Publication

Anomalous dissipation via spontaneous stochasticity with a two-dimensional autonomous velocity field

2026

Abstract

We study anomalous dissipation in the context of passive scalars and we construct a two-dimensional autonomous divergence-free velocity field in Cα (with α∈(0,1) arbitrary but fixed) which exhibits anomalous dissipation. Our proof employs the fluctuation-dissipation formula, which links spontaneous stochasticity with anomalous dissipation. Therefore, we address the issue of anomalous dissipation by showing that the variance of stochastic trajectories, in the zero noise limit, remains positive. Based on this result, we answer Questions 2.2 and Question 2.3 in Bruè and De Lellis regarding anomalous dissipation for the forced three-dimensional Navier–Stokes equations.

Official source

https://infoscience.epfl.ch/entities/publication/ad62356f-e983-4a0c-bff8-d25bcdc63770

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/ad62356f-e983-4a0c-bff8-d25bcdc63770

About this result

Related concepts (19)

Attractor

In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space.

Fluctuation-dissipation theorem

The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance (to be intended in their general sense, not only in electromagnetic terms) of the same physical variable (like voltage, temperature difference, etc.

Supersymmetric theory of stochastic dynamics

Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches.