Holder Continuous Solutions of Active Scalar Equations

We consider active scalar equations $\partial_t \theta + \nabla \cdot (u \, \theta) = 0$, where $u = T[\theta]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator with symbol $m$. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with H\"older regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in ${\cal D}'$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier $m$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected.Comment: 61 page

On the Endpoint Regularity in Onsager's Conjecture

Onsager's conjecture states that the conservation of energy may fail for 3D incompressible Euler flows with Hölder regularity below 1/3. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the 3D incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents [0,1/3). Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [IO16b] to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include the fact that singularites of positive space-time Lebesgue measure are necessary for any energy non-conserving solution to exist while having critical regularity of an integrability exponent greater than three

A heat flow approach to Onsager's conjecture for the Euler equations on manifolds

We give a simple proof of Onsager's conjecture concerning energy conservation for weak solutions to the Euler equations on any compact Riemannian manifold, extending the results of Constantin-E-Titi and Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, our approach yields an alternative proof of the sharp result of the latter authors. Our method builds on a systematic use of a smoothing operator defined via a geometric heat flow, which was considered by Milgram-Rosenbloom as a means to establish the Hodge theorem. In particular, we present a simple and geometric way to prove the key nonlinear commutator estimate, whose proof previously relied on a delicate use of convolutions.Comment: 15 pages. Improved exposition, corrected typos. Added a criterion for energy conservation in terms of the H\"older norm in Theorem 1.

Nonuniqueness and existence of continuous, globally dissipative Euler flows

We show that Hölder continuous, globally dissipative incompressible Euler flows (solutions obeying the local energy inequality) are nonunique and contain examples that strictly dissipate energy. The collection of such solutions emanating from a single initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is C^0 dense in the space of continuous, divergence free vector fields on the torus T^3

H\"older Continuous Euler Flows in Three Dimensions with Compact Support in Time

Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the H\"{o}lder class $C_{t,x}^{1/5 - \epsilon}$. By slightly modifying the proof, we show that every smooth solution to incompressible Euler on $(-2, 2) \times {\mathbb T}^3$ coincides on $(-1, 1) \times {\mathbb T}^3$ with some H\"{o}lder continuous solution that is constant outside $(-3/2, 3/2) \times {\mathbb T}^3$. We also propose a conjecture related to our main result that would imply Onsager's conjecture that there exist energy dissipating solutions to Euler whose velocity fields have H\"{o}lder exponent $1/3 - \epsilon$.Comment: Minor corrections throughout text and some added detail

A Proof of Onsager's Conjecture

For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class C_tC_^xα that have nonempty, compact support in time on R × T^3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α < 1/3 due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class L_t^∞C_x^α. The previous best results were solutions in the classC_tC_x^α for α < 1/5, due to [Isett], and in the class L_t^1C_x^α for α < 1/3 due to [Buckmaster, De Lellis, Székelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Székelyhidi]. The present proof combines the method of convex integration and a new “Gluing Approximation” technique. The convex integration part of the proof relies on the “Mikado flows” introduced by [Daneri, Székelyhidi] and the framework of estimates developed in the author's previous work