Local high-Sobolev and smooth electrovacuum

The constraints in the Statement are the normal projections and Maxwell Gauss constraints of the model [L1] under the stated $K,e,b$ conventions. Initialization [L3] gives $H=0$ and the compatible derivative variables $p={\bar{\mathrm{\nabla }}}_{0}g$, $q_i={\bar{\mathrm{\nabla }}}_{i}g$ on the initial slice. The initial state lies in $H^s$: $h\in H^{s+1}$ gives $q_i\in H^s$, $K,e,b\in H^s$, and the harmonic normal derivatives are in $H^s$ by the smooth composition estimates [F7] on the compact range of the uniformly positive $h$. Compactness of ${\mathbb{T}}^3$ and $h\ge c_{0}I$ place the initial state in a compact subset of the open admissible set.

(Endpoint product and composition estimates.) Put $⟨k⟩=(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. For integer $m\in \{2,3\}$, the inequality $⟨k{⟩}^m\le 2^{m-1}(⟨\mathrm{\ell }{⟩}^m+⟨k-\mathrm{\ell }{⟩}^m)$ and the convolution formula for Fourier coefficients give $\mathrm{\parallel }uv{\mathrm{\parallel }}_{H^m}\le C_m(\mathrm{\parallel }⟨k{⟩}^m\widehat{u}{\mathrm{\parallel }}_{{\mathrm{\ell }}^2}\mathrm{\parallel }\widehat{v}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}+\mathrm{\parallel }\widehat{u}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}\mathrm{\parallel }⟨k{⟩}^m\widehat{v}{\mathrm{\parallel }}_{{\mathrm{\ell }}^2})\mathrm{.}$ Cauchy–Schwarz gives $\mathrm{\parallel }\widehat{u}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}\le ({\sum }_k⟨k{⟩}^{-2m}{)}^{1\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}$, and the weight sum is finite in dimension three for $m\ge 2$; the analogous bound holds for $v$. Young's convolution inequality (or the triangle inequality in ${\mathrm{\ell }}^2$ followed by summation in ${\mathrm{\ell }}^1$) therefore proves the algebra estimate in both $H^2$ and $H^3$. For a smooth finite-dimensional map $G$ on a neighborhood of a compact value set, Fourier approximation and the chain rule through order three show that $G(Z)\in H^3$ when $Z\in H^3$: the terms are bounded-coefficient multiples of $D^{\alpha }Z$, products $DZ{D}^{2}Z$, or products of at most three first derivatives; $H^3\hookrightarrow C^1$ places first derivatives in $L^{\mathrm{\infty }}$ and the highest derivatives in $L^2$. The same expansions show continuity under strong $H^3$ convergence, since $H^3\hookrightarrow C^1$ gives uniform convergence of the coefficients and first derivatives while higher derivatives converge in $L^2$. This also covers smooth dependence on $(t,x)$ on a compact coefficient-time slab. Together with the supplied estimates at orders $H^k$ for $k\ge 4$ in [F7], these are the endpoint estimates needed below.

Compose the length-frame state components with $D_{L_0}$. Replacing each ${\mathrm{\partial }}_{X^i}$ by $L_0^{-1}{\mathrm{\partial }}_{{\theta }^i}$ multiplies its symmetric spatial matrix by the fixed positive scalar $L_0^{-1}$ and preserves smoothness and the positive time matrix. By [F1], the metric and Maxwell principal blocks are symmetric with positive time matrices and have no derivative coupling. At the initial orthogonal state $g^{00}<0$, $(g^{ij})>0$ and $\alpha >0$; these strict inequalities persist on a compact admissible state neighborhood, so the complete coefficient family meets the positivity and compact-state assumptions of [L5]. Applying that draft supplier conditionally gives the reduced solution and its $L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$ and ${\mathrm{\partial }}_{t}U\in L^{\mathrm{\infty }}{H}^{s-1}$ bounds. The actual inspected generic existence theorem [F4] states the same periodic coefficient interface and is marked published with precheck pending; it is independent current support, not acceptance of the CP01 draft supplier.

The first-order metric equations impose ${\bar{\mathrm{\nabla }}}_{0}g=p$ and ${\bar{\mathrm{\nabla }}}_0{q}_i-{\bar{\mathrm{\nabla }}}_{i}p={\mathcal{C}}_i$, where ${\mathcal{C}}_i=[{\bar{\mathrm{\nabla }}}_0,{\bar{\mathrm{\nabla }}}_i]g$ is the fixed background-curvature commutator. For $Z_i=q_i-{\bar{\mathrm{\nabla }}}_{i}g$, subtract the commutator identity for $g$ from the equation for $q_i$. The derivative-slot connection terms cancel and give a homogeneous linear ODE along each coordinate time line for $Z_i$; its initial value is zero by [L3], so uniqueness for a linear ODE gives $Z_i=0$. Thus $q_i={\bar{\mathrm{\nabla }}}_{i}g$ throughout the interval.

(Third-order spacetime regularity.) The Fourier norm identity $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{s+1}}^2\asymp \mathrm{\parallel }g{\mathrm{\parallel }}_{H^s}^2+{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_{i}g{\mathrm{\parallel }}_{H^s}^2$, together with step 3.1 and the smooth background coefficients, gives $g\in L^{\mathrm{\infty }}{H}^{s+1}$. The same identity at one lower order and the $C{H}^{s-1}$ continuity of the first-order state give $g\in C{H}^s$. The tetrad reconstruction of $F$ is a smooth algebraic function of $(g,e,b)$ on the compact admissible state set; [F7] gives $F\in L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$. Solve the reduced system as $U_t=\mathcal{R}(t,x,U,{\mathrm{\partial }}_{i}U)$ with $\mathcal{R}$ smooth on the compact admissible range. For every $s\ge 5$, $U\in C{H}^{s-1}$ and ${\mathrm{\partial }}_{i}U\in C{H}^{s-2}$; the composition estimates of [F7] at orders at least four and step 1.2 at orders two and three give $U_t\in C{H}^{s-2}$. Differentiating this identity in time, justified by Fourier approximation and the same chain/product estimates, gives $U_{tt}={\mathrm{\partial }}_t\mathcal{R}(t,x,U,\mathrm{\partial }U)+D_U\mathcal{R}(t,x,U,\mathrm{\partial }U)U_t+D_{\mathrm{\partial }U}\mathcal{R}(t,x,U,\mathrm{\partial }U)\mathrm{\partial }{U}_t\mathrm{.}$ Its coefficient maps lie in $C{H}^{s-2}$, while $\mathrm{\partial }{U}_t\in C{H}^{s-3}$; the algebra estimates of [F7] for orders at least four and of step 1.2 for orders two and three imply $U_{tt}\in C{H}^{s-3}$. At the base order $s=5$, these spaces are $H^3$ and $H^2$, and for $s=6$ the lower product order is $H^3$, also covered by step 1.2. The identities $p={\bar{\mathrm{\nabla }}}_{0}g$ and $q_i={\bar{\mathrm{\nabla }}}_{i}g$ then give all third spatial derivatives of $g$ from two spatial derivatives of $q$, the $t^{2}x$ derivatives from one spatial derivative of $p_t$, and $t^3$ derivatives from $p_{tt}$; the other derivatives of total order at most three follow directly from $p,q\in C{H}^4$ and $p_t\in C{H}^3$. Since $H^2({\mathbb{T}}^3)\hookrightarrow C^0$, all these derivatives are continuous. The same smooth composition argument gives $F\in C^2$ from $(g,e,b)$, so the classical regularity required by [F2] and [F3] holds.

Maxwell propagation [F2] now gives $dF=d{\star }_{g}F=0$. The exact Hodge-to-divergence identity in [L4] converts the second equation to ${\mathrm{\nabla }}_a{F}^{ab}=0$, so the stress supplier applies with $j=0$ and gives ${\mathrm{\nabla }}^a{T}_{ab}=0$. Thus [F3] applies with the initial Einstein constraints and $H=0$ from [L3]; hence the reduced metric equation is the full Einstein equation. Smooth persistence [F5] applies to smooth data on the same interval. Finally, [F6] applies to any two reduced solutions on a common interval with these uniform compact-state and $H^s$ bounds and yields both uniqueness and the claimed $C{H}^{s-1}$ continuous-dependence estimate.

(Local smooth data on an arbitrary slice.) Fix a point of a smooth initial three-manifold and a relatively compact slice chart. Choose local product coordinates $(x^0,x^i)$ with $x^0=0$ on the slice and ${\mathrm{\partial }}_0=n$ there. Choose a smooth positive spatial background $\gamma$ in this chart and put $Q^i:=h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk})$. Prescribe the local metric jet by $g_{ij}=h_{ij}$, $g_{00}=-1$, $g_{0i}=0$, ${\mathrm{\partial }}_0{g}_{ij}=2K_{ij}$, ${\mathrm{\partial }}_0{g}_{00}=-2{\mathrm{tr}}_{h}K$, and ${\mathrm{\partial }}_0{g}_{0i}=h_{ij}{Q}^j$, relative to the fixed product background $\bar{g}=-(d{x}^0{)}^2+{\gamma }_{ij}(x)d{x}^{i}d{x}^j$. The Christoffel formula gives $H^0=-{\mathrm{tr}}_{h}K+{\mathrm{tr}}_{h}K=0$ and $H^i=-Q^i+h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk})=0$ on the slice. These smooth local data, including the auxiliary derivative variables, take values after shrinking in a convex finite-dimensional neighborhood whose closure lies in the open symmetric-hyperbolic state set. Extend $\gamma$ to a smooth positive metric on a periodic three-torus, agreeing on a smaller ball, by interpolating with a flat metric through the convex cone of positive matrices. Extend the state by a smooth cutoff to a constant admissible state; convexity keeps all values in that neighborhood. Extend the coefficient matrices by the same spatial cutoff, interpolating $A^0$ to the identity and each symmetric $A^i$ and the lower-order term to zero. The convex cone of positive matrices preserves a positive time matrix, and the spatial matrices remain symmetric. The conditional local solver in [F1] therefore gives a smooth reduced solution on a torus cylinder, which satisfies the original reduced equations wherever the cutoff is one. The derivative-definition relations remain valid along the time lines there, by the same homogeneous commutator ODE as step 3.1. The Maxwell divergence residuals vanish near the chosen point by the local flow argument of [F2]. The Einstein constraints and $H=0$ hold on the initial ball; the local cone-energy proof of [F3] therefore gives $H=0$ in a smaller domain of dependence. On that neighborhood the reduced solution solves the full source-free system. This proves the additional local existence clause without imposing constraints on the auxiliary extension outside the slice ball.