Integrability of the energy weak Navier–Stokes formulation

The Bochner $L^2(0,T;V)$ hypothesis makes $u$ strongly measurable in $V$. The continuous trilinear bound [F2] makes $v\mapsto c(v;v,\cdot )$ a continuous map $V\to V^{\ast }$, so applying it to simple-function approximants of $u$ shows that $t\mapsto c(u(t);u(t),\cdot )$ is strongly measurable. Since $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2$ is essentially bounded and $\mathrm{\nabla }u\in L^2(0,T;L^2)$, [F2] gives ${\int }_0^T\mathrm{\parallel }c(u;u,\cdot ){\mathrm{\parallel }}_{V^{\ast }}^{4\mathrm{/}3}dt\le C\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}{L}^2}^{2\mathrm{/}3}{\int }_0^T\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^{2}dt<\mathrm{\infty }$.

The time derivative term pairs $u\in L^{\mathrm{\infty }}(0,T;H_{\sigma })$ with ${\mathrm{\partial }}_t\phi \in L^{\mathrm{\infty }}(0,T;L^2)$; the viscosity term is bounded by $2\nu \mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2{L}^2}\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{L^2{L}^2}$. Since $\mathrm{\mid }\mathrm{div}\phi \mathrm{\mid }\le \sqrt{d}\mathrm{\mid }\mathrm{\nabla }\phi \mathrm{\mid }$, the pressure term is finite by Holder from $p\in L^{4\mathrm{/}3}{L}^2$ and $\phi \in L^{\mathrm{\infty }}V$. The forcing term is finite because $f\in L^2{L}^2$ and $\phi$ is bounded in $L^{\mathrm{\infty }}{L}^2$.

Each spatial pairing is therefore finite, and step 1.1 gives the stated Bochner integrability of the nonlinear functional; the time-tested weak formulation is well-defined.