CaltechAUTHORS: Monograph
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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenFri, 11 Oct 2024 19:20:23 -0700Nonuniqueness 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:20180628-113011634