Knowlpedia

Theorem

Global kinetic-energy identity

Smooth incompressible flow balances energy change and viscous dissipation against work by the force.

Statement

For a smooth incompressible Navier–Stokes solution on a flat torus, with periodic pressure and force, or on Rn\mathbb R^n with a fixed compact spatial support for the velocity on each compact time interval, the kinetic-energy identity is

12∥u(t)∥22+ν∫st∥∇u(r)∥22 dr=12∥u(s)∥22+∫st∫f(r,x)⋅u(r,x) dx dr.\frac12\|u(t)\|_2^2+\nu\int_s^t\|\nabla u(r)\|_2^2\,dr =\frac12\|u(s)\|_2^2+\int_s^t\int f(r,x)\cdot u(r,x)\,dx\,dr.

It follows by integrating the . All fields are smooth where the compact-support integrals are taken.

Cancellation and scope

Divergence terms have zero integral by the stated support or periodicity conditions. Differentiating the identity gives 12ddt∥u∥22+ν∥∇u∥22=⟨f,u⟩\tfrac12\frac d{dt}\|u\|_2^2+\nu\|\nabla u\|_2^2=\langle f,u\rangle. Whole-space smoothness and an L2L^2 bound alone do not justify discarding flux at infinity; decay, approximation, or a separate cutoff argument is needed in that setting.

References