# Analysis of nonlinear PDE's

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].

Researchers in this area: Elliott, Kolokoltsov, MacKay, Rodrigo, Robinson, Stinner, Stuart, Topping