Energy: the energy equality, dissipation and enstrophy

Take the dot product of the Navier-Stokes equations with \(\mathbf{u}\) and integrate over \(\mathbb{R}^3\) (with enough decay) or the torus \(\mathbb{T}^3\). Three things happen. The time derivative gives \(\frac{d}{dt}\int\tfrac12|\mathbf{u}|^2\). The nonlinear term vanishes: \[ \int \mathbf{u}\cdot(\mathbf{u}\cdot\nabla)\mathbf{u} = \int u_j\,\partial_j\!\left(\tfrac12|\mathbf{u}|^2\right) = -\int(\nabla\cdot\mathbf{u})\tfrac12|\mathbf{u}|^2 = 0. \] The pressure term vanishes too, \(\int\mathbf{u}\cdot\nabla p = -\int p\,\nabla\cdot\mathbf{u} = 0\). Integrating the viscous term by parts gives \(-\nu\int|\nabla\mathbf{u}|^2\). So every smooth solution satisfies \[ \frac{d}{dt}\,\frac12\int|\mathbf{u}|^2\,dx = -\nu\int|\nabla\mathbf{u}|^2\,dx, \] and integrating in time, the energy equality \[ \frac12\|\mathbf{u}(t)\|_{L^2}^2 + \nu\int_0^t\|\nabla\mathbf{u}(s)\|_{L^2}^2\,ds = \frac12\|\mathbf{u}_0\|_{L^2}^2. \] The nonlinearity moves energy around but never creates it; viscosity only removes it.

The rate of loss is the dissipation. For divergence-free fields, \(\int|\nabla\mathbf{u}|^2 = \int|\boldsymbol{\omega}|^2 = 2\int S:S\), so dissipation can be read as total squared vorticity or total squared strain. For the Taylor-Green field \((\sin x\cos y, -\cos x\sin y)\) on \([0,2\pi]^2\), the energy is \(\pi^2\) and \(\int|\nabla\mathbf{u}|^2 = 4\pi^2\), consistent with its exact decay like \(e^{-4\nu t}\). On the torus with zero mean, the Poincaré inequality \(\|\nabla\mathbf{u}\|_{L^2} \ge \|\mathbf{u}\|_{L^2}\) turns the energy equality into exponential decay, \(\|\mathbf{u}(t)\|^2 \le \|\mathbf{u}_0\|^2 e^{-2\nu t}\).

The energy equality gives two bounds valid for all time and all data: \(\sup_t\|\mathbf{u}(t)\|_{L^2} \le \|\mathbf{u}_0\|_{L^2}\) and \(\int_0^\infty\|\nabla\mathbf{u}\|_{L^2}^2\,dt \le \|\mathbf{u}_0\|^2_{L^2}/(2\nu)\). In three dimensions they are the basic a priori bounds for general data, and the other known global bounds are derived from them. They are strong enough to construct weak solutions, but, as the scaling lesson showed, they are weak at small scales: energy scales like \(\lambda^{-1}\).

In two dimensions there is a second conserved structure. The enstrophy \(\tfrac12\int\omega^2\) obeys, from \(D\omega/Dt = \nu\Delta\omega\), \[ \frac{d}{dt}\,\frac12\int\omega^2 = -\nu\int|\nabla\omega|^2 \le 0, \] because there is no stretching term. Since \(\|\nabla\mathbf{u}\|_{L^2} = \|\omega\|_{L^2}\), this is a uniform bound on the velocity gradient, a quantity that in 2D is subcritical (it scales like \(\lambda\), so it controls small scales), and it leads to global smooth solutions. In 3D the same calculation leaves \(\int\boldsymbol{\omega}\cdot S\boldsymbol{\omega}\) on the right, which can be positive; estimating it only gives \(\frac{d}{dt}\|\nabla\mathbf{u}\|^2 \le C\nu^{-3}\|\nabla\mathbf{u}\|^6\), an inequality that bounds the gradient for a time depending on the size of the data, and, once the dissipation term is kept, for all time if the data are small, but not for large data.

Picture it: a bank account of kinetic energy that the nonlinearity can move between scales but never top up, with viscosity charging a fee proportional to the squared gradients. Think it: the energy equality is an identity for smooth solutions, obtained by testing the equation against the solution itself; for weak solutions only an inequality survives, and whether equality holds is tied to their smoothness.