Solenoidal energy weak velocity for incompressible Navier–Stokes

Assume Countable Choice as inherited from Energy weak solutions of incompressible Navier–Stokes. Let $D$, $V$, $Q$, $V_{\sigma }$, $H_{\sigma }$, $a_{\nu }$, and $c$ be as in Energy weak solutions of incompressible Navier–Stokes; the Stokes velocity and pressure spaces are $V=H_0^1(D{)}^d$ and $Q=L_0^2(D)$ as specified in Stokes velocity and pressure spaces and weak forms. For $f\in L^2(0,T;L^2(D{)}^d)$ and $u_0\in H_{\sigma }$, a solenoidal energy weak velocity is a velocity $u\in L^{\mathrm{\infty }}(0,T;H_{\sigma })\cap L^2(0,T;V_{\sigma })$ that is weakly continuous into $H_{\sigma }$, satisfies $\mathrm{div}u=0$ almost everywhere, and obeys the scalar-test weak equation $-{\int }_0^T(u(t),v{)}_{L^2}{\eta }^{\mathrm{\prime }}(t)dt+{\int }_0^T(a_{\nu }(u(t),v)+c(u(t);u(t),v))\eta (t)dt={\int }_0^T(f(t),v{)}_{L^2}\eta (t)dt+(u_0,v{)}_{L^2}\eta (0)$ for every fixed $v\in V_{\sigma }$ and $\eta \in C^1([0,T])$ vanishing near $T$. It also satisfies, for almost every $t\in (0,T)$, $\frac{1}{2}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^2}^2+\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_{L^2}^{2}ds\le \frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2+{\int }_0^t(f(s),u(s){)}_{L^2}ds\mathrm{.}$ This solenoidal definition does not itself assert a pressure exists in $L^{4\mathrm{/}3}(0,T;L_0^2(D))$; that is a separate regularity conclusion.