@article{5eb95548a3bb4858910035ad413323ee,

title = "CONVEX INTEGRATION CONSTRUCTIONS IN HYDRODYNAMICS",

abstract = "We review recent developments in the field of mathematical fluid dynamics which utilize techniques that go under the umbrella name convex integration. In the hydrodynamical context, these methods produce paradoxical solutions to the fluid equations which defy physical laws. These counterintuitive solutions have a number of properties that resemble predictions made by phenomenological theories of fluid turbulence. The goal of this review is to highlight some of these similarities while maintaining an emphasis on rigorous mathematical statements. We focus our attention on the construction of weak solutions for the incompressible Euler, Navier–Stokes, and magneto-hydrodynamic equations which violate these systems{\textquoteright} physical energy laws.",

author = "TRISTAN BUCKMASTER and VLAD VICOL",

note = "Funding Information: Received by the editors August 25, 2020. 2020 Mathematics Subject Classification. Primary 35Q35. The first author was supported by the NSF grant DMS-1900149 and a Simons Foundation Mathematical and Physical Sciences Collaborative Grant. The second author was supported by the NSF grant CAREER DMS–1911413. 1Some of these phenomenological theories may be traced back to the works of O. Reynolds, L. Prandtl, T. von Karman, G. I. Taylor, L. F. Richardson, W. Heisenberg, A. Kolmogorov, A. Obhukov, L. Onsager, L. Landau, E. Hopf, G. Batchelor, R. H. Kraichnan or P. G. Saffman, and many others. The topic is too vast to review here, and we refer the reader to [9,53,74,82,133] for further references. 2By this we mean quantitative predictions about hydrodynamic turbulence, which are confirmed experimentally both in a laboratory setting and in computer simulations to the point that these predictions are undoubted in the physics community. Examples of such experimental facts include the anomalous dissipation of energy in the infinite Reynolds number limit [158, 172] or Publisher Copyright: {\textcopyright} 2020. All Rights Reserved.",

year = "2021",

doi = "10.1090/bull/1713",

language = "English (US)",

volume = "58",

pages = "1--44",

journal = "Bulletin of the American Mathematical Society",

issn = "0273-0979",

publisher = "American Mathematical Society",

number = "1",

}