Mean-zero pressure as a distribution

Write $T=u\otimes u$. Since $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(0,\mathrm{\infty };L_x^2)$, it and $T$ are locally integrable on $U$; on each compact time interval this follows from finite spatial measure and $\mathrm{\parallel }u(t)\otimes u(t){\mathrm{\parallel }}_{L_x^1}=\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L_x^2}^2$. For any fixed compact time support $K$, the regular pairings of $u$ and $T$ are bounded by their local $L^1$ norms times $p_{K,0}$, so their absolute-value seminorms restrict continuously to every fixed-support test space and belong to the family defining the test topology. They are spacetime distributions. Their distributional derivatives therefore have the declared transpose meaning. The map $a\mapsto a\otimes a$ is continuous from $L_x^2$ to $L_x^1$, since $\mathrm{\parallel }a\otimes a-b\otimes b{\mathrm{\parallel }}_1\le (\mathrm{\parallel }a{\mathrm{\parallel }}_2+\mathrm{\parallel }b{\mathrm{\parallel }}_2)\mathrm{\parallel }a-b{\mathrm{\parallel }}_2$; hence the Fourier coefficients $\widehat{T}(k,t)$ are measurable and bounded on finite time intervals. For $k\mathrm{\ne }0$ set $\widehat{\pi }(k,t)=-{\sum }_{a,b=1}^d\frac{{\kappa }_{k,a}{\kappa }_{k,b}}{\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2}{\widehat{T}}_{ab}(k,t),\widehat{\pi }(0,t)=0.$ The coefficients are bounded in $k$ by $C_d\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2$. For a smooth spacetime test $\varphi$, repeated spatial integration by parts makes $\widehat{\varphi }(k,t)$ decay faster than any power of $\mathrm{\mid }k\mathrm{\mid }$, uniformly with each time derivative on its compact support. Therefore $⟨\pi ,\varphi ⟩=\int {\sum }_k\widehat{\pi }(k,t)\widehat{\varphi }(-k,t)dt$ converges absolutely and is bounded by a constant times one finite-order seminorm $p_{K,m}$. Its absolute-value seminorm belongs to the family defining the test topology, so it defines a spacetime distribution with zero spatial mean.

At each nonzero frequency, the Fourier multiplier identity gives $i{\kappa }_k\widehat{\pi }(k)=-(I-P_k)i{\sum }_b{\kappa }_{k,b}{\widehat{T}}_{\cdot b}(k)$; the zero-frequency identity is also valid because both sides vanish. The weak identity in [F1] is exactly ${\mathrm{\partial }}_{t}u+P\mathrm{div}T-\nu \mathrm{\Delta }u=0$ after testing against solenoidal fields. Adding the displayed gradient identity yields ${\mathrm{\partial }}_{t}u+\mathrm{div}T+\mathrm{\nabla }\pi -\nu \mathrm{\Delta }u=0$ for every smooth periodic vector test, which is the claimed equation.

If ${\pi }_1,{\pi }_2$ are two such mean-zero distributions, their difference $Q$ has zero spatial gradient. Its $k$th spatial Fourier coefficient, a distribution in time, therefore vanishes for every $k\mathrm{\ne }0$ since $i{\kappa }_k\mathrm{\ne }0$; its zero coefficient vanishes by the mean-zero normalization. For any compactly time-supported smooth periodic test $\varphi (t,x)$, repeated integration by parts in $x$ makes its spatial Fourier coefficients and all their time derivatives decay faster than every power of $\mathrm{\mid }k\mathrm{\mid }$, uniformly on the compact time support. Thus its rectangular Fourier partial sums converge to $\varphi$ in the smooth test-function topology. Continuity of $Q$ and vanishing of all its spatial Fourier coefficients give $⟨Q,\varphi ⟩=0$. Hence $Q=0$ as a spacetime distribution, proving uniqueness.