Random Perturbation of PDEs and Fluid Dynamic Models [electronic resource] : École d’Été de Probabilités de Saint-Flour XL – 2010 / by Franco Flandoli.

1. Introduction to Uniqueness and Blow-up -- 2. Regularization by Additive Noise -- 3. Dyadic Models -- 4. Transport Equation -- 5. Other Models. Uniqueness and Singularities.

This volume deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.