Convex integration and phenomenologies in turbulence

TL;DR

This review surveys how convex integration yields wild weak solutions to the Euler and Navier–Stokes equations, connecting rigorous constructions to turbulence phenomenology like Onsager's energy conservation/dissipation threshold and intermittency. It outlines elementary C0 Euler solutions dissipating energy, advances toward the Onsager 1/3 threshold via 1/3− Hölder constructions (including Isett and later dissipative refinements), and extends intermittent convex integration to the Navier–Stokes setting to produce non-unique weak solutions with finite energy. The work highlights the role of Reynolds stress decomposition, Beltrami and Mikado building blocks, and time-space localization (gluing and intermittency) in achieving precise control of energy and regularity. Open problems point to extending these methods to 2D, refining energy inequalities, and probing the limits of dissipation and nonuniqueness in turbulent regimes.

Abstract

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash's fundamental ideas on $C^1$ flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies the phenomenological theories of hydrodynamic turbulence. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative $C^{0+}_{x,t}$ weak solutions of the Euler equations, first proven by De Lellis-Székelyhidi Jr.. Second, we present Isett's recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work of De Lellis-Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class $C^{\frac 13-}_{x,t}$ are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result, which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class $C^0_t L^{2+}_x \cap C^0_t W^{1,1+}_x$. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

Figures (10)

• Figure 1: Experimental and numerical evidence for the anomalous dissipation of energy.
• Figure 2: Mild deviations of $\zeta_p$ from $p/3$. [Chen, Dhruva, Kurien, Sreenivasan, Taylor ChenEtAl05]. The graphic plots $p$ versus $(\zeta_{|p|}-\frac{p}{3}) \frac{3}{p}$ for various nonzero moments $p$, as gathered from experiments (circles) and direct numerical simulations (crosses and stars). The solid line is the outcome of the $p$-model of [Meneveau-Sreenivasan MeneveauSreenivasan87]. The dashed and dotted lines are predictions of the $\beta$ model of [Yakhot Yakhot01].
• Figure 3: Experimental evidence for $S_{3}^{\|}(\ell) = - \frac{4}{5} (\varepsilon \ell)$. The quantity $- S_3^{\|}(\ell)/(\varepsilon \ell )$ is plotted as a function of $\ell$. Squares denote experimental observations of centerline in pipe flow at $\mathrm{Re\,} = 230000$. Circles indicate data from a $512^3$ DNS of homogenous turbulence at $\mathrm{Re\,} = 220$. Dots indicate the $4/5$ law. [K. R. Sreenivasan et. al. SreenivasanEtAl96]
• Figure 4: Kolmogorov's anomalous energy dissipation rate $\varepsilon$ and Onsager's energy flux $\Pi_\kappa$ appear to agree, at least for $\kappa$ in the inertial range. [Kaneda et al KanedaEtAl03]: $2048^3$ DNS runs on Earth Simulator computing system.
• Figure 5: [Meneveau & Sreenivasan MeneveauSreenivasan91]. Two typical signals, with $\varepsilon' = (du_1/dt)^2$, serving as the surrogate of the energy dissipation rate, are plotted here upon normalizing by their mean values. Graph (a) was in a laboratory boundary layer at a moderate Reynolds number. Graph (b) was obtained in the atmospheric surface layer at high Reynolds number ($10^4$). The increased intermittency at the higher Reynolds number is clear.

Theorems & Definitions (29)

• Definition 3.1
• Conjecture 3.2: Onsager's conjecture
• Theorem 3.3: Theorem 1, Isett16
• Theorem 3.4: Theorem 1.1, BDLSV17
• Definition 3.5
• Definition 3.6
• Theorem 3.7: Theorem 1.2, BV
• Definition 3.8
• Theorem 3.9: Theorem 1.1, BCV18
• Theorem 3.10: Theorem 1.3, BV