Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

Abstract

For any " > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space L 1 t .C 1=3 " x/; namely, x 7 ! v.x; t / is . 1 =3 "/-Hölder continuous in space at a.e. time t and the integral R OEv. ; t / 1=3 " dt is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class L 1 t .C1=3 " x /.

On Nonperiodic Euler Flows with Hölder Regularity

“…There Onsager stated that energy is conserved by periodic solutions in the class L ∞ t C α x if α > 1/3, and conjectured that energy conservation may fail for such solutions if α < 1/3 (see [15,17] for detailed expositions). The conservation of energy stated by Onsager was proven in [7,18], and this result was refined in [6] to show that energy conservation holds for energy class solutions in the space L 3 t B 1/3 3,c 0 (N) on either I × T n or I × R n (see also [16,21] for further proofs). On the other hand, the proof of energy conservation fails for the space L 3 t B 1/3 3,∞ , and an example in [6] suggests that anomalous dissipation of energy may be possible in this class.…”

“…The conservation of energy stated by Onsager was proven in [7,18], and this result was refined in [6] to show that energy conservation holds for energy class solutions in the space L 3 t B 1/3 3,c 0 (N) on either I × T n or I × R n (see also [16,21] for further proofs). On the other hand, the proof of energy conservation fails for the space L 3 t B 1/3 3,∞ , and an example in [6] suggests that anomalous dissipation of energy may be possible in this class. The Besov regularityḂ 1/3 p,∞ carries a special significance in turbulence theory as it agrees with the p = 3 case of the scaling |v(x + x) − v(x)| p 1/ p ∼ ε 1 3 | x| 1 3 predicted by Kolmogorov's theory [26].…”

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

“…In the case of the 1954 result, Nash's new ideas have led to powerful generalizations such as convex integration and the h-principle, 6 both formulated by Gromov, as well as many applications explained in detail in his survey. The second result is forever tied to the so-called Nash hard implicit function theorem (often called the Nash-Moser implicit function theorem), a stunning, out of the blue, perturbation technique for providing solutions to complicated nonlinear partial differential equations.…”

“…It is not known if the result can be extended to C 2 metrics; see discussion in [31] and [18]. 6 I understand the h-principle here, loosely, as high flexibility of the space of solutions to a system of partial differential equations underlying a geometric or a physical problem. This high flexibility is particularly striking, as is the case of Nash's theorem, when it is due to limited regularity rather than indeterminacy of the system.…”

“…The result was then improved by Isett to α < 1/5 (see [44] and the shorter proof in [5]). Further progress has been made in [6] where the authors show that h-principles are available for solutions that are integrable in time with values in a Holder space, i.e., L 1 t (C α ) with α < 1/3. Moreover, in [7] it has been very recently proved that the h-principle holds for weak solutions in L ; see [45].…”