SPD

Abstract

Several examples of PDEs are investigated under the influence of a bilinear multiplicative noise of transport type. The examples include linear models (transport equations, systems related to inviscid vorticity equations, systems of parabolic equations) and nonlinear ones (2D point vortex dynamics, 1D Vlasov-Poisson point charge dynamics, aggregation models, nonlinear parabolic models). The main questions addressed are uniqueness and no-blow-up due to noise - when the deterministic equations may lack uniqueness or have blow-up. This relatively wide variety of examples allows to make a first list of intuitive and rigorous reasons why noise may be an obstruction to the emergence of pathologies in PDEs.