Hamiltonian compactness and dissipation for the generalized SQG equation in the inviscid limit

TL;DR

This paper proves that anomalous dissipation is prevented in the inviscid limit of the generalized SQG equation due to strong solution compactness, applicable across various regimes and external forces.

Contribution

It introduces a robust mechanism ensuring solution compactness and global existence for a broad class of initial data in the generalized SQG equation.

Findings

Anomalous dissipation is prevented by solution compactness.

Strong compactness applies regardless of criticality and external forcing.

Global existence is established for initial data with critical integrability.

Abstract

We consider the dissipative generalized Surface Quasi-Geostrophic equation with dissipation given by any fractional power of the Laplacian. In the inviscid limit, it is proved that anomalous dissipation of the Hamiltonian is prevented by the strong compactness of the solutions in the lowest norm that makes the nonlinearity well-defined. In fact, only the dynamics at certain frequencies matters. The argument is quite robust as it applies regardless of the criticality regime and of the presence of a, possibly noncompact, external forcing. This reveals a more general mechanism behind some recent results obtained for the Navier-Stokes and the critical dissipative Surface Quasi-Geostrophic equations. Because of nonuniqueness issues, in our broader context it is important to work with Leray solutions enjoying suitable higher-order bounds. The existence of such solutions is shown and it might be of independent interest. Finally, we prove that the strong compactness is guaranteed for any initial datum with critical integrability, from which global existence of conservative, although Onsager's supercritical, weak solutions of the inviscid problem is deduced. This offers the largest class of initial data for which global existence is known so far, matching with the one considered by Delort at the endpoint.