Uniqueness and nonuniqueness of turbulent solutions from singular initial data

Alexei Mailybaev
Institute of Pure and Applied Mathematics (IMPA)

We analyze solutions starting from singular initial conditions in shell models of turbulence. Such initial conditions may result from a finite-time blowup, developed turbulent states, or unstable discontinuities. First, we consider an example with nonunique solutions, which are all physically relevant: An infinite number of solutions arise depending on a way the viscosity approaches zero. Next, we argue that despite of the nonuniqueness of specific physical realizations, a probability distribution for the whole set of possible solutions is unique, i.e., there is a unique spontaneously stochastic solution. This uniqueness is explained as the ordinary deterministic chaos developing in a renormalized system. The results are fully supported by numerical simulations. If time permits, I will show how these ideas can be applied in the inviscid limit of the Rayleigh-Taylor instability.

Presentation (PDF File)

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