Random Perturbation of PDEs and Fluid Dynamic Models: École d’Été de Probabilités de Saint-Flour XL – 2010 (Lecture Notes in Mathematics Book 2015)

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. Annotation. The book 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 Front Matter....Pages i-ix Introduction to Uniqueness and Blow-Up....Pages 1-16 Regularization by Additive Noise....Pages 17-69 Dyadic Models....Pages 71-99 Transport Equation....Pages 101-131 Other Models: Uniqueness and Singularities....Pages 133-159 Back Matter....Pages 161-176