The Fourth Workshop on Fluids and PDE was held at the National Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, from Monday 26 May to Friday 30 May 2014. This workshop is held every two to three years in Brazil. The fourth edition of the workshop was the closing event of a Thematic Program on Incompressible Fluids Dynamics, to be held at IMPA next Spring. Hence, the focus of the workshop will be incompressible fluid mechanics .

We present two types of ordinary differential equations in spaces of trajectories which encode the global attractors of the 2D Navier-Stokes equations (NSE), and the 1D damped driven nonlinear Schrodinger equation (NLS). The idea is derived from the concept of determining modes, determining nodal values, etc., and hence the resulting ODE is called a determining form. In one approach, based on Fourier modes, the trajectories on the global attractor are identified with traveling wave solutions of the determining form. In the other, which can be used with general interpolating operators such as nodal projections, the attractor is identified with steady states of the determining form. While determining forms were originally developed for the NSE, the analysis for the NLS requires a different approach, as it is dispersive and not strongly dissipative. This talk regards joint work with C. Foias, R. Kravchenko, T. Sadigov, and E. Titi .