The Concentrated Loss

In viscous fluids, energy dissipation is distributed — friction acts everywhere, smearing the loss across the flow. As viscosity decreases, you might expect dissipation to decrease too. In the inviscid limit, with zero viscosity, the loss should vanish entirely. Euler’s equations conserve energy.

Instead, dissipation collapses to points.

In the inviscid limit of two-dimensional fluids, energy dissipation concentrates on atomic (point-supported) measures. The limit does not produce zero dissipation distributed uniformly. It produces finite dissipation concentrated infinitely — all the loss that was spread across the flow gathers into singularities when the spreading mechanism (viscosity) is removed.

The mathematical structure is precise. When temporal oscillations are controlled, the dissipation measures converge to sums of Dirac masses. The energy lost equals the energy lost in the viscous case, but its spatial distribution has collapsed from a smooth density to a collection of points. The total stays; the distribution becomes singular.

This is not an anomaly of fluid mechanics. It is a structural feature of limits that remove distributed resistance. Removing the mechanism that spreads a cost does not remove the cost — it concentrates it. The loss has to go somewhere. When it can no longer be distributed, it localizes.

The pattern appears wherever distributed friction is eliminated. Financial deregulation does not eliminate risk; it concentrates it at singular failure points. Removing predators from an ecosystem does not eliminate mortality; it concentrates collapse at carrying capacity. Lossless compression does not eliminate error; it pushes it to boundaries.

Removing friction does not remove loss. It creates singularities where the loss accumulates. The cost is conserved; only its geometry changes.