Behind the mirror: the hidden dissipative singular solutions of ideal reversible fluids on log-lattices

TL;DR

This paper introduces a novel dynamic approach to identify singular solutions of Euler equations using log-lattice models, revealing a phase transition and linking these solutions to turbulence scaling exponents.

Contribution

It proposes a reversible framework with an efficiency measure to construct and analyze singular solutions on log-lattices, bridging ideal fluid theory and turbulence observations.

Findings

Identifies a phase transition at a critical efficiency separating regular and singular solutions.

Shows singular solutions exhibit self-similar blow-ups with specific exponents.

Demonstrates dissipative solutions with power-law exponents similar to those in real turbulence.

Abstract

Empirical observations show that turbulence exhibits a broad range of scaling exponents, characterizing how the velocity gradients diverge in the inviscid limit. These exponents are thought to be linked to singular solutions of the Euler equations. In this work, we propose a dynamic approach to construct concept of these solutions directly from the fluid equations, using a reversible framework and introducing the efficiency $\cal{E}$, a non-dimensional number that quantifies the amount of energy stored within the flow due to an applied force. To circumvent the computational burden of tracking singularities at finer and finer scale, we test this approach on fluids on log-lattices, which allow for high effective resolutions at a moderate cost, while preserving the same symmetries and global conservation laws as ordinary fluids. We observe a phase transition at a given efficiency, separating regular, viscous solutions (hydrodynamic phase), from singular, inviscid solutions (singular phase). The singular solutions experience self-similar blow-ups with exponents corresponding to non-dissipative solutions. By applying a stochastic regularization, we are able to go past the blow-up, and show that the resulting solutions converge to power-law solutions with exponents characterizing dissipative solutions. Overall, the range of scaling exponents observed for log-lattice solutions is comparable to those of ordinary fluids.