Quadratic phase drift on a signed stable boundary passage

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 $(P,Q,r_1a,r_2b,r_3c)$. For $y=(m,u,v,w)\in\mathbb R^4$, define

$$K(u,v,w)=u^2+v^2+w^2-2(uv+vw+wu),\qquad \kappa=u(v+w-u),$$ $$F_m=-v^2+w^2+u(v-w)-m\kappa,\qquad F_u=\tfrac12(1-m^2-(v-w)^2+u^2)u,$$ $$F_v=(m-1-\kappa)v,\qquad F_w=(-m-1-\kappa)w,\qquad F=(F_m,F_u,F_v,F_w),$$

and $E(m,v)=(m-1)^2+v^2$.

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

$$\psi(p,z)=\frac{\sqrt3\,z}{1+p},\qquad \Pi(x)=(\psi(p,s),\psi(p,a),\psi(p,b),\psi(p,c)).$$

For $y=(m,u,v,w)$, set $D(y)=3+m^2+K(u,v,w)$ and

$$\Xi_A(y)=\left(A-1,\frac{Am}{\sqrt3},\frac{Au}{\sqrt3}, \frac{Av}{\sqrt3},\frac{Aw}{\sqrt3}\right),\qquad \Xi(y)=\Xi_{6/D(y)}(y).$$

Write $N(x)=(P,Q,r_1a,r_2b,r_3c)$ for the normalized vector field and $\mathcal C(x)=C(p,s,a,b,c)$ for its constraint.

The normalized and projective state spaces are $\mathbb R^5$ and $\mathbb R^4$ with their finite-product norms. The maps $\Pi,\Xi,D,F$ act on these state spaces.

The projective vector field in normalized time is

$$G(y)=\frac6{D(y)}F(y).$$

Write $X=\mathbb R^5$, $Y=\mathbb R^4$, and let $\alpha_{x,t_0}$ be the maximal solution of $\alpha'=N(\alpha)$ with initial value $x$ at $t_0$ and domain $I_{\max}(x,t_0)$. Put

$$J_{x,t_0}(t)=d_x\alpha_{x,t_0}(t),\qquad \Psi_{t_0}(y,t)=\Pi(\alpha_{\Xi(y),t_0}(t)),$$ $$\mathcal J_{y,t_0}(t)=d\Pi_{\alpha_{\Xi(y),t_0}(t)} \circ J_{\Xi(y),t_0}(t)\circ d\Xi_y.$$

On the invariant plane with zero active wall, use $y=(m,u,w)\in\mathbb R^3$, the inclusion $\iota(m,u,w)=(m,u,0,w)$ and the restriction $\pi_s(m,u,v,w)=(m,u,w)$. Define

$$f(y)=\left(mu^2-(m+1)uw+w^2, \frac{1-m^2-w^2+u^2}{2}u,-(m+1+uw-u^2)w\right),$$ $$\Delta(y)=3+m^2+(u-w)^2,\qquad f_o(y)=6f(y)/\Delta(y).$$

Write $J=df$, $H=d^2f$, $J_o=df_o$, $H_o=d^2f_o$ for their explicit Jacobian and Hessian expressions. The denominator derivatives are $d\Delta=(2m,2(u-w),2(w-u))$ and $d^2\Delta(v,z)=2v_0z_0+2(v_1-v_2)(z_1-z_2)$. The exact original-time boundary flow and its projected variation are

$$\Phi(y,t)=\pi_s\Psi_0(\iota y,t),\qquad L(y,t)=\pi_s\mathcal J_{\iota y,0}(t)\iota.$$

For a fixed scale $K$, set $\Phi_K(y,s)=\Phi(y,Ks)$, $L_K(y,s)=L(y,Ks)$ and $Q_K(y,s)=d_yL_K(y,s)$. For directions $v,z$, define

$$S_K(y,s;v,z)=K\left[H_o(\Phi_K)(L_Kv,L_Kz)+J_o(\Phi_K)Q_K(v,z)\right].$$

All vector norms are sup norms, and operator norms are induced by them. Write $b(y)=|u|+|w|$, $E(y)=e^{512b(y)}$, $v_s=(v_1,v_2)$, and

$$\mathcal H(y;v,z)=b(y)\|v\|\|z\|+\|v_s\|\|z\|+\|v\|\|z_s\|,$$ $$B_2(y)=192b(y)(40000E(y)^3)+10000E(y)^2.$$

The closed boundary region is $\mathcal S_c=\{(m,u,0,w):|m-c|\leq1/10,\ |u|,|w|\leq1/100\}$. The boundary projection $P(m,u,w)$ is the limit of $\Phi(m,u,w;t)_0$ as $t\to\infty$, using the total limit convention of the formal definitions. Let $a\leq t$, $c\geq2$, $Y'=hF(Y)$ on $[a,t]$, with continuous $h\geq0$. Suppose $z'=(c+1)h$ and $Y(r)\in\mathcal S_c$ throughout this interval. Then

$$|Y_0(t)-Y_0(a)|\leq\frac{8(|Y_1(a)|+|Y_3(a)|)^2}{c+1}.$$

import Definitions.Def_BianchiIX_Algebra_Projective
import Definitions.Def_BianchiIX_Dynamics_ProjectiveState
import Definitions.Def_BianchiIX_CompactPhaseExpansionData
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.Deriv.Inv
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Analysis.SpecialFunctions.Sqrt
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.Arcosh
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.ODE.ExistUnique
import Mathlib.Analysis.Calculus.FDeriv.Partial
import Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Data.Matrix.Mul

noncomputable section
open BianchiIX
open Set MeasureTheory

theorem BianchiIX.scaled_stable_boundary_drift {Y : ℝ → ProjectiveState} {a z : ℝ → ℝ}
    {s t c : ℝ} (hst : s ≤ t) (hc : 2 ≤ c)
    (hY : ∀ r ∈ Icc s t, HasDerivAt Y (a r • projectiveField (Y r)) r)
    (hz : ∀ r ∈ Icc s t, HasDerivAt z ((c + 1) * a r) r)
    (hac : ContinuousOn a (Icc s t)) (ha : ∀ r ∈ Icc s t, 0 ≤ a r)
    (hv : ∀ r ∈ Icc s t, Y r 2 = 0)
    (hm : ∀ r ∈ Icc s t, |Y r 0 - c| ≤ 1 / 10)
    (hu : ∀ r ∈ Icc s t, |Y r 1| ≤ 1 / 100)
    (hw : ∀ r ∈ Icc s t, |Y r 3| ≤ 1 / 100) :
    |Y t 0 - Y s 0| ≤ 8 * (|Y s 1| + |Y s 3|) ^ 2 / (c + 1) := by sorry