Finite solenoidal Galerkin coefficient dynamics

Assume the Axiom of Countable Choice: for every sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets there is a function $f$ on $\mathbb{N}$ such that $f(n)\in X_n$. For $d\in \{1,2,3\}$, let $Q_d={\mathbb{R}}^d\mathrm{/}(2\pi \mathbb{Z}{)}^d$ carry the probability measure $d\mu =(2\pi {)}^{-d}dx$. For $k\in {\mathbb{Z}}^d$ define $e_k:Q_d\to \mathbb{C}$ by $e_k([x])=e^{ik\cdot x}$; this is independent of the representative since $e^{2\pi ik\cdot n}=1$ for $n\in {\mathbb{Z}}^d$. For an integer $N\ge 1$ and a Borel-measurable map $q:Q_d\to {\mathbb{R}}^N$ with ${\int }_{Q_d}\mathrm{\mid }q(x)\mathrm{\mid }d\mu (x)<\mathrm{\infty }$ in the Euclidean norm, considered up to equality $\mu$-almost everywhere, define its componentwise Fourier coefficients by $\widehat{q}(k)={\int }_{Q_d}q(x)e^{-ik\cdot x}d\mu (x)$; this integral is finite componentwise because $\mathrm{\mid }{e}_k(x)\mathrm{\mid }=1$. The supplied space $H^0(Q_d;{\mathbb{R}}^N)$ is the completion of finite real trigonometric polynomials in the norm $\mathrm{\parallel }q{\mathrm{\parallel }}_{H^0}^2={\sum }_{k\in {\mathbb{Z}}^d}\mathrm{\mid }\widehat{q}(k){\mathrm{\mid }}^2$, identified with its unique $L^2$ representative; real classes satisfy $\widehat{q}(-k)=\overline{\widehat{q}(k)}$. For vector fields with these representatives, set $⟨v,w{⟩}_2={\int }_{Q_d}v\cdot wd\mu$. For an integer $r\ge 1$, an admissible finite solenoidal family consists of classes $v_1,\dots ,v_r\in H^0(Q_d;{\mathbb{R}}^d)$ whose representatives are smooth $2\pi$-periodic fields satisfying $\mathrm{div}{v}_i={\sum }_{m=1}^d{\mathrm{\partial }}_{x_m}{v}_i^m=0$ and $⟨v_i,v_j{⟩}_2={\delta }_{ij}$, where ${\delta }_{ij}=1$ if $i=j$ and $0$ otherwise. Given such a family, a parameter $\nu \ge 0$, and a coefficient vector $a=(a_1,\dots ,a_r)\in {\mathbb{R}}^r$, put $u_a={\sum }_{i=1}^r{a}_i{v}_i$ and define

$A_{ij}={\int }_{Q_d}{\sum }_{\mathrm{\ell },m=1}^d({\mathrm{\partial }}_{x_m}{v}_j^{\mathrm{\ell }})({\mathrm{\partial }}_{x_m}{v}_i^{\mathrm{\ell }})d\mu ,c_{ijk}={\int }_{Q_d}{\sum }_{\mathrm{\ell },m=1}^d{v}_j^m({\mathrm{\partial }}_{x_m}{v}_k^{\mathrm{\ell }})v_i^{\mathrm{\ell }}d\mu \mathrm{.}$

The unforced finite solenoidal Galerkin coefficient dynamics is the initial-value system for $a:[0,T)\to {\mathbb{R}}^r$,

$a_i^{\mathrm{\prime }}(t)+{\sum }_{j,k=1}^r{c}_{ijk}{a}_j(t)a_k(t)+\nu {\sum }_{j=1}^r{A}_{ij}{a}_j(t)=0,a_i(0)=a_i^0,i=1,\dots ,r,$

where $T\in (0,\mathrm{\infty }]$ is the right endpoint of the interval on which a solution is defined and $a^0\in {\mathbb{R}}^r$ is the initial coefficient vector. A solution on $[0,T)$ is continuous on $[0,T)$ and continuously differentiable on $(0,T)$, has a right derivative at $0$, and its derivative extends continuously to $0$ with that value; the ODE holds for $0<t<T$ and at $t=0$ using this right derivative. When $T<\mathrm{\infty }$, no value or derivative at $T$ is required because $T$ is excluded; when $T=\mathrm{\infty }$, no endpoint condition is imposed at infinity. The displayed integrals are finite because their integrands are smooth on the compact torus. This is a finite-dimensional polynomial ODE; this definition asserts no convergence to a solution of the infinite-dimensional PDE.

Periodic fluid Fourier Sobolev spaces

Axiom of Countable Choice

Wiedemann, §3.1.2, Definition 3.1 and §3.2, Theorem 3.4 proof Steps 1–2

Ryzhik, §5, equations (5.24)–(5.35)