Tuesday, January 20, 2026 — 3:00 pm — aula M1.5, edificio Matematica
On the Navier-Stokes equations: a possible gap related to the energy equality of a weak solution
Abstract: It is well known that a weak solution a priori enjoys an energy inequality. Hence, we investigate on the possible gap:
$$\|v(s)\|_2^2 \:- \: 2 \int_s^t \| \nabla v(\tau) \|_2^2 \, d \tau \:- \: \|v(t)\|_2^2 \geq 0 \: . \qquad (1)$$
We look for the existence of a Leray’s weak solution enjoying the energy equality. We are unable to fully prove the result. However, we show that if there exists a possible gap for the energy equality, i.e. at $(1) \: > 0$, then the gap is represented by means of a suitable additional dissipation. The additional dissipation is given in terms of the “kinetic energy”.
We are not able to detect a turbulence character for these quantities, or, more in general, to understand the physical meaning of the anomalous energy dissipation.
The gap vanishes in the case of a further “small regularity” of the weak solution.
This result is an existence result, we do not achieve the result for all possible the Leray-Hopf weak solutions.