt
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.FunProp
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Tactic.FieldSimp
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Topology.Instances.Matrix
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.Algebra.Module.Determinant
import Mathlib.Analysis.Calculus.Deriv.Pi
import Mathlib.LinearAlgebra.Determinant
import Mathlib.Data.Real.Basic
import Mathlib.LinearAlgebra.Alternating.Curry

noncomputable section
open BianchiIX

theorem BianchiIX.mainWallChart_signed_flux {h : ℝ} {y : SectionCoordinates}
    (hh : 0 < h) (hy : y ∈ mainWallChartDomain h) :
    normalizedSectionFlux (mainWallChart h y) (mainWallChartFrame h y) = -mainWallChartDensity h y := by sorry