Local Sobolev stability and continuation for the reduced electrovacuum system

By [F5], the reduced equation has the form $A^0(t,x,U){\mathrm{\partial }}_{t}U+A^i(t,x,U){\mathrm{\partial }}_{i}U+B(t,x,U)=0$ in the sense of Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions. Its time matrix is $\mathrm{diag}(I,-g^{00}I,g^{ij}I,{\alpha }^{-1}{I}_6)$, positive definite throughout the admissible open set. Every spatial matrix is symmetric: the metric wave blocks and the Maxwell curl blocks are symmetric, while the shift adds a scalar identity to the latter. The algebraic stress and the frame derivatives, expressed using $p,q$, create only smooth zero-order terms, so there is no off-diagonal derivative coupling. Enclose the fixed compact state range in the interior of a larger compact subset of this open set; on that subset and a compact coefficient-time slab containing $T$ in its interior, smoothness and strict positivity give bounded coefficient derivatives and a uniform positive time-matrix lower bound. These verify the concrete hypotheses required by [L1] and [F3] at the restart. Since $\mathrm{\parallel }U{\mathrm{\parallel }}_{H^s}$ is uniformly bounded and $H^{s-1}$ is an algebra for $s\ge 5$, the equation and [F4] also give $\mathrm{\parallel }{\mathrm{\partial }}_{t}U{\mathrm{\parallel }}_{H^{s-1}}\le C_R$ near $T$.

For each finite Fourier projection $P_N$, every coefficient of $P_{N}U$ is absolutely continuous and its derivative is the corresponding coefficient of $P_N{\mathrm{\partial }}_{t}U$. Thus $\mathrm{\parallel }{P}_{N}U(t)-P_{N}U(t^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^{s-1}}\le C_R\mathrm{\mid }t-t^{\mathrm{\prime }}\mathrm{\mid }$. Letting $N\to \mathrm{\infty }$ and using Fourier convergence in $H^{s-1}$ gives the same estimate for $U(t)-U(t^{\mathrm{\prime }})$. Hence $U(t)$ has a strong $H^{s-1}$ limit $U_T$ as $t\uparrow T$.

For each finite frequency set $\{\mathrm{\mid }k\mathrm{\mid }\le N\}$, strong convergence of its coefficients gives ${\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }{\widehat{U}}_T(k){\mathrm{\mid }}^2={\lim }_{t\uparrow T}{\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }\widehat{U}(t,k){\mathrm{\mid }}^2\le R^2,$ where $R={\sup }_{t<T}\mathrm{\parallel }U(t){\mathrm{\parallel }}_{H^s}$. Taking the supremum over $N$ in the Fourier definition [F4] proves $U_T\in H^s$ and $\mathrm{\parallel }{U}_T{\mathrm{\parallel }}_{H^s}\le R$. By the embedding in [F4], $U(t)\to U_T$ uniformly in space. Since each $U(t,x)$ lies in the fixed compact state set, the closedness of that set gives $U_T(x)$ in the same set for every $x$.

For the full-development conclusion, assume the original solution has the compatibility, harmonic and full-equation properties stipulated in the Statement. The derivative identities from [F1] then pass to $U_T$: $q_T={\bar{\mathrm{\nabla }}}_i{g}_T$ because these relations are continuous in the strong $H^{s-1}$ limit. Since $q_T\in H^s$, the Fourier norm identity gives $g_T\in H^{s+1}$; likewise $p_T,e_T,b_T\in H^s$. At the endpoint order $H^3$ needed when $s=5$, the Fourier convolution estimate $⟨k{⟩}^3\le 4(⟨\mathrm{\ell }{⟩}^3+⟨k-\mathrm{\ell }{⟩}^3)$ and $\mathrm{\parallel }\widehat{u}{\mathrm{\parallel }}_{{\mathrm{\ell }}^1}\le ({\sum }_k⟨k{⟩}^{-6}{)}^{1\mathrm{/}2}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^3}$ prove $\mathrm{\parallel }uv{\mathrm{\parallel }}_{H^3}\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{H^3}\mathrm{\parallel }v{\mathrm{\parallel }}_{H^3}$; the same argument with exponent $2$ proves the $H^2$ product estimate. The chain rule through order three, $H^3\hookrightarrow C^1$, and Fourier approximation give continuity of smooth composition in $H^3$ on compact value sets: first derivatives converge uniformly, higher derivatives converge in $L^2$, and every chain-rule product therefore converges in $L^2$. In particular, matrix inversion is continuous in $H^3$ on a compact set of positive-definite metrics. For $s\ge 6$ the supplied algebra/composition estimates at $H^{s-2}\ge H^4$ apply directly, and for $s=5$ these local $H^3$ estimates cover the curvature and connection products. The constraints involve at most two spatial derivatives of $h_T$ and one derivative of $K_T,e_T,b_T$, which converge in $H^{s-2}$ from the first-order state and its derivative identities; each inverse-metric factor and each product in the constraint expressions is thus continuous at the endpoint. The harmonic vector is a smooth function of $(g_T,p_T,q_T)$, so $H_T=0$ as the limit of $H(t)=0$. Therefore $U_T$ determines constrained data of precisely the regularity required by [F1].

Restart the same harmonic-reduced coefficient system of [F5] at time $T$ with the actual complete state $U_T$, keeping the fixed background and coordinates unchanged. The coefficients are smooth on the same time-state domain, $T$ is interior to its time interval, and step 3.1 supplies an $H^s$ datum with compact admissible range and bounded norm. Applying [L1] (equivalently [F3] after translating time) gives a reduced solution $V$ on $(T-\epsilon ,T+\epsilon )$ with $V(T)=U_T$. This gives a reduced extension candidate for every state in the Statement, without reinitialization to lapse one, shift zero, or the special metric jet of the original initial slice. For the full-development conclusion, assume now the additional properties used in step 4.1. The reduction's equations propagate $q_i-{\bar{\mathrm{\nabla }}}_{i}g=0$ by the homogeneous background-commutator ODE, so $V$ has the required metric derivative identities. It has the same $H^s$ bounds and compact-state control as the local solver class; the spacetime-regularity argument in proof step 4.1 of [F1], which uses only those bounds, the derivative identities and the fixed smooth coefficients, applies to $V$ after translating time and yields $g_V\in C^3$ and $F_V\in C^2$. Its zero initial Maxwell divergence residuals from step 4.1 propagate by the normal-form transport argument of [F1]; the coordinate time foliation is smooth and its normal is $C^1$ at this regularity. Thus both full Maxwell equations hold, and the explicit classical stress identity carried by [F1] gives conservation at the restarted $C^3$ metric/$C^2$ field regularity; no smooth-only stress theorem is applied. The harmonic residual identity from [F1] gives the homogeneous wave equation for $H$. Initially $H=0$ by step 4.1, so its tangential derivatives vanish. In any orthonormal frame with future unit time vector $n$, the Einstein residual is $E_{ab}=\frac{1}{2}({\mathrm{\nabla }}_a{H}_b+{\mathrm{\nabla }}_b{H}_a-g_{ab}{\mathrm{\nabla }}_c{H}^c)$, hence $E(n,n)=\frac{1}{2}{\mathrm{\nabla }}_n{H}_n$ and $E(n,{\mathsf{e}}_i)=\frac{1}{2}{\mathrm{\nabla }}_n{H}_i$ there. The endpoint Einstein constraints from step 4.1 give ${\mathrm{\nabla }}_{n}H=0$. The same local wave-energy uniqueness argument of [F1], valid for this positive symmetric wave block with bounded coefficient derivatives, yields $H=0$ near the endpoint slice. Therefore $V$ is a full solution.

On the overlap before $T$, compare $V$ and the old solution $U$ using the difference-energy estimate [F2] on $[t,t^{\mathrm{\prime }}]$ with $t<t^{\mathrm{\prime }}<T$, in the reverse time direction. The compact state sets and $H^s$ norms of both solutions have common bounds, so its constant is uniform as $t^{\mathrm{\prime }}\uparrow T$. Their strong $H^{s-1}$ traces both tend to $U_T$, giving $\mathrm{\parallel }U(t)-V(t){\mathrm{\parallel }}_{H^{s-1}}\le C{e}^{C(t^{\mathrm{\prime }}-t)}\mathrm{\parallel }U(t^{\mathrm{\prime }})-V(t^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^{s-1}}\to 0$. Thus the solutions agree on the overlap, and $V$ extends the original reduced solution through $T$, and the full development when the additional hypotheses hold. The same estimate identifies any two continuations. Time reversal gives the left-endpoint case. On each common interval the same [F2] estimate gives the stated $H^{s-1}$ continuous dependence, with constants depending on the coefficient time slab, compact state set, norm bound and $s$.