![]() ( Wayback Machine) It appears as an article in the Encyclopedia of Mathematical Physics. As for the quick reference, you may find helpful the short survey on turbulence theories by Ricardo Rosa. This is probably not what one would expect from a truly turbulent flow.Įdit 2. ![]() In other words, these solutionsĪctually solve the linear Stokes equation and don't "see" the nonlinearity of the full Navier-Stokes system. The thing is that the nonlinear term $v\cdot \nabla v$ is equal to $0$ for most known classical explicit solutions. Concerning explicit solutions to the Navier-Stokes equations, I don't think any of them really exhibit turbulence features. In spite of the vast popularity of their paper, even the existence of such attractors is still unknown.Įdit 1. Of global attractors with sensitive dependence of motion on the initial conditions in the phase space of the Navier–Stokes equations ( link). In 1970 Ruelle and Takens formulated the conjecture that turbulence is the appearance As you may be aware, this idea is 'regularity theory'. The weakest one says that the maximum of the dimensions of minimal attractors of the Navier–Stokes equations grows along with the Reynolds number Re. Even if your putative solutions only a priori belong a 'big' space which includes lots of coarse functions, it is often the case that being a member of this big space and being a weak or distributional solution to your PDE actually implies much greater smoothness. ![]() Kolmogorov suggested to study minimal attractors of the Navier-Stokes equations and formulated several conjectures as plausible explanations of turbulence. ![]() In the Ptolemaic Landau–Hopf theory turbulence is understood as a cascade of bifurcations from unstable equilibriums via periodic solutions ( the Hopf bifurcation) to quasiperiodic solutions with arbitrarily large frequency basis.Īccording to Arnold and Khesin, in the 1960's most specialists in PDEs regarded the lack of global existence and uniqueness theorems for solutions of the 3D Navier–Stokes equation as the explanation of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-dependent themselves. There is probably no universally accepted mathematical definition of turbulence. ![]()
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