The equation of everything

posted by @ulaulaman about #EulerEquation #CedricVillani #mathematics #physics

Following Cedric Villani, let you imagine a pond. Everything is calm and quiet, without a breath of wind, but at acertain point the pond surface is perturbed by a wave, without any external force. The next minute everything returns calm and quiet.
Villani used this poetical picture in order to describe some results about the Euler equation for an incompressible fluid: \[\frac{\partial v}{\partial t} + \nabla \cdot \left ( v \times v \right ) + \nabla p = f\] \[\nabla \cdot v = 0\] About this equation (or this sistem of wave equations), Scheffer (in 1993) found that the equation permit the spontaneous creation of energy form nothing, while Shnirelman (in 1997), found a new proof of this fact and proposed to enforce a criterion for physically prevent this kind of mathematical solutions. Sheffer and Shnirelman Theorem:

There is a weak solution of the Euler incompressible equation in 2-dimensions, without forcing ($f \equiv 0$) with compact support in space-time.

Between 2007 and 2008 De Lellis and Szekelyhidi proofed a more general theorem that establishes, among other things, that the criterion of Shnirelman is not valid, or, in other world, that the solutions of the Euler incompressible equation without forcing are physically acceptable.

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References:
Cedric Villani, Paradoxe de Scheffer-Shnirelman revu sous l'angle de l'integration convexe, Seminaire Bourbaki (2008)
Scheffer V. (1993). An inviscid flow with compact support in space-time, Journal of Geometric Analysis, 3 (4) 343-401. DOI: 10.1007/BF02921318
Shnirelman A. (1997). On the nonuniqueness of weak solution of the Euler equation, Communications on Pure and Applied Mathematics, 50 (12) 1261-1286. DOI: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6

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Equation of everything:

In theoretical physics, particularly mathematical physics, an equation of everything is an equation that would unify the four fundamental forces of nature into a single equation. In principle, this equation should be able to explain any natural phenomena (i.e, black holes, elementary particles). The four forces, (gravity, electromagnetism, strong force, and weak force) should emerge naturally from this equation. Some physicists such as Stephen Hawking think that it would be impossible to construct a real equation of everything due to the incompleteness theorem of mathematics. Nevertheless, varying theories of everything such as string theory and quantum gravity hope to produce an equation of everything.