Seminaire de Simon Thalabard

11/10/2019 Sophia

Spontaneous stochasticity of shear-layer instabilities

In the late 60's, Edward Lorenz formulated the idea that atmospheric scales of motion could very well be intrinsically unpredictable: No matter how small an initial observational error, one cannot predict the future of the system beyond a prescribed  time, unless in probabilistic terms. 
My intention in this talk is to connect this idea to the more recent  concept of ``spontaneous stochasticity'', and non-uniqueness of solutions for certain systems of ordinary or partial differential equations.
For example, it is known that Lagrangian particles moving in turbulent fluids are also intrinsically unpredictable: Particles that initially coincide almost surely separate in finite time, and the corresponding Lagrangian trajectories then need to be described in probabilistic terms. The origin of the phenomenon is the rough nature of the advecting velocity fields.
Going one step further, we examine the possibility that  velocity dynamics at high-Reynolds number  be deterministic only at a formal level only. In that case intrinsic probabilistic description could arise due  to either  finite-time blow or singular initial conditions.
A physical example is that of   randomly perturbed  Kelvin-Helmholtz interface,  characterized by  a jump in velocity field and infinite vorticity. Using numerics, we substantiate the idea that this kind of singular initial configuration is ``spontaneously unstable'', meaning that  its  future evolution  is determined in a statistical sense only.  

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