Periodic torus calculus

For an integer $d\ge 1$, set ${\mathbb{T}}^d:={\mathbb{R}}^d\mathrm{/}(2\pi \mathbb{Z}{)}^d$ and let $Q_d:=[0,2\pi {]}^d$ be its chosen fundamental cube. A scalar or vector field on ${\mathbb{T}}^d$ is represented by its $(2\pi \mathbb{Z}{)}^d$-periodic lift to ${\mathbb{R}}^d$. For $k\ge 0$ and $m\ge 1$, $C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k({\mathbb{T}}^d;{\mathbb{R}}^m)$ consists of such lifts whose componentwise partial derivatives through order $k$ exist, are continuous and periodic, and are bounded, uniformly continuous, and Riemann integrable on $Q_d$. For a scalar field $f$ and a vector field $v=(v_1,\dots ,v_d)$, define

$\mathrm{\nabla }f:=({\mathrm{\partial }}_{1}f,\dots ,{\mathrm{\partial }}_{d}f),\mathrm{div}v:={\sum }_{j=1}^d{\mathrm{\partial }}_j{v}_j,\mathrm{\Delta }f:={\sum }_{j=1}^d{\mathrm{\partial }}_j^{2}f\mathrm{.}$

The integral of an integrable scalar field over ${\mathbb{T}}^d$ means its Riemann integral over $Q_d$. For a matrix field $A=(A_{ij})$, $\mathrm{\mid }A{\mathrm{\mid }}^2:={\sum }_{i,j}\mathrm{\mid }{A}_{ij}{\mathrm{\mid }}^2$. The notation $C^1([0,T];C_{\mathrm{p}\mathrm{e}\mathrm{r}}^k)$ means that the field is continuously differentiable in time in the supremum norms of its spatial derivatives through order $k$. These are dimensionless mathematical conventions; they assign no physical units.