Classical uniqueness from difference energy

Statement

For two smooth periodic incompressible Navier–Stokes solutions with common force and viscosity, or smooth whole-space solutions $u,v\in C([0,T];H^3(\mathbb R^3))$ with common $f\in C([0,T];L^2)$ , let $w=v-u$ . Then

\|w(t)\|_2^2\le\|w(0)\|_2^2 \exp\!\left(2\int_0^t\|\nabla u(s)\|_\infty\,ds\right).

In particular, the same initial velocity gives the same velocity on $[0,T]$ . This is the classical difference-energy uniqueness estimate obtained from Gronwall's inequality.

Energy calculation

The difference equation and integration cancellations give

\frac12\frac d{dt}\|w\|_2^2+\nu\|\nabla w\|_2^2 =-\int(w\cdot\nabla)u\cdot w \le\|\nabla u\|_\infty\|w\|_2^2.

On the torus, smoothness and periodicity justify the integration. In the whole-space case, $H^3$ controls $u,\nabla u,v,\nabla v$ in $L^\infty$ ; the quadratic tensor lies in $L^2$ , and the identified canonical pressure lies in $L^2$ . Smooth approximation and expanding cutoffs justify the pairing and cancellation. The reference gradient is bounded on the compact time interval, so Gronwall applies. Equality of velocities determines pressure only up to a function of time.