Other Models: Uniqueness and Singularities

This chapter contains a number of other examples, presented for different purposes. Not only the uniqueness problem but also emergence of singularities is discussed. First, we give a few examples where noise does not change the difficulties related to these two issues; a little bit improperly, we call them “negative” examples (in spite of the fact that they are very interesting). Then we show two examples where singularities are prevented by noise: continuity equation and vortex point motion. We call them “positive” examples. The next section on nonlinear Schrödinger equation describes theoretical and numerical results both of positive and negative type. Finally, we summarize the attempts made on the 3D stochastic Navier–Stokes equations, in the direction of understanding uniqueness and singularities.