Dissipative anomalies in singular Euler flows

Gregory Eyink, Baltimore

Among the more fascinating—and perhaps unexpected—applications of the Euler equations is high-Reynolds-number turbulence. Experimental and theoretical investigations before World War II (Hugh Dryden, Geoffrey Taylor) suggested that energy dissipation in turbulent flow governed by the incompressible Navier-Stokes equation in many circumstances does not vanish in the limit of zero viscosity, or infinite Reynolds number. Lars Onsager (1945, 1949) observed that inviscid Euler equations may not conserve energy if the velocity field is sufficiently singular (Hoelder exponent ≤ 1/3). Onsager used this exact result to predict (independently of Kolmogorov) the -5/3 energy spectrum in turbulent flow and suggested that the inertial-range energy cascade is described by singular solutions of the Euler equations. We present a simple explanation of Onsager's result, in terms of conservation properties of “coarse-grained” fluid equations for a continuous range of length-scales. The same considerations apply not only to energy, but also to other inviscid conservation laws, such as helicity in three space dimensions and enstrophy in two, or the conservation of circulations in any dimension. Onsager's point of view continues to suggest new properties of turbulent flow that can be tested by experiment and simulation. Despite many mathematical advances, however, there is still no physically satisfactory proof of existence of such dissipative Euler solutions, nor of their uniqueness and regularity. A solution of this outstanding problem could shed great light on turbulent flow dynamics and, possibly, other problems where singular Euler solutions may arise (e.g. cosmology).

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