Finite-slab geometric Cauchy stability

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.

The finite-slab estimate [F1] with zero source difference gives $\mathrm{\parallel }{z}_n-z{\mathrm{\parallel }}_{C(J;H^{s-1})}\to 0$ from convergence of the initial data. Since $s\ge 5$, [F2] gives uniform convergence in space of the metric components and their first spatial derivatives; the first-order state includes $p_{ab}={\mathrm{\partial }}_0{g}_{ab}$ and $q_{abi}={\mathrm{\partial }}_i{g}_{ab}$, so those derivatives converge uniformly as well. Thus the metric components converge in $C^1$ on the compact spacetime slab.

The classical first-order states are $C^1$, and ${\mathrm{\partial }}_{0}g=p$, ${\mathrm{\partial }}_{i}g=q_i$, so their metric components are $C^2$. The coordinate formula in [F3] also applies at this regularity: substitution in ${\mathrm{\nabla }}_b{g}_{cd}=0$ and ${\mathrm{\Gamma }}^a{}_{bc}={\mathrm{\Gamma }}^a{}_{cb}$ verifies compatibility and torsion freedom using only first metric derivatives, and the same algebra determines the coefficients uniquely. Thus the Christoffel functions are formed from the metric and its first derivatives by finite sums, products and inversion of the metric matrix. The metric matrices lie in a compact subset of the nonsingular Lorentzian matrices, so the adjugate-over-determinant formula bounds their inverses; step 1.1 therefore gives $\mathrm{\Gamma }(g_n)\to \mathrm{\Gamma }(g)$ uniformly. The uniform $H^s$ state and time-derivative bounds in the hypotheses, together with [F2], bound all second coordinate derivatives of the metric on every compact chart neighborhood: purely spatial derivatives come from the $H^s$ bounds for $q$, and derivatives involving $x^0$ come from ${\mathrm{\partial }}_{t}p$ and ${\mathrm{\partial }}_{t}q$, which are bounded in $H^{s-1}$. Thus the Christoffel functions and geodesic vector fields have uniformly bounded first derivatives on a compact neighborhood of the lift set. A finite cover by convex coordinate balls, together with boundedness of the vector fields for pairs outside a common ball, gives one Lipschitz constant on the compact lift set. The geodesic vector fields $G_n(x,v)=(v,-\mathrm{\Gamma }(g_n)(x)[v,v])$ also converge uniformly there to $G$.

Let $Y_n=(x_n,v_n)$ and $Y=(x,v)$ be the position-velocity lifts. The coordinate covariant-acceleration formula in [F4], obtained by differentiating the velocity and adding ${\mathrm{\Gamma }}^k{}_{ij}{\dot{x}}^i{\dot{x}}^j$, uses only $C^2$ curves and continuous Christoffel functions; hence their chart components satisfy $\dot{x}=v$, $\dot{v}=-\mathrm{\Gamma }(x)[v,v]$. Their integral equations and the common Lipschitz constant $L$ give $\mathrm{\mid }{Y}_n(\lambda )-Y(\lambda )\mathrm{\mid }\le \mathrm{\mid }{Y}_n(a)-Y(a)\mathrm{\mid }+(b-a)\mathrm{\parallel }{G}_n-G{\mathrm{\parallel }}_{\mathrm{\infty }}+L{\int }_a^{\lambda }\mathrm{\mid }{Y}_n(r)-Y(r)\mathrm{\mid }dr\mathrm{.}$ The first two terms form a constant $A_n\to 0$. Apply [F5] to $u(\lambda )=\mathrm{\mid }{Y}_n(\lambda )-Y(\lambda )\mathrm{\mid }$ to get ${\sup }_I\mathrm{\mid }{Y}_n-Y\mathrm{\mid }\le A_n\exp (L(b-a))\to 0$. Since ${\dot{x}}_n=v_n$, this is exactly $C^1$ convergence of the coordinate curves.

On the common compact phase set, the velocities are uniformly bounded. Uniform metric convergence from step 1.1 and uniform curve/velocity convergence from step 3.1 imply $g_n({\dot{\gamma }}_n,{\dot{\gamma }}_n)⟶g(\dot{\gamma },\dot{\gamma })$ uniformly on $I$. The limit therefore satisfies $g(\dot{\gamma },\dot{\gamma })\le -\eta$; because all curves use the same time orientation, the limit remains future timelike. The uniform margin places all numbers $-g_n({\dot{\gamma }}_n,{\dot{\gamma }}_n)$ and their limit in a bounded interval $[\eta ,M]$ with $\eta >0$. The square-root function is Lipschitz on this interval, so the proper-time integrands converge uniformly. Their integrals on the finite interval converge, and [F6] identifies them with the stated proper times.