Absence of Anomalous Dissipation of Energy in Forced Two Dimensional Fluid Equations

Anomalous dissipation of energy in three dimensional turbulence is one of the basic statements of physical theory [55]. It has been verified experimentally to a large degree [75], but not mathematically. The statement is about the average behavior of the energy dissipation rate = ν〈|∇u|〉 as ν → 0. Here ν is kinematic viscosity, u is the velocity (assumed to have mean zero), ∇ are spatial gradients and 〈. . . 〉 represents an ensemble average or space-time average. The assertion of turbulence theory is that is a positive number, and that it does not vanish with viscosity, in the limit of zero viscosity. The term “anomalous dissipation” was imported from field theory in physics and it refers to the fact that, in the limit of vanishing viscosity, there still is remanent dissipation, even though the limit equation conserves energy. There are two distinct approaches to the question of anomalous dissipation. In the first, the limit of zero viscosity is taken on solutions of the initial value problem with fixed initial data. Under appropriate conditions this leads to a solution of the corresponding initial value problem of the inviscid equation. This equation conserves energy if solutions are smooth, but might dissipate energy if solutions are not sufficiently smooth. This circle of ideas, and specifically the precise degree of smoothness needed, goes by the name of “Onsager conjecture” [49, 47, 48, 14, 9, 36, 37, 38, 39, 35, 59]. This