Periodic high-Sobolev state spaces

For the geometric data convention, orient ${\mathbb{T}}_{2\pi }^3$ by the ordered dimensionless coordinates $({\theta }^1,{\theta }^2,{\theta }^3)$; the corresponding length-coordinate frame $({\mathrm{\partial }}_{X^1},{\mathrm{\partial }}_{X^2},{\mathrm{\partial }}_{X^3})$ has the same orientation.

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as required by The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s and The Fourier basis and Parseval's identity on the finite torus; no use of the full Axiom of Choice is made here. Let ${\mathbb{T}}_{2\pi }^3$, its normalized measure, the componentwise real Sobolev spaces $H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$, and their weak-derivative and Fourier-coefficient conventions be exactly those of The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s. In particular, $H^s$ is defined for integer $s\ge 0$, the weak-derivative norm is equivalent to the Fourier-weight norm by Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms, and the trigonometric characters are a complete orthonormal system by The Fourier basis and Parseval's identity on the finite torus. A field is a function with the domain and codomain declaration of A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain; Sobolev fields are its $L^2$-equivalence classes, not pointwise representatives.

For finite $m\ge 1$, open $O\subseteq {\mathbb{R}}^m$, compact $K\subseteq O$, $R\ge 0$, and integer $s\ge 5$, define the bounded admissible initial-state class

${\mathcal{S}}_s(K,R):=\{u_0\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m):u_0\text{ has its continuous representative valued in }K,\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^s}\le R\}\mathrm{.}$

The representative is unique because two continuous functions equal almost everywhere on the torus agree everywhere. For a smooth periodic first-order reduction of the electrovacuum equations, its specified finite list of real state components is identified with one such ${\mathbb{R}}^m$-valued function; the state-component convention is part of that reduction. In the geometric data convention below, the component arrays are specified separately. This is a definition of admissible function-space classes only; it asserts no existence result for a coupled Einstein-Maxwell reduction.

For the geometric data convention, write the dimensionless torus coordinate as ${\theta }^i\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and introduce a fixed length scale $L_0>0$ in metres. The corresponding length coordinates are $X^i=L_0{\theta }^i$ on ${\mathbb{T}}_{2\pi L_0}^3:={\mathbb{R}}^3\mathrm{/}(2\pi L_0\mathbb{Z}{)}^3$; the Sobolev variable remains $\theta$, and $X$ is not a second name for a coordinate on ${\mathbb{T}}_{2\pi }^3$. The arrays $h_{ij}(\theta )$ and $K_{ij}(\theta )$ are the covariant metric and second-fundamental-form components in the length-coordinate frame ${\mathrm{\partial }}_{X^i}$, with $h_{ij}=h_{ji}$ and $K_{ij}=K_{ji}$. Let ${\mathsf{e}}_A(h)$, $A=1,2,3$, be the positively oriented $h$-orthonormal frame obtained by Gram-Schmidt from the ordered global frame $({\mathrm{\partial }}_{X^1},{\mathrm{\partial }}_{X^2},{\mathrm{\partial }}_{X^3})$. For spatial vectors $e,b$, the electromagnetic component arrays in this convention are $e_A:=h(e,{\mathsf{e}}_A)$ and $b_A:=h(b,{\mathsf{e}}_A)$; these are the orthonormal observer components used in the Einstein-Maxwell first-order reduction. The frame coefficients are smooth functions of the positive matrix $(h_{ij})$. If $h\in H^{s+1}$ has range in a compact subset of the positive-definite matrices, the composition supplier gives these coefficients in $H^{s+1}$; since $s\ge 5$, the Sobolev-algebra supplier gives their products with coordinate vector components in $H^s$. The inverse frame matrix has the same property, so the coordinate and orthonormal component arrays describe the same $H^s$ vector fields.

Thus the data component spaces mean $(h_{ij})\in H^{s+1}({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^9)$ with values in symmetric positive matrices, $(K_{ij})\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^9)$ with values in symmetric matrices, and $(e_A),(b_A)\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^3)$. A smooth periodic first-order state also includes the specified component functions of that reduction and the first spatial metric derivatives; in the above length-coordinate convention $v_{kij}:={\mathrm{\partial }}_{X^k}{h}_{ij}=L_0^{-1}{\mathrm{\partial }}_{{\theta }^k}{h}_{ij}$ and $K_{ij}$ carry one inverse-length factor. For a fixed field scale $F_0>0$ in tesla, the dimensionless physical component arrays are $h_{ij}$, $L_0{K}_{ij}$, $L_0{v}_{kij}$, $e_A\mathrm{/}{F}_0$, and $b_A\mathrm{/}{F}_0$. Each fixed rescaling is invertible and preserves the finite-component Sobolev topology. These are component and unit conventions, not additional field equations.