Infinitely many non-conservative solutions for the three-dimensional Euler equations with arbitrary initial data in $C^{1/3-ε}$

Abstract

Let 0 < β < β < 1/3. We construct infinitely many distributional solutions in C β x,t to the three-dimensional Euler equations that do not conserve the energy, for a given initial data in C β . We also show that there is some limited control on the increase in the energy for t > 1.

Construction of solutions to the 3D Euler equations with initial data in $H^β$ for $β>0$

“…In contrast, the solutions that we construct must increase the energy. The authors of the present paper with Ye have recently [KMY22] shown non-uniqueness for all data in C 1/3− by allowing the energy to rise. The present paper can be thought of as a continuation of this work, although the calculations are different and the proof idea differs in some key areas, which we discuss further in Subsection 2.4.…”

“…It follows that the natural space for the error Rq which arises from the quadratic nonlinearity is L 1 . This is the cause for the differences between this paper and [KMY22], where Hölder estimates are used for both v q and Rq . One such difference is the size of certain estimates.…”

“…if v in is itself a stationary Euler flow. It would be interesting (as also in [KMY22]) to improve this to bounds that are independent of certain parameters, or to find ways to constrain the energy for all times t ≥ 0.…”

“…Then in Section 4, we use some intermittent Mikado flows as in [BV21] (although we only use spatial intermittency; see the definition of W (k) ), with some modified cutoffs from our earlier work [KMY22] to control the energy at positive times. In Section 5, we define and estimate the resulting new stress error Rq+1 .…”

“…The other main difference is how we achieve the initial data in our construction. In [KMY22], in order to achieve the C 1/3− regularity, we incrementally improved the initial data at the gluing step. The gluing step is not as well-behaved in the L 2 case, and in any case what we can prove is far from what should be the optimal regularity.…”