The Onsager Conjecture: A Pedagogical Explanation

In 1949, a Nobelist Last Onsager considered liquid flows with velocities changing as r for spatial points at distance r, and conjectured that the threshold value α = 1/3 separates the two possible regimes: for α > 1/3 energy is always preserved, while for α < 1/3 energy is possibly not preserved. In this paper, we provide a simple pedagogical explanation for this conjecture. 1 Formulation of the Problem The equations that describe the velocity filed v(t, x) of an incompressible nonviscous liquid go back to Euler – and are known as Euler equations. In 1949, a Nobelist Lars Onsager considered solutions v(t, x) to Euler’s equation for which, for some constant C, we have |v(t, x) − v(t, x′)| ≤ C · r, where r denotes the distance between the points x and x′ [2]. He conjectured that: • when α > 1/3, then all the corresponding solutions v(t, x) preserve energy, while • for α < 1/3, there exist solutions that do not preserve energy. This conjecture remains one of the central open problems in the foundations of hydrodynamics; see, e.g., [1] and references therein. How can we explain this technical conjecture in simple physical terms? The main objective of this paper is to provide such an explanation.