IAS Physics Group Meeting

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Abstract: Consider smooth solutions to the 3d Navier-Stokes for divergence-free vector fields $u^\varepsilon$:
\[
\partial_t u^\varepsilon + {\rm div}\, (u^\varepsilon \otimes u^\varepsilon ) + \nabla p^\varepsilon = \varepsilon \Delta u^\varepsilon
\]
While the balance of the energy is
\[
\frac{d}{dt} \int |u^\varepsilon|^2 (x,t)\, dx = - 2 \varepsilon \int |Du^\varepsilon|^2 (x,t)\, dx\, ,
\]
it is a tenet of the theory of fully developed turbulence that in a variety of situations the size of the left hand side should typically be independent of $\varepsilon$. However this ``anomalous'' dissipation should not be triggered by an hypothetical initial datum $u^\varepsilon (\cdot, 0)$ in which we ignite lots of oscillations: a rule of thumb would be that, if the {\em linear} Stokes equations exhibit a similar behavior, then the roughness of the initial data is certainly excessive.

Producing rigorous mathematical examples of this prediction is hard: a tight control on the smoothness of the initial data forces the solutions to converge to classical solutions of the Euler equation and therefore rules out any possible dissipation, unless one shows a quite severe blow-up for smooth solutions of Euler; for sufficiently rough initial data this obstruction is absent, but proving something about the solutions of Navier-Stokes becomes very challenging.

In a recent joint work with Elia Brué we study what happens if we introduce a forcing term $f^\varepsilon$. The problem is compatively easier and it can be proved that there is an exact regularity threshold below which anomalous dissipation happens in some examples and above which it would only be possible through a blow-up scenario. Surprisingly one side of the problem is linked with some fundamental questions about solving ODEs with rough coefficients.