Weak heat solutions on the continuum torus

Assume the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), meaning every at most countable family of nonempty sets has a choice function. Let $d\ge 1$, let ${\mathbb{T}}^d=(\mathbb{R}\mathrm{/}\mathbb{Z}{)}^d$ carry the normalized product Haar measure $dx$ with $dx({\mathbb{T}}^d)=1$, and let ${\rho }_0$ be a real measurable function on ${\mathbb{T}}^d$ with $0\le {\rho }_0\le 1$ almost everywhere. For a finite $T\ge 0$, a weak heat solution with datum ${\rho }_0$ is a family $(u_t{)}_{0\le t\le T}$ of real essentially bounded measurable-function classes (elements of $L^{\mathrm{\infty }}({\mathbb{T}}^d,dx;\mathbb{R})$), with $u_0={\rho }_0$ as a function class, such that for every complex-valued smooth periodic test function $G$, the map $t\mapsto {\int }_{{\mathbb{T}}^d}{u}_t(x)G(x)dx$ is continuous and

${\int }_{{\mathbb{T}}^d}{u}_t(x)G(x)dx={\int }_{{\mathbb{T}}^d}{\rho }_0(x)G(x)dx+{\int }_0^t{\int }_{{\mathbb{T}}^d}{u}_s(x)\mathrm{\Delta }G(x)dxds$

for every $0\le t\le T$. Here $\mathrm{\Delta }={\sum }_{i=1}^d{\mathrm{\partial }}_{x_i}^2$ is the periodic Laplacian; equality of function classes means equality almost everywhere with respect to $dx$. For the Fourier convention on this torus, set $e_k(x)=\exp (2\pi ik\cdot x)$ for $k\in {\mathbb{Z}}^d$ and $\widehat{f}(k)={\int }_{{\mathbb{T}}^d}f(x)e_{-k}(x)dx$ for $f\in L^1({\mathbb{T}}^d,dx)$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

Fourier coefficients and trigonometric polynomials on the torus