The evolution-space energy identity and continuous H trace

For a smooth $V$-valued function $w$, the continuous inclusion into $H$ and the ordinary product rule give $\frac{d}{dt}\mathrm{\parallel }w(t){\mathrm{\parallel }}_H^2=2(w^{\mathrm{\prime }}(t),w(t){)}_H=2⟨w^{\mathrm{\prime }}(t),w(t){⟩}_{V^{\ast },V}$ by the canonical embedding Evolution triples and the parabolic energy space.

For smooth $w$, choose $s$ with $\mathrm{\parallel }w(s){\mathrm{\parallel }}_H^2\le T^{-1}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(H)}^2$. Integrating step 1.1 from $s$ to any $t$ gives ${\sup }_t\mathrm{\parallel }w(t){\mathrm{\parallel }}_H^2\le T^{-1}{c}_i^2\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(V)}^2+2\mathrm{\parallel }{w}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(V^{\ast })}\mathrm{\parallel }w{\mathrm{\parallel }}_{L^2(V)}\mathrm{.}$ Apply this to differences of the approximants in [F2]; they are Cauchy in $C([0,T];H)$. Their uniform limit is continuous and agrees with $u$ almost everywhere by their $L^2(H)$ convergence. The same bound passes to the limit.

For approximants $u_n$ from [F2], integrate step 1.1 between arbitrary $s,t$. Uniform $H$ convergence passes the two endpoint norms to those of the continuous representative. The integrands converge in $L^1$ because $\mathrm{\parallel }⟨u_n^{\mathrm{\prime }},u_n⟩-⟨u^{\mathrm{\prime }},u⟩{\mathrm{\parallel }}_{L^1}\le \mathrm{\parallel }{u}_n^{\mathrm{\prime }}-u^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(V^{\ast })}\mathrm{\parallel }{u}_n{\mathrm{\parallel }}_{L^2(V)}+\mathrm{\parallel }{u}^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^2(V^{\ast })}\mathrm{\parallel }{u}_n-u{\mathrm{\parallel }}_{L^2(V)},$ whose right-hand side tends to zero. This proves the energy identity for all $s,t$.

Apply the energy identity from step 2.2 to $u+v$ and $u-v$, subtract, and use the real polarization identity $\mathrm{\parallel }u+v{\mathrm{\parallel }}_H^2-\mathrm{\parallel }u-v{\mathrm{\parallel }}_H^2=4(u,v{)}_H$. Bilinearity gives the asserted integration-by-parts identity.

The trace estimate is the smooth bound in step 2.1 passed through the uniform approximation in [F2], with $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(H)}\le c_i\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(V)}$. If two continuous representatives agree almost everywhere, their difference is continuous and zero on a dense subset of $[0,T]$, hence zero everywhere; the representative is unique.