PDEs are ubiquitous in applications of mathematics. Rigorous theory has contributed immensely to the development of tools needed to understand the behaviour of solutions. Exemplary topics are: (a) The PDE theory around the Navier-Stokes equations necessary to establish a mathematical framework for a range of data assimilation problems arising in fluid mechanics. (b) Mass conserving nonlinear-reaction diffusion equations modelling biological aggregation and population dynamics. For (a) the natural viewpoint is that of Bayesian statistics, with the desired posterior probability measure on a space of functions. Of interest is the study of problems in which either Eulerian orLagrangian observations are made, and the objective is to make inference about the underlying velocity field, [A, P, S]. Within (b) it is of fundamental interest to study the formation, structure and evolution of singularities. Since the equations conserve mass, they can be seen as representing the evolution of a probability density enabling the use of measure value solutions and the understanding of the solution beyond the first singularity [A, P].