A quantitative continuation and bootstrap contract

Fix one evolution problem, its state space and its declared classical solution class. For the general-relativity application here, use one fixed periodic symmetric-hyperbolic system on ${\mathbb{T}}_{2\pi }^3$ with integer regularity $s\ge 5$, a fixed coordinate gauge and reconstruction, smooth coefficient maps on an open state domain $O$, symmetric principal matrices and a positive time matrix. The relevant fixed-gauge context is the one defined by def-gr-fx-fixed-gauge-data-topology: one coefficient family and one matter branch, compact state sets $K_0\subset \mathrm{int}{K}_1\subset \mathrm{int}{K}_2\subset O$, a uniform positive lower and finite upper bound for $A^0$ on $K_2$, and separate bounds $R_z,R_t$ for the state and its time derivative. The state components are divided by their fixed unit scales before their Euclidean-valued Sobolev norms are formed; these are coordinate norms and do not quotient by diffeomorphisms. On each compact time slab, the resulting normalized state $u$ belongs to $L^{\mathrm{\infty }}{H}^s\cap C^0{H}^{s-1}$ and has ${\mathrm{\partial }}_{t}u\in L^{\mathrm{\infty }}{H}^{s-1}$.

A forward solution is given on $[0,T_{\max })$, where $T_{\max }\in (0,\mathrm{\infty }]$. It is maximal in this solution class if, when $T_{\max }<\mathrm{\infty }$, no solution on $[0,T_{\max }+\rho )$ for any $\rho >0$ agrees with it on $[0,T_{\max })$. Let $E_{\mathrm{r}\mathrm{a}\mathrm{w}}$ be a specified nonnegative scalar functional of the solution, continuous on every compact subinterval of $[0,T_{\max })$. If $E_{\mathrm{r}\mathrm{a}\mathrm{w}}$ has physical units, choose a fixed positive scale $E_{\ast }$ with the same units and set $E=E_{\mathrm{r}\mathrm{a}\mathrm{w}}\mathrm{/}{E}_{\ast }$. If $E_{\mathrm{r}\mathrm{a}\mathrm{w}}$ is dimensionless, set $E=E_{\mathrm{r}\mathrm{a}\mathrm{w}}$. In either case $E$ is dimensionless. In the general-relativity convention the time coordinate is $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in metres; choose a fixed length scale $L_{\ast }>0$ and use the dimensionless time $t=x^0\mathrm{/}{L}_{\ast }$. Thus ${\mathrm{\partial }}_t=L_{\ast }{\mathrm{\partial }}_{x^0}$. The fixed-gauge bound $R_t$ is on ${\mathrm{\partial }}_{x^0}u$ and has units ${\mathrm{m}}^{-1}$; the corresponding bound with respect to $t$ is $L_{\ast }{R}_t$. The constants $A$, $ϵ$ and $\eta$ below are dimensionless as well. In a purely mathematical application, take these variables as real scalars without a physical interpretation.

Specify a finite list of dimensionless margin functions $m_j:[0,T_{\max })\to \mathbb{R}$ and thresholds ${\mu }_j>0$. Write $\mathcal{M}(T)$ when $m_j(t)\ge {\mu }_j$ for every listed $j$ and every $0\le t<T$. The continuation criterion is a separate result for the chosen evolution problem: if a solution has a finite right endpoint $T$ with $E(t)\le \eta$ for every $0\le t<T$ and $\mathcal{M}(T)$, then it extends past $T$ in the same solution class.

For constants $A>0$, $ϵ>0$ and $\eta >0$ with $2A{ϵ}^2<\eta$, a bootstrap estimate is the implication, required on every finite $T<T_{\max }$,

$[E(t)\le 2A{ϵ}^2\text{ for all }0\le t\le T]⟹[E(t)\le A{ϵ}^2\text{ for all }0\le t\le T]\mathrm{.}$

This contract records the continuity and continuation inputs needed by an application. It does not itself prove the bootstrap estimate, the margin bounds, or any Einstein evolution estimate.

Fixed-gauge finite-slab data and solution topology

Finite-slab stability in a fixed hyperbolic gauge

Metivier, Chapter 3, §§3.3–3.5

Wang, §6.3

Klainerman and Szeftel, abstract and §1.2.1

Jaramillo, Valiente Kroon and Gourgoulhon, §§1.2.1 and 2.2

The fixed-gauge context is the one in Fixed-gauge finite-slab data and solution topology. The finite-slab supplier Finite-slab stability in a fixed hyperbolic gauge assumes Countable Choice and Dependent Choice and proves only the stated fixed-gauge finite-time comparison and neighborhood results under its compact-state, regularity, constraint, matter, gauge and reconstruction hypotheses. It does not supply the improvement implication or continuation margins in this definition, and its proof is not uniform as the finite slab length tends to infinity.