Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Abstract

Motivated by the theory of turbulence in fluids, the physicist and chemist Lars Onsager conjectured in 1949 that weak solutions to the incompressible Euler equations might fail to conserve energy if their spatial regularity was below 1/3-Hölder. This book uses the method of convex integration to achieve the best-known results regarding nonuniqueness of solutions and Onsager's conjecture. Focusing on the intuition behind the method, the ideas introduced now play a pivotal role in the ongoing study of weak solutions to fluid dynamics equations. The construction itself—an intricate algorithm with hidden symmetries—mixes together transport equations, algebra, the method of nonstationary phase, underdetermined partial differential equations (PDEs), and specially designed high-frequency waves built using nonlinear phase functions. The powerful “Main Lemma”—used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem—has been extended to a broad range of applications that are surveyed in the appendix. Appropriate for students and researchers studying nonlinear PDEs, this book aims to be as robust as possible and pinpoints the main difficulties that presently stand in the way of a full solution to Onsager's conjecture.

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…”

“…See [19,29] for further discussion. Recently there has also been a series of advances towards the negative direction of Onsager's conjecture that we will discuss further below [1,12,14,23].…”

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

“…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]). 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.…”

“…He felt that being an expert in a particular subject would inhibit his originality. 44 In my conversations with him, I was always struck by his ability to see any topic of discussion, even the most mundane ones, from an unusual, unexpected angle.…”