Navier–Stokes energy and weak solutions

For unforced Navier–Stokes with positive viscosity nu, add nu times the Laplacian of velocity to the Euler right-hand side. On a periodic torus, dot the smooth equation with velocity and integrate. Incompressible transport and pressure give no net contribution, while integration by parts of the Laplacian gives minus nu times the squared gradient norm. With kinetic energy defined as half the squared L² norm, the coefficient is nu, not two nu; the latter appears only when the equality is written for the squared norm itself.

The integrated identity bounds velocity uniformly in L² and its gradient in L² over every finite time interval. In two dimensions these bounds interact with stronger estimates to support global smooth evolution from smooth data. In three dimensions the energy budget alone does not bound the high regularity used to continue a classical solution. The distinction is analytic, not a failure of the formal cancellation: the cancellation is exact, but its controlled quantity is too weak to supply all desired derivative bounds.

On the three-dimensional torus, take divergence-free L² initial data. A standard Leray–Hopf existence theorem gives a global distributional solution that is weakly continuous in L², belongs to L-infinity in time with values in L², and to L² in time with values in H¹ on every finite interval. It attains the initial data strongly in L² and satisfies the energy inequality from the initial time, together with the standard almost-every-starting-time form. The pressure can be recovered distributionally; divergence-free test fields remove it from the basic weak formulation.

One construction projects onto finitely many divergence-free Fourier modes. The finite-dimensional equations retain the energy cancellation, giving bounds uniform in the projection size. A time-derivative bound in a suitable negative space, together with spatial compactness on the torus, yields strong local-in-time L² convergence along a subsequence. Mere weak convergence of u would not identify u tensor u. Strong convergence of the needed factors lets the nonlinear term pass to the limit, while lower semicontinuity turns the approximate energy identities into an inequality.

A weak existence theorem supplies neither uniqueness nor regularity merely by producing a limit. A weak–strong uniqueness theorem says that, while a sufficiently regular strong solution exists, an energy-class Leray–Hopf solution with the same data agrees with it. This compares classes on an interval where the strong hypotheses hold; it does not extend that interval by definition. Additional spacetime bounds such as Prodi–Serrin control can bridge the regularity gap, and their exponents will be checked in the next lesson.

The setting also matters for research claims. External forcing contributes a work term to the energy balance, and a bounded-domain problem adds boundary conditions that affect estimates. Whole-space and periodic formulations have related but distinct compactness and pressure issues. The equations in this lesson are explicitly unforced and periodic. A dated announcement about another regularity or breakdown formulation should not be substituted for their energy-class theorem, nor should the energy-class theorem be mistaken for an assessment of that announcement. The logical content here is global weak existence under the stated assumptions and the limits of the resulting bounds.