Anomalous dissipation for the forced Navier-Stokes equations

Abstract: Consider smooth (or at least Leray) solutions to 3d Navier-Stokes \[ \left\{ \begin{array}{ll} \partial_t u^\varepsilon + {\rm div}\, (u^\varepsilon \otimes u^\varepsilon ) + \nabla p^\varepsilon = \varepsilon \Delta u^\varepsilon\\ \\ u^\varepsilon (0, \cdot) = u^\varepsilon_0\, . \end{array}\right. \] While the balance of the energy is \[ \frac{d}{dt} \frac{1}{2} \int |u|^2 (x,t)\, dx = - \varepsilon \int |Du|^2 (x,t)\, dx\, , \] it is a tenet of the theory of fully developed turbulence that in a variety of situations the left hand side should typically be independent of $\varepsilon$: the mechanism is not supposed to be ignited by high oscillations in the initial data, which would trigger an ``immediate'' dissipation through the viscosity, but it is rather thought to be an effect of the quadratic nonlinearity. It is on the other hand very challenging to produce rigorous examples. If the bounds on $u_0^\varepsilon$ are too strong, the well-posedness theory for Euler obstructs the anomalous dissipation up until the first blow-up time of Euler. With sufficiently coarse bounds Euler can be shown to have a variety of dissipative solutions, but it is very difficult to prove the convergence of Navier-Stokes to any of them. In a recent joint work with Elia Bruè we study the forced version of the equation and we can prove rigorously that, as soon as the regularity of the force drops below the known thresholds for the well-posedness of classical Euler, it is in fact possible to show anomalous dissipation.

mathematical physics

Audience: researchers in the discipline

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