Energy weak solutions of incompressible Navier–Stokes

Assume Countable Choice, as in the Stokes-space conventions of Stokes velocity and pressure spaces and weak forms. Let $D\subset {\mathbb{R}}^d$ be a nonempty bounded connected Lipschitz domain, $d\in \{2,3\}$, and fix $0<T<\mathrm{\infty }$ and $\nu >0$. Use $V=H_0^1(D;{\mathbb{R}}^d)$ and $Q=L_0^2(D)$ from Stokes velocity and pressure spaces and weak forms, the viscosity form $a_{\nu }(v,w)=2\nu {\int }_D\epsilon (v):\epsilon (w)$, and the form $c$ from The skew convective form for incompressible flow. Set $V_{\sigma }=\{v\in V:\mathrm{div}v=0\text{ in }{L}^2(D)\},H_{\sigma }={\overline{V_{\sigma }}}^{L^2(D)}\mathrm{.}$ For $f\in L^2(0,T;L^2(D;{\mathbb{R}}^d))$ and $u_0\in H_{\sigma }$, an energy weak solution is a velocity $u\in L^{\mathrm{\infty }}(0,T;H_{\sigma })\cap L^2(0,T;V_{\sigma })$, weakly continuous into $H_{\sigma }$, and a pressure $p\in L^{4\mathrm{/}3}(0,T;Q)$ with zero spatial mean a.e., such that $\mathrm{div}u=0$ a.e. and, for every $\phi \in C^1([0,T];V)$ vanishing near $T$, $-{\int }_0^T(u,{\mathrm{\partial }}_t\phi {)}_{L^2}dt+{\int }_0^T(a_{\nu }(u,\phi )+c(u;u,\phi )+b(\phi ,p))dt={\int }_0^T(f,\phi {)}_{L^2}dt+(u_0,\phi (0){)}_{L^2}\mathrm{.}$ It also satisfies, for almost every $t\in (0,T)$, $\frac{1}{2}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^2}^2+\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_{L^2}^{2}ds\le \frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2+{\int }_0^t(f(s),u(s){)}_{L^2}ds\mathrm{.}$ No uniqueness is included in this definition. The formulation uses the real duality convention of Banach evolution triples and weak time derivatives.