CaltechAUTHORS: Combined
https://feeds.library.caltech.edu/people/Isett-Philip/combined.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 14 Oct 2024 13:19:38 -0700Anomalous dissipation for 1/5-Hölder Euler flows
https://resolver.caltech.edu/CaltechAUTHORS:20180627-082316639
Year: 2015
DOI: 10.4007/annals.2015.182.1.3
Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field.
In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent — albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class C^(1/5−ε).https://resolver.caltech.edu/CaltechAUTHORS:20180627-082316639Hölder Continuous Solutions of Active Scalar Equations
https://resolver.caltech.edu/CaltechAUTHORS:20180627-075939255
Year: 2015
DOI: 10.1007/s40818-015-0002-0
We consider active scalar equations ∂_tθ + ∇ ⋅ (uθ)=0, where u = T[θ] 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 Hölder regularity C^(1/9−)_(t,x). In fact, every integral conserving scalar field can be approximated in 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.https://resolver.caltech.edu/CaltechAUTHORS:20180627-075939255On Nonperiodic Euler Flows with Hölder Regularity
https://resolver.caltech.edu/CaltechAUTHORS:20180626-155143019
Year: 2016
DOI: 10.1007/s00205-016-0973-3
In (Isett, Regularity in time along the coarse scale flow for the Euler equations, 2013), the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L^∞_t(B^(1/3)_(3,∞) due to low regularity of the energy profile. This paper is the first and main paper in a series of two, the results of which may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5 . The main result of the present paper shows that any given smooth Euler flow can be perturbed in C^(1/5 − ϵ)_(t,x) on any pre-compact subset of ℝ×ℝ^3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than C^(1/5 − ϵ)_(t,x). As a corollary of this theorem, we show the existence of nonzero C^(1/5 − ϵ)_(t,x) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, Hölder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.https://resolver.caltech.edu/CaltechAUTHORS:20180626-155143019A heat flow approach to Onsager's conjecture for the Euler equations on manifolds
https://resolver.caltech.edu/CaltechAUTHORS:20180627-084240647
Year: 2016
DOI: 10.1090/tran/6549
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 T^d or 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.https://resolver.caltech.edu/CaltechAUTHORS:20180627-084240647Hölder continuous Euler flows in three dimensions with compact support in time
https://resolver.caltech.edu/CaltechAUTHORS:20180702-123946440
Year: 2017
DOI: 10.48550/arXiv.1211.4065
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. In this book, Philip Isett 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.https://resolver.caltech.edu/CaltechAUTHORS:20180702-123946440On the kinetic energy profile of Hölder continuous Euler flows
https://resolver.caltech.edu/CaltechAUTHORS:20180626-154151950
Year: 2017
DOI: 10.1016/j.anihpc.2016.05.002
In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding 1/3. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space L_t^∞(B_(3,∞)^(1/3) due to low regularity of the energy profile.
The present paper is the second in a series of two papers whose results may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and Hölder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C_(t,x)^(1/5 − ϵ). The exponent 1/2 is sharp in view of a regularity result of Isett [8]. The proof employs an improved greedy algorithm scheme that builds upon that in Buckmaster–De Lellis–Székelyhidi [1].https://resolver.caltech.edu/CaltechAUTHORS:20180626-154151950Nonuniqueness and existence of continuous, globally dissipative Euler flows
https://resolver.caltech.edu/CaltechAUTHORS:20180626-160542363
Year: 2018
DOI: 10.48550/arXiv.1710.11186
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.https://resolver.caltech.edu/CaltechAUTHORS:20180626-160542363On the Endpoint Regularity in Onsager's Conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20180626-161143819
Year: 2018
DOI: 10.48550/arXiv.1706.01549
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.https://resolver.caltech.edu/CaltechAUTHORS:20180626-161143819Regularity in time along the coarse scale flow for the incompressible Euler equations
https://resolver.caltech.edu/CaltechAUTHORS:20180628-113011634
Year: 2018
DOI: 10.48550/arXiv.1307.0565
One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon that coarse scale averages of the velocity carry the fine scale features of the flow. In this paper, we study time-regularity properties of Euler flows which are connected to this phenomenon and the observation that each frequency level has a natural time scale when it is viewed along the coarse scale flow. We assume only that our velocity field is Hölder continuous in the spatial variables, which is well-motivated by problems related to turbulence.
We show that any periodic Euler flow in the class C_tC^α_x also belongs to C^α_(t,x), and that the pressure belongs to C^(2α−)_(t,x). We also show that, when α ≤ 1/3, the energy profile of the solution has H\"older regularity 2α/(1−α) in time, even though it might fail to be conserved in view of Onsager's conjecture. We demonstrate improved regularity for advective derivatives of the velocity and pressure. In particular, we recover in our context the celebrated result of Chemin that the particle trajectories of classical solutions to Euler are smooth, and establish existence of smooth trajectories in any case where the velocity field has borderline regularity.
The analysis demonstrates that many of the main analytic features of solutions constructed by convex integration methods are consequences of the Euler equations rather than artifacts of the constructions. The proof proceeds by estimating frequency increments associated to the various physical quantities of interest. Several types of commutator estimates play a role in the proof, including the commutator estimate of Constantin, E and Titi for the relevant Reynolds stress and a more flexible proof of this estimate.https://resolver.caltech.edu/CaltechAUTHORS:20180628-113011634A Proof of Onsager's Conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20180627-075519807
Year: 2018
DOI: 10.4007/annals.2018.188.3.4
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.https://resolver.caltech.edu/CaltechAUTHORS:20180627-075519807