IDOS Energy, Coherence, and Supporting Dynamics — Part 72

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$H^1$, the operator domain is $H^2$, and the spatial principal part is uniformly elliptic. The conserved derivative energies are

$$\|\partial_t^{j+1}v\|_2^2+ \|\mathcal T\partial_t^jv\|_2^2=\text{constant}.$$

Time derivatives commute with the equation. Elliptic recovery of spatial derivatives costs the undifferentiated $L^2$ displacement only once, giving $\|S(t)\|_{H^{s+1}\times H^s}\le C_s(1+|t|)$. The skew $G$ need not commute with $A$.

The common phase is a zero mode, and far-out normalized phase bumps have $\|Av_R\|_2\to0$. Removing any finite number of symmetry modes leaves this zero threshold. Neither persistence proof below assumes a spectral gap, a compact resolvent or a bounded inverse of $A$.

A Quantitative Nonlinear Prefix Interval

Fix integer $s\ge10$ and define the relative norm

$$B_s(t)=\|X(t)\|_{H^{s+1}_{\rm uloc}}+ \|X_t(t)\|_{H^s_{\rm uloc}}.$$

There are $c_*,C_*>0$ for the fixed action, carrier and prefix such that

$$B_s(0)\le\epsilon,\qquad \epsilon(1+T)^{12}\le c_* \quad\Longrightarrow\quad B_s(t)\le C_*\epsilon(1+|t|)^9\quad (|t|\le T). \tag{P1}$$

The conclusion includes existence, uniqueness and positive relative support and density on all space. Data are smooth compatible bounded data or the smooth-data closure specified by the local theorem. Spare derivatives justify the calculation; their magnitudes do not replace the stated lower-order smallness. Constants may depend on the finite carrier, prefix, units and positive chart margins.

Here is the nonlinear estimate behind the powers. The exact rotated perturbation action is $\mathcal L_2+N_3$, with

$$\mathcal L_2=\frac12|v_t|^2+v_t\cdot B_0(x)v -\frac12|\mathcal Tv|^2,\qquad N_3=w_0 n_3(x,X,DX).$$

The reference is on shell, so removing the first variation subtracts a divergence. The rotating conserved generator is $E-\Omega Q_{\rm common}$; the perturbed charge need not equal the reference charge. The remainder is cubic in relative jets, and its physical Hessian and time derivative are $O(B_s)$ on the fixed tube.

Use the estimate weight $\chi_{\delta,y}=e^{-2\delta\langle x-y\rangle}$, $0<\delta\le1$, and add $\delta^2\|\partial_t^jv\|_{L^2(\chi)}^2$ to the energies for $0\le j\le s$. This changes the estimated norm, not the field equation. If $F(v)=0$ is the full Euler equation, then

$$F'(v)\partial_t^jv=R_j,\quad R_1=0,\qquad \sum_{j=1}^s\|R_j\|_{L^2(\chi)}\le C_sB_s\mathcal J_s.$$

The complete Jacobi operator $F'(v)$ retains coefficient variations with respect to both $v$ and $Dv$. Remaining chain-rule factors have total order at most $j+1$; no uncontrolled derivative of order $s+2$ is left. Descending elliptic estimates with the fixed reference operator give

$$\mathcal J_s\le C_s(E_0+\|v\|_{L^2(\chi)})+C_sB_s\mathcal J_s \le C_s\delta^{-1}\mathcal E_{s,\delta,y}.$$

Only the undifferentiated displacement costs $\delta^{-1}$. Under $B_s\le c_0\delta^2$ the true Jacobi and zeroth nonlinear energies are equivalent to the regularized positive reference energy. Their exact canonical flux identity yields

$$D^+\mathcal E_{s,\delta,y} \le C_E[\delta+B_s\delta^{-2}]\mathcal E_{s,\delta,y}. \tag{P2}$$

The reference gyro cancels from the energy. The estimate has no hidden fixed exponential growth rate. Weighted local Sobolev multiplication handles intermediate derivatives; it does not assume every factor is bounded pointwise.

Integrability of the actual ten-dimensional tail gives

$$\sup_y\left[\frac{\int\chi_{\delta,y}w_0\,dx}{w_0(y)}\right]^{1/2} \le C\delta^{-8}.$$

For centers within distance $O(\delta^{-1})$ use the finite total integral; for farther centers split at $|x-y|=|y|/2$ and use the exponential factor. This avoids multiplying a worst density ratio by a separate full collar volume. One further displacement factor gives

$$B_s(t)\le C\epsilon\delta^{-9} \exp\left(C_E\delta|t|+ C_E\delta^{-2}\int_0^{|t|}B_s(\tau)\,d\tau\right).$$

Set $\delta=(1+T)^{-1}$. The bootstrap $B_s\le 2A\epsilon\delta^{-9}$ bounds the nonlinear exponent by $2AC_E\epsilon\delta^{-12}$. Fixing that quantity sufficiently small also keeps the energy positive and the chart strict, so continuation proves (P1). The linear relative group obeys the corresponding $C_s(1+|t|)^9$ bound.

The sufficient horizon and terminal size are therefore

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