[
{
"id": "authors:bm5e1-e1120",
"collection": "authors",
"collection_id": "bm5e1-e1120",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180627-075519807",
"type": "article",
"title": "A Proof of Onsager's Conjecture",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
}
],
"abstract": "For any \u03b1 < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class C_tC_^x\u03b1 that have nonempty, compact support in time on R \u00d7 T^3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for \u03b1 < 1/3 due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent \u03b1 = 1/3 marks the threshold for conservation of energy for weak solutions in the class L_t^\u221eC_x^\u03b1. The previous best results were solutions in the classC_tC_x^\u03b1 for \u03b1 < 1/5, due to [Isett], and in the class L_t^1C_x^\u03b1 for \u03b1 < 1/3 due to [Buckmaster, De Lellis, Sz\u00e9kelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, Sz\u00e9kelyhidi]. 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, Sz\u00e9kelyhidi] and the framework of estimates developed in the author's previous work.",
"doi": "10.4007/annals.2018.188.3.4",
"issn": "0003-486X",
"publisher": "Princeton University",
"publication": "Annals of Mathematics",
"publication_date": "2018-11",
"series_number": "3",
"volume": "188",
"issue": "3",
"pages": "871-963"
},
{
"id": "authors:13zvk-3hy98",
"collection": "authors",
"collection_id": "13zvk-3hy98",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180628-113011634",
"type": "monograph",
"title": "Regularity in time along the coarse scale flow for the incompressible Euler equations",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
}
],
"abstract": "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 H\u00f6lder continuous in the spatial variables, which is well-motivated by problems related to turbulence. \n\nWe show that any periodic Euler flow in the class C_tC^\u03b1_x also belongs to C^\u03b1_(t,x), and that the pressure belongs to C^(2\u03b1\u2212)_(t,x). We also show that, when \u03b1 \u2264 1/3, the energy profile of the solution has H\\\"older regularity 2\u03b1/(1\u2212\u03b1) 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. \n\nThe 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.",
"doi": "10.48550/arXiv.1307.0565",
"publisher": "arXiv",
"publication_date": "2018-07-03"
},
{
"id": "authors:bpdne-0ep45",
"collection": "authors",
"collection_id": "bpdne-0ep45",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-160542363",
"type": "monograph",
"title": "Nonuniqueness and existence of continuous, globally dissipative Euler flows",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
}
],
"abstract": "We show that H\u00f6lder 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.",
"doi": "10.48550/arXiv.1710.11186",
"publication_date": "2018-06-26"
},
{
"id": "authors:0hr4y-96b77",
"collection": "authors",
"collection_id": "0hr4y-96b77",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-161143819",
"type": "monograph",
"title": "On the Endpoint Regularity in Onsager's Conjecture",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
}
],
"abstract": "Onsager's conjecture states that the conservation of energy may fail for 3D incompressible Euler flows with H\u00f6lder 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 H\u00f6lder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents [0,1/3). \nOur 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. \nWe 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.",
"doi": "10.48550/arXiv.1706.01549",
"publication_date": "2018-06-26"
},
{
"id": "authors:dk4bg-pw844",
"collection": "authors",
"collection_id": "dk4bg-pw844",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-154151950",
"type": "article",
"title": "On the kinetic energy profile of H\u00f6lder continuous Euler flows",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
},
{
"family_name": "Oh",
"given_name": "Sung-Jin",
"clpid": "Oh-Sung-Jin"
}
],
"abstract": "In [8], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with H\u00f6lder 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^\u221e(B_(3,\u221e)^(1/3) due to low regularity of the energy profile. \n\nThe 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 H\u00f6lder exponent less than 1/5. The main result of this paper shows that any non-negative function with compact support and H\u00f6lder regularity 1/2 can be prescribed as the energy profile of an Euler flow in the class C_(t,x)^(1/5 \u2212 \u03f5). 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\u2013De Lellis\u2013Sz\u00e9kelyhidi [1].",
"doi": "10.1016/j.anihpc.2016.05.002",
"issn": "0294-1449",
"publisher": "Elsevier",
"publication": "Annales de l'Institut Henri Poincare (C) Non Linear Analysis",
"publication_date": "2017-05",
"series_number": "3",
"volume": "34",
"issue": "3",
"pages": "711-730"
},
{
"id": "authors:3vvf9-nxe16",
"collection": "authors",
"collection_id": "3vvf9-nxe16",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180702-123946440",
"type": "book",
"title": "H\u00f6lder continuous Euler flows in three dimensions with compact support in time",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
}
],
"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-H\u00f6lder. 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.\n\nThe construction itself\u2014an intricate algorithm with hidden symmetries\u2014mixes 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\"\u2014used here to construct nonzero solutions with compact support in time and to prove nonuniqueness of solutions to the initial value problem\u2014has 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.",
"doi": "10.48550/arXiv.1211.4065",
"isbn": "978-0-691-17483-9",
"publisher": "Princeton University Press",
"place_of_publication": "Princeton, NJ",
"publication_date": "2017"
},
{
"id": "authors:yf2xg-byp79",
"collection": "authors",
"collection_id": "yf2xg-byp79",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180627-084240647",
"type": "article",
"title": "A heat flow approach to Onsager's conjecture for the Euler equations on manifolds",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
},
{
"family_name": "Oh",
"given_name": "Sung-Jin",
"clpid": "Oh-Sung-Jin"
}
],
"abstract": "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.\nOur 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.",
"doi": "10.1090/tran/6549",
"issn": "0002-9947",
"publisher": "American Mathematical Society",
"publication": "Transactions of the American Mathematical Society",
"publication_date": "2016-09",
"series_number": "9",
"volume": "368",
"issue": "9",
"pages": "6519-6537"
},
{
"id": "authors:cq1tg-ymd45",
"collection": "authors",
"collection_id": "cq1tg-ymd45",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180626-155143019",
"type": "article",
"title": "On Nonperiodic Euler Flows with H\u00f6lder Regularity",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
},
{
"family_name": "Oh",
"given_name": "Sung-Jin",
"clpid": "Oh-Sung-Jin"
}
],
"abstract": "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 H\u00f6lder 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^\u221e_t(B^(1/3)_(3,\u221e) 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 H\u00f6lder 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 \u2212 \u03f5)_(t,x) on any pre-compact subset of \u211d\u00d7\u211d^3 to violate energy conservation. Furthermore, the perturbed solution is no smoother than C^(1/5 \u2212 \u03f5)_(t,x). As a corollary of this theorem, we show the existence of nonzero C^(1/5 \u2212 \u03f5)_(t,x) solutions to Euler with compact space-time support, generalizing previous work of the first author (Isett, H\u00f6lder continuous Euler flows in three dimensions with compact support in time, 2012) to the nonperiodic setting.",
"doi": "10.1007/s00205-016-0973-3",
"issn": "0003-9527",
"publisher": "Springer",
"publication": "Archive for Rational Mechanics and Analysis",
"publication_date": "2016-08",
"series_number": "2",
"volume": "221",
"issue": "2",
"pages": "725-804"
},
{
"id": "authors:4vkp6-yvp72",
"collection": "authors",
"collection_id": "4vkp6-yvp72",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180627-075939255",
"type": "article",
"title": "H\u00f6lder Continuous Solutions of Active Scalar Equations",
"author": [
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
},
{
"family_name": "Vicol",
"given_name": "Vlad",
"clpid": "Vicol-Vlad"
}
],
"abstract": "We consider active scalar equations \u2202_t\u03b8 + \u2207 \u22c5 (u\u03b8)=0, where u = T[\u03b8] 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\u00f6lder regularity C^(1/9\u2212)_(t,x). In fact, every integral conserving scalar field can be approximated in D\u2032 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.",
"doi": "10.1007/s40818-015-0002-0",
"issn": "2524-5317",
"publisher": "Springer",
"publication": "Annals of PDE",
"publication_date": "2015-12",
"series_number": "1",
"volume": "1",
"issue": "1",
"pages": "Art. No. 2"
},
{
"id": "authors:tsfe8-xxa66",
"collection": "authors",
"collection_id": "tsfe8-xxa66",
"cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180627-082316639",
"type": "article",
"title": "Anomalous dissipation for 1/5-H\u00f6lder Euler flows",
"author": [
{
"family_name": "Buckmaster",
"given_name": "Tristan",
"clpid": "Buckmaster-Tristan"
},
{
"family_name": "De Lellis",
"given_name": "Camillo",
"clpid": "De-Lellis-Camillo"
},
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Philip"
},
{
"family_name": "Sz\u00e9kelyhidi",
"given_name": "L\u00e1szl\u00f3, Jr.",
"clpid": "Sz\u00e9kelyhidi-L\u00e1szl\u00f3-Jr"
}
],
"abstract": "Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in H\u00f6lder 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.\nIn 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 H\u00f6lder exponent \u2014 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 \u03b5, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the H\u00f6lder class C^(1/5\u2212\u03b5).",
"doi": "10.4007/annals.2015.182.1.3",
"issn": "0003-486X",
"publisher": "Princeton University",
"publication": "Annals of Mathematics",
"publication_date": "2015-07",
"series_number": "1",
"volume": "182",
"issue": "1",
"pages": "127-172"
}
]