Energy bound from a time-integrable L2 force

Statement

Suppose a solution satisfies the energy identity on $[0,T)$ , and $F(t)=\int_0^t\|f(r)\|_2\,dr$ is finite. Then

\|u(t)\|_2\le\|u(0)\|_2+F(t),

and

\|u(t)\|_2^2+2\nu\int_0^t\|\nabla u(r)\|_2^2\,dr \le\bigl(\|u(0)\|_2+F(t)\bigr)^2.

If $F(T)<\infty$ , these give uniform energy and finite total dissipation up to the possibly excluded endpoint.

Regularizing a vanishing norm

Set $Y_\delta(t)=(\|u(t)\|_2^2+\delta^2)^{1/2}$ . Drop dissipation in the differential identity and use Cauchy–Schwarz to obtain $Y_\delta'\le\|f\|_2$ . Integration and $\delta\downarrow0$ prove the first bound. Insert it into the integrated energy identity: the work is at most $\int_0^t F'(r)(\|u(0)\|_2+F(r))\,dr$ . This gives the squared bound. Monotone convergence handles total dissipation as $t\uparrow T$ , without assuming regularity at $T$ .