Finite-slab stability in a fixed hyperbolic gauge

Let $r=s-1$. By [F1] and compactness of $J\times {\mathbb{T}}_{2\pi }^3\times K_2$, all coefficient derivatives needed below are bounded and $A^0(u)$ is uniformly symmetric positive definite; [F4] gives equivalent weighted and unweighted $H^r$ norms.

Put $w=u-v$ and $F=F_u-F_v$. Subtracting the two equations and adding and subtracting each coefficient evaluated at $u$ gives $A^0(u){\mathrm{\partial }}_{t}w+{\sum }_i{A}^i(u){\mathrm{\partial }}_{i}w=F+Q$, where $Q=-(A^0(u)-A^0(v)){\mathrm{\partial }}_{t}v-{\sum }_i(A^i(u)-A^i(v)){\mathrm{\partial }}_{i}v-B(u)+B(v)$. This is an identity: the coefficient-difference terms multiplying derivatives of $v$ and the difference of the zero-order terms are exactly the residual after the common principal operator is applied to $w$.

Fix a smooth reference solution $\bar{u}$ with state range in $K_0$. Let $d>0$ be the distance of its compact range to the complement of $\mathrm{int}{K}_1$, put $M_0={\sup }_J\mathrm{\parallel }\bar{u}{\mathrm{\parallel }}_{H^s}$ and $R_{\ast }=M_0+2$, and let $K_{\mathrm{m}\mathrm{i}\mathrm{d}}$ be the closed $d\mathrm{/}3$ tube about that range. It is a compact subset of $\mathrm{int}{K}_1$. The coefficient time interval is open and contains compact $J$, so a compact enlarged time interval exists. The proof of [F5] on this enlarged interval, with datum values in $K_{\mathrm{m}\mathrm{i}\mathrm{d}}$, output values in $K_1$ and datum norm at most $R_{\ast }$, gives a common local time ${\tau }_{\ast }>0$ and bound $M_{\mathrm{l}\mathrm{o}\mathrm{c}}$ for every restart time in $J$: the coefficient bounds, positive time-matrix bound and state margin are uniform. Decrease ${\tau }_{\ast }$ if necessary to stay in that enlarged interval. Smooth restart data give smooth local solutions by [F6].

The composition-difference estimates in [F3], the state bound $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^{\mathrm{\infty }}{H}^s}\le R_z$, and the time-derivative bound $\mathrm{\parallel }{\mathrm{\partial }}_{t}v{\mathrm{\parallel }}_{L^{\mathrm{\infty }}{H}^r}\le R_t$ from [F1] imply $\mathrm{\parallel }Q(t){\mathrm{\parallel }}_{H^r}\le C\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^r}$. Solving the displayed difference equation for ${\mathrm{\partial }}_{t}w$ with $(A^0(u){)}^{-1}$, whose operator norm is at most $a^{-1}$ by [F4], and using multiplier bounds at order $r-1$ from [F3] gives $\mathrm{\parallel }{\mathrm{\partial }}_{t}w(t){\mathrm{\parallel }}_{H^{r-1}}\le C(\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^r}+\mathrm{\parallel }F(t){\mathrm{\parallel }}_{H^{r-1}})$; here $\mathrm{\parallel }{\mathrm{\partial }}_{i}w{\mathrm{\parallel }}_{H^{r-1}}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}$.

Define $E(t)={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}{\int }_{{\mathbb{T}}_{2\pi }^3}(D^{\alpha }w{)}^{\mathsf{T}}{A}^0(t,x,u)D^{\alpha }wdx$. Put $a=A^0(t,x,u)$, $a_i=A^i(t,x,u)$, $y_{\alpha }=D^{\alpha }w$, and $h={\mathrm{\partial }}_{t}w$. For $E_N:={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}\int (P_N{y}_{\alpha }{)}^{\mathsf{T}}a(P_N{y}_{\alpha })$, the finite Fourier sums $P_{N}w$ are $C^1$ in time by classical regularity, and differentiation gives $E_N^{\mathrm{\prime }}={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}(2⟨P_N{D}^{\alpha }h,a{P}_N{y}_{\alpha }⟩+⟨P_N{y}_{\alpha },{\mathrm{\partial }}_{t}a{P}_N{y}_{\alpha }⟩)$. The one-order tail estimate [F10] gives $P_N{y}_{\alpha }\to y_{\alpha }$ uniformly in $L^2$, since $w$ is bounded in $H^{r+1}$. Self-adjointness changes the first pairing to $⟨D^{\alpha }h,P_N(a{P}_N{y}_{\alpha })⟩$. The product $a{y}_{\alpha }$ is uniformly bounded in $H^1$ by the Leibniz rule and uniform $C^1$ bound on $a$, so $\mathrm{\parallel }{P}_N(a{P}_N{y}_{\alpha })-a{y}_{\alpha }{\mathrm{\parallel }}_2\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{P}_N{y}_{\alpha }-y_{\alpha }{\mathrm{\parallel }}_2+\mathrm{\parallel }(P_N-I)(a{y}_{\alpha }){\mathrm{\parallel }}_2\to 0$ uniformly on $J$. Because $D^{\alpha }h$ is uniformly bounded in $L^2$, the first pairings converge in $L^1(J)$; the quadratic pairings converge uniformly since ${\mathrm{\partial }}_{t}a$ is bounded. Thus $E_N\to E$ uniformly and $E_N^{\mathrm{\prime }}\to \mathcal{G}$ in $L^1$, where $\mathcal{G}={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}(2⟨D^{\alpha }h,a{y}_{\alpha }⟩+⟨y_{\alpha },{\mathrm{\partial }}_{t}a{y}_{\alpha }⟩)$. Passing to the limit in $E_N(t)-E_N(t^{\mathrm{\prime }})={\int }_{t^{\mathrm{\prime }}}^t{E}_N^{\mathrm{\prime }}$ shows $E$ is absolutely continuous and $E^{\mathrm{\prime }}=\mathcal{G}$ almost everywhere. Differentiating the difference equation in space is an $L^2$ identity almost everywhere because $w\in L^{\mathrm{\infty }}{H}^{r+1}$, $h\in L^{\mathrm{\infty }}{H}^r$, and $F+Q\in L^{\mathrm{\infty }}(J;H^r)$ by steps 1.2 and 2.1; periodicity gives no boundary term.

The derivatives of $A^{\mu }(t,x,u)$ are bounded on the fixed compact coefficient set. The $Q$ term is bounded by [step 2.1]; the commutator terms by [F3] and the $H^{r-1}$ bound on ${\mathrm{\partial }}_{t}w$ from that step; and the forcing pairing ${\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le r}⟨D^{\alpha }w,D^{\alpha }F⟩$ by $\mathrm{\parallel }w{\mathrm{\parallel }}_{H^r}\mathrm{\parallel }F{\mathrm{\parallel }}_{H^r}$. Absorbing the quadratic lower-order terms with $2xy\le x^2+y^2$ while retaining the forcing pairing in mixed form, and using [F4] for energy equivalence, gives $E^{\mathrm{\prime }}(t)\le \alpha E(t)+\beta \sqrt{E(t)}\mathrm{\parallel }F(t){\mathrm{\parallel }}_{H^r}$ and $a\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^r}^2\le E(t)\le b\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^r}^2$ almost everywhere, for separate positive constants $\alpha ,\beta$. The convergence established in step 3.1 justifies all time-derivative and commutator pairings.

For $\epsilon >0$, set $Y_{\epsilon }=(E+\epsilon {)}^{1\mathrm{/}2}$. Step 4.1 gives $Y_{\epsilon }^{\mathrm{\prime }}\le (\alpha \mathrm{/}2)Y_{\epsilon }+(\beta \mathrm{/}2)\mathrm{\parallel }F{\mathrm{\parallel }}_{H^r}$. Multiplication by $e^{-\alpha t\mathrm{/}2}$ and integration yield $Y_{\epsilon }(t)\le e^{\alpha t\mathrm{/}2}{Y}_{\epsilon }(0)+(\beta \mathrm{/}2){\int }_0^t{e}^{\alpha (t-r^{\mathrm{\prime }})\mathrm{/}2}\mathrm{\parallel }F(r^{\mathrm{\prime }}){\mathrm{\parallel }}_{H^r}d{r}^{\mathrm{\prime }}$. Let $\epsilon \downarrow 0$ and use energy equivalence. The stated estimate follows with $\lambda =\alpha \mathrm{/}2$, $C_0=\sqrt{b\mathrm{/}a}$ and $C_F=\beta \mathrm{/}(2\sqrt{a})$. When $F=0$, it is the homogeneous estimate.

For the neighborhood assertion, compare smooth $u$ to $\bar{u}$ and bootstrap $\mathrm{\parallel }u-\bar{u}{\mathrm{\parallel }}_{H^s}<{\delta }_{\ast }$, where $0<{\delta }_{\ast }<\min \{1,d\mathrm{/}(4C_{\mathrm{e}\mathrm{m}\mathrm{b}})\}$ and $C_{\mathrm{e}\mathrm{m}\mathrm{b}}$ is the $H^s\to C^0$ constant from [F7]. Then $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}<M_0+1<R_{\ast }$ and its values lie in $K_{\mathrm{m}\mathrm{i}\mathrm{d}}$. In the difference identity of step 1.2 with $v=\bar{u}$ and $F=0$, one has $\mathrm{\parallel }Q{\mathrm{\parallel }}_{H^s}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^s}$: extend each coefficient smoothly by a cutoff near $K_1$ as in [F3], write its difference as ${\int }_0^1{D}_{z}A(\bar{u}+\theta w)wd\theta$, and apply F3 and the $H^s$ algebra bound [F14]. The smooth reference has bounded ${\mathrm{\partial }}_t\bar{u}$ and ${\mathrm{\partial }}_i\bar{u}$ in $H^s$. At order $s-1$, inversion of the difference equation gives $\mathrm{\parallel }{\mathrm{\partial }}_{t}w{\mathrm{\parallel }}_{H^{s-1}}\le C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^s}$. Now repeat the spatial energy computation of step 4.1 at order $s$; F3 at its top order bounds the commutators with ${\mathrm{\partial }}_{t}w$ and ${\mathrm{\partial }}_{i}w$ by $C\mathrm{\parallel }w{\mathrm{\parallel }}_{H^s}$. Smoothness justifies differentiation. Thus $E_s^{\mathrm{\prime }}\le C{E}_s$ and $\mathrm{\parallel }w(t){\mathrm{\parallel }}_{H^s}\le C_{\ast }{e}^{C_{\ast }t}\mathrm{\parallel }w(0){\mathrm{\parallel }}_{H^s}$ in fixed numerical units, with $C_{\ast }$ independent of the partial interval. Choose the initial $H^s$ radius less than ${\delta }_{\ast }\mathrm{/}(2C_{\ast }{e}^{C_{\ast }T})$. A first exit from the bootstrap bound is impossible. To extend, restart at an interior time less than ${\tau }_{\ast }\mathrm{/}2$ before the current right endpoint; step 1.3 gives another smooth solution, and uniqueness [F2] identifies the solutions on the overlap. The same strict bootstrap estimate holds on their union. Successive interior restarts increase the covered interval by at least ${\tau }_{\ast }\mathrm{/}2$, so finitely many cover $J$. The resulting solutions have range in $K_1$ and a common $H^s$ bound. This argument also covers $s=5$ and invokes no lower-norm-controlled refinement at order five.

For the fixed harmonic-gauge reduction in the geometric clause, let $H$ be its gauge residual and $E$ its Einstein residual. Define $P\subseteq [0,T]$ to consist of times at which $H=0$ and the normal Einstein constraints vanish on the whole slice $\{t\}\times {\mathbb{T}}_{2\pi }^3$. The given initial gauge condition and constraints put $0$ in $P$. At any $t_0\in P$, the reduced matter solution has conserved stress by the stated fixed-model hypotheses, so [F13] gives an open neighborhood of that entire slice on which $H=0$ and $E=0$. The torus slice is compact, hence this neighborhood contains all slices with $\mathrm{\mid }t-t_0\mathrm{\mid }<\epsilon$ for some $\epsilon >0$; thus $P$ is relatively open. If $t_n\in P$ converges to $t_0$, smoothness of the reduced metric, matter fields and their residuals makes $H=0$ and the normal constraint residuals zero at $t_0$, so $P$ is closed. At an endpoint use [F5] to extend the smooth reduced solution a short distance, and [F2] to identify it with the original solution on the overlap; this makes the same open-set argument valid there. Since $[0,T]$ is connected, $P=[0,T]$. The neighborhoods from [F13] therefore cover the slab, proving the original Einstein-matter equations throughout it.

The constant and the number of restarts depend on the chosen finite $T$ and the fixed compact state margin. Neither the estimate nor the restart argument is uniform as $T\to \mathrm{\infty }$; [F11] is only an existence/maximality and isometry-uniqueness statement. The strict-fluid and dust qualifications in [F9] remain active: shocks or vacuum boundaries and dust flow crossings lie outside the corresponding local branch.