Periodic solenoidal Fourier spaces

Let $d\ge 2$ be an integer, and let $L>0$ be the numerical period in a fixed length unit. Let ${\mathbb{T}}_L^d=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^d$ carry normalized Lebesgue measure $L^{-d}dx$. For $k\in {\mathbb{Z}}^d$ put ${\kappa }_k=2\pi k\mathrm{/}L$ and $e_k(x)=\exp (i{\kappa }_k\cdot x)$. For a complex scalar or vector $L^2$ class $f$, let $\widehat{f}(k)=L^{-d}{\int }_{[0,L{]}^d}f(x)e^{-i{\kappa }_k\cdot x}dx$; real fields are the classes satisfying $\widehat{f}(-k)=\overline{\widehat{f}(k)}$.

The periodic scalar test functions are $C^{\mathrm{\infty }}({\mathbb{T}}_L^d;\mathbb{C})$, identified with smooth $L$-periodic lifts to ${\mathbb{R}}^d$ and equipped with the seminorms $p_m(\phi )={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_{x\in {\mathbb{T}}_L^d}\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\phi (x)\mathrm{\mid }(m\in {\mathbb{N}}_0)\mathrm{.}$ A periodic distribution is a continuous complex-linear functional on this test space, with bilinear pairing. Its derivative is defined by $⟨{\mathrm{\partial }}_{j}T,\phi ⟩=-⟨T,{\mathrm{\partial }}_j\phi ⟩\mathrm{.}$ For $f\in L^2({\mathbb{T}}_L^d;\mathbb{C})$, its associated regular distribution is $T_f(\phi )={\int }_{{\mathbb{T}}_L^d}f\phi d\mu$; vector fields are treated componentwise. This is continuous since $\mathrm{\mid }\int f\phi d\mu \mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_2{p}_0(\phi )$. For a vector distribution $U=(U_1,\dots ,U_d)$, define $⟨\mathrm{div}U,\phi ⟩=-{\sum }_{j=1}^d⟨U_j,{\mathrm{\partial }}_j\phi ⟩\mathrm{.}$ Thus, for $u\in L^2({\mathbb{T}}_L^d;{\mathbb{R}}^d)$, $\mathrm{div}u=0$ distributionally means ${\int }_{{\mathbb{T}}_L^d}u(x)\cdot \mathrm{\nabla }\phi (x)L^{-d}dx=0(\phi \in C^{\mathrm{\infty }}({\mathbb{T}}_L^d;\mathbb{C}))\mathrm{.}$

For $s\ge 0$, let $H^s({\mathbb{T}}_L^d)=\{f\in L^2:{\sum }_{k\in {\mathbb{Z}}^d}(1+L^2\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2<\mathrm{\infty }\},$ with norm given by the square root of this sum; the weight is dimensionless. Under Countable Choice the coefficient map is injective, so this is a norm. The real solenoidal spaces are $H=\{u\in L^2({\mathbb{T}}_L^d;{\mathbb{R}}^d):k\cdot \widehat{u}(k)=0\text{ for every }k\mathrm{\ne }0\},V=H\cap H^1({\mathbb{T}}_L^d;{\mathbb{R}}^d)\mathrm{.}$ The zero Fourier mode is unrestricted. Under Countable Choice, the supplied Fourier-basis theorem makes the coefficient map a surjective isometry from $L^2$ onto ${\mathrm{\ell }}^2$; the coefficient-space norm identifications and the equivalence between Fourier divergence and distributional divergence in this definition use that hypothesis. Under the same assumption, the orthogonal solenoidal projection $P$ is well-defined on $L^2$ and has multiplier $P_0=I$ and $P_k=I-{\kappa }_k{\kappa }_k^{\mathsf{T}}\mathrm{/}\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2$ for $k\mathrm{\ne }0$. For integer $N\ge 0$, rectangular truncation ${\mathrm{\Pi }}_N$ keeps precisely $\mathrm{\mid }k{\mathrm{\mid }}_{\mathrm{\infty }}\le N$. A weak derivative in coordinate $j$ has Fourier coefficient $i{\kappa }_{k,j}\widehat{f}(k)$. For $r>0$, $H^{-r}$ is the Hilbert completion of finite complex Fourier coefficient sequences in the norm squared ${\sum }_k(1+L^2\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2{)}^{-r}\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2$; its real subspace is the completion of the conjugate-symmetric sequences.

Leonid Ryzhik, Math 256B Lecture Notes, §5.2.2, pp. 42–43.

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions, §2.1–2.2, pp. 4–5.