December 7th, 2019

(no subject)

Любопытная [полу]популярная статья про диссипативную аномалию:
De Lellis, Camillo; Székelyhidi, László (2019). On turbulence and geometry: from nash to onsager. Notices of the American Mathematical Society, (05):677-685

"The Scheffer–Shnirelman paradox. In [54] V. Scheffer constructed a non-trivial weak solution to the Euler equations in ℝ2 with compact support in space and time. Subsequently A. Shnirelman in [56] gave an entirely different proof in 𝕋2. Such a result is hard to interpret physically, as it would correspond to a perfect incompressible fluid which can start and stop moving by itself, without the action of external forces. As such, for a long time Scheffer’s Theorem remained some sort of “paradox” in the PDE literature [64], cited mostly as a warning example of unphysical behaviour if the notion of solution is too weak, with emphasis more on the non-uniqueness aspect rather than the violation of energy conservation...The first result in connection with Theorem 1(b) was obtained by Shnirelman in [57]—he showed the existence of a solution on 𝕋3 with finite and strictly monotone decreasing energy on some time interval. Nevertheless, this solution is not continuous, and, since it was obtained using a generalized flow model with sticky particles, it seemed to have no connection to the energy cascade picture postulated by Kolmogorov and Onsager"
𝕋 это вроде бы просто пространство с периодичeскими граничными условиями.

Насколько я понял, АШ пытается каким-то образом применять "нестандартный анализ" к этим задачам.