IDOS Energy, Coherence, and Supporting Dynamics — Part 78

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$$\begin{aligned} \mathfrak a((X,\chi),(Y,\zeta))=\int w\bigg[ \sum_i\langle\partial_iX,D_g\partial_iY\rangle+ \langle X,M(x)Y\rangle+ \kappa(\nabla\chi+PX)\cdot(\nabla\zeta+PY)\bigg]dx. \end{aligned} \tag{N9}$$

For any $f$ with $\int w(1+|x|^2)|f|^2dx<\infty$, the weak equation $\mathfrak a((X,\chi),(Y,\zeta))=\int w\langle f,Y\rangle dx$ has a unique solution modulo constant phase satisfying

$$\int w\chi^2dx+ \int w(1+|x|^2)(|X|^2+|\nabla X|^2+|\nabla\chi|^2)dx \le C\int w(1+|x|^2)|f|^2dx. \tag{N10}$$

For (N8), $f=(2\gamma,\varepsilon B\gamma)$ is affine in physical position. Its required fourth moment is finite, since the exterior radial power is $9+4-16=-3$.

First construct the solution in the homogeneous space $X\in H^1(wdx)$, $\chi\in H^1_{\rm loc}$, $\nabla\chi\in L^2(wdx)$, fixing the ordinary mean of $\chi$ on a unit ball to zero. Its squared norm $N_h$ is $\int w(|X|^2+|\nabla X|^2+|\nabla\chi|^2)dx$. Anchored Poincare on every bounded ball proves completeness; it does not assert a weighted global Poincare inequality.

The necessary Hardy estimate must precede cutoff density. On finite rays outside a fixed radius $R_0$, integrate

$$\frac d{dr}(r^{-8}\chi(r\vartheta)^2) =-8r^{-9}\chi^2+2r^{-8}\chi\,\partial_r\chi.$$

The outer term after rearrangement is $-T^{-8}\chi(T\vartheta)^2\le0$. Integrate over directions, discard that favorable term, and apply Young's inequality to the product of $r^{-9}\chi^2$ and $r^{-7}|\partial_r\chi|^2$. The inner trace is controlled on a fixed collar by the anchored local $H^1$ bound. Monotone convergence then proves

$$\int\frac{w\chi^2}{1+|x|^2}dx \le C\int w|\nabla\chi|^2dx. \tag{N11}$$

The argument applies to local $H^1$ representatives by approximation on finite annuli. Thus it assumes neither the left-hand integral nor a boundary condition at infinity. Smooth cutoffs now converge in the homogeneous norm: their gradient errors are controlled by the tails of (N11) and $\int w|\nabla\chi|^2$. Local mollification proves density of smooth compact representatives modulo constants. Positivity of $D_g,M,\kappa$ and $\nabla\chi=(\nabla\chi+PX)-PX$ make (N9) coercive in this space. The source has no phase component and hence is well-defined on the quotient. The Hilbert variational theorem supplies $N_h\le C\int w|f|^2dx$. It has not yet supplied $\chi\in L^2(wdx)$.

For that step use the exact constant matrices

$$A_0=M_\infty+\kappa P^*P,\qquad C_0=A_0^{-1}\kappa P^*,\qquad K=\kappa(I-PC_0)=(\kappa^{-1}I+PM_\infty^{-1}P^*)^{-1}>0.$$

Choose $s=s_0+x\cdot K^{-1}x\asymp1+|x|^2$, $s_0>0$, and

$$F_R=\frac{R^2s}{R^2+s},\quad c_R=\frac{dF_R}{ds}=\frac{R^4}{(R^2+s)^2},\qquad (Y,\zeta)=(F_RX-C_0\chi\nabla F_R,F_R\chi). \tag{N12}$$

Subtract the ball average of $\zeta$. For fixed $R$ this test belongs to the homogeneous space: $\nabla F_R=O_R(r^{-3})$ and $\nabla^2F_R=O_R(r^{-4})$, and (N11) controls every occurrence of $\chi$. Expanding (N9), the identity $A_0C_0=\kappa P^*$ cancels the constant massive-phase cross term. The result is the leading energy

$$\int wF_R\left[\sum_i\langle\partial_iX,D_g\partial_iX\rangle+ \langle X,MX\rangle+\kappa|\nabla\chi+PX|^2\right]dx +\int w\chi\nabla\chi\cdot K\nabla F_R\,dx$$

plus exactly

$$\begin{aligned} E_1&=\int w\sum_i\langle\partial_iX,D_gX\rangle\partial_iF_R\,dx,\\ E_2&=-\int w\sum_i\left\langle\partial_iX,D_gC_0 [(\partial_i\chi)\nabla F_R+\chi\partial_i\nabla F_R]\right\rangle dx,\\ E_3&=\int w\chi\langle X,V(x)C_0\nabla F_R\rangle dx. \end{aligned}$$

Since $K\nabla F_R=2c_Rx$, integration by parts makes the phase term $\int w\chi^2Z_Rdx$, where

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