Energy conservation and Onsager's conjecture for the Euler equations

Abstract

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in R 3 conserve energy only if they have a certain minimal smoothness, (of order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space B 1/3 3,c(N) . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood-Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to B 2/3 3,c(N) conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

Lower bounds of potential blow-up solutions of the three-dimensional Navier-Stokes equations in Ḣ32

“…Remark 1.1. The lower bound for theḢ 3 2 -norm of blow-up solutions was also presented in papers by Montero [9] and McCormick, Olson, Robinson, Rodrigo, Vidal-Lopez, and Zhou [8], which both appeared shortly after this paper.…”

“…where the velocity u(x, t) and the pressure p(x, t) are unknowns, ν > 0 is the kinematic viscosity coefficient, the initial data u 0 (x) ∈ L 2 (Ω), and the spatial domain Ω may have periodic boundary conditions or Ω = R 3 . The question of the regularity of solutions to ( 1.1) remains open and is one of the Clay Mathematics Institute Millennium Prize problems.…”

On Nonperiodic Euler Flows with Hölder Regularity

“…The aforementioned stability result derives from the following estimate for the difference of the energy profiles e 1 , e 2 of two weak solutions to Euler in the class v 1 , v 2 ∈ L ∞ tḂ 1/3 3,∞ with domain I × T n or I × R n , which was observed in [24,Section 3] by extending the argument of [6,7]:…”

“…As a consequence, we are forced in our construction to work with approximate solutions that likewise satisfy the laws of conservation of linear and angular momentum (6), which are linear and thus survive under weak limits. In particular, our corrections must maintain the conservation of angular momentum (in addition to the divergence-free property and the conservation of linear momentum), which is a new feature compared to the construction on the periodic torus.…”

On Nash’s unique contribution to analysis in just three of his papers

“…One can make sense in this way of solutions which are merely in L 2 (R × T 3 ). 11 The sharpest result available in [11] shows the energy conservation holds true for flows in the method of convex integration to construct continuous solutions which dissipate energy (see [22]) and, by tweaking the result a bit, to go up to Hölder exponents α < 1/10 (see [23]). The result was then improved by Isett to α < 1/5 (see [44] and the shorter proof in [5]).…”