Dissipative continuous Euler flows

Abstract

We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy

On Nonperiodic Euler Flows with Hölder Regularity

“…such that the energy profile of v is equal to T 3 |v| 2 (t, x)dx =ē(t) for all t ∈ R. [2,12,14] for prescribing smooth energy profiles the periodic setting and on the organizational framework developed in [23]. We remark that our arguments also allow one to achieve an energy profile that does not have compact support provided the norm e C γ t = sup t |e(t)| + sup t sup | t| =0…”

“…On the periodic torus, the construction of (1/5 − ε)-Hölder solutions that fail to conserve energy was first achieved in [23] improving on initial constructions of (1/10 − ε)-Hölder solutions in [12,14] (see also [1,2] for a shorter proof closer to the scheme of [12,14]). We also note the construction of solutions with compact time support in the class C 0 t,x ∩ L 1 t C…”

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

“…We have a very general result, which applies to large classes of equations, which establishes well-posedness for sufficiently large a. In the case of the incompressible Euler equation, the best well-posedness result 22 is C 1,α , α > 0, in Hölder spaces and H a , a > n/2+1, in Sobolev spaces H a , where n is the space dimension. Both bounds are sharp, in view of a result of Bourgain and Li; see [4].…”

“…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]).…”

“…These results played a fundamental role in establishing large data global regularity results in critical dimensions. 26 In the case of quasilinear equations, such as the Einstein field equation in three space dimensions, it is known that well-posedness holds for a ≥ a * + 1 2 (see 22 Well-posedeness also holds in W a p spaces for a > n/p + 1. This result is also sharp.…”