On Turbulence and Geometry: from Nash to Onsager

Abstract

Turbulence: a challenge for mathematicians. There is a huge literature on turbulent incompressible flows in applied mathematics, physics, and engineering. The outcome of such tremendous effort has been the derivation of several theories, which often allow quite accurate predictions of many phenomena. There is also a quite broad consensus on which fundamental partial differential equations

Convex integration and phenomenologies in turbulence

“…Section 4: In this section we summarize some of the key aspects of Nash-style convex integration schemes in fluid dynamics. A number of excellent surveys articles on this topic are already available in the literature, by De Lellis and Székelyhidi Jr. [47,188,50,51]. These surveys discuss in detail the Nash-Kuiper theorem [152,118], Gromov's h-principle [92], convex integration constructions for flexible differential inclusions inspired by the work of Müller andŠverák [151], the Scheffer [173]-Shnirelman [179] constructions, leading to the constructions of non-conservative Hölder continuous solutions of the Euler equations.…”

“…Inspired by the work [151,110], and building on the plane-wave analysis introduced by Tartar [189,190,55], De Lellis and Székelyhidi Jr., in [48], applied convex integration in the context of weak L ∞ solutions of the Euler equations, yielding an alternative proof of Scheffer's [173] and Schnirelman's [179] famous nonuniqueness results. The work [48], has since been extended and adapted by various authors to various problems arising in mathematical physics [46,41,181,201,27], see the reviews [47,188,50,51] and references therein.…”

“…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 with the phenomenological theories of hydrodynamic turbulence [47,188,50,51]. 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. [49,48].…”

Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

“…We will discuss three types of results here: blow-up criteria, infinite-time singularity formation, and model problems. We will not be discussing weak solutions in any detail but we refer the reader to the recent review papers [14] and [4].…”

Some preliminary observations on a defect Navier–Stokes system

“…The constraints provided by the proposed model (17) to the classical system consisting of (17a), (17b) are the same as in Case III. For the classical system, non-unique, dissipative, Hölder continuous weak solutions have recently been constructed, as described in the excellent review [DLSJ19], rigorously establishing Onsager's conjecture on anomalous dissipation in the incompressible Euler equations (i.e., there exists (continuous) velocity fields whose total kinetic energy strictly decreases in time if the velocity field satisfies the condition |v(x, t) − v(y, t)| < C|x − y| h ∀x, y, t (x, y are locations, t is time) with h < 1 3 , where C > 0 is a constant independent of x, y, t), as well as far-reaching connections to the geometric isometric embedding problem.…”