Energy-critical fourth order Schrödinger equations are investigated. We establish local well-posedness and stability in a general setting, and we prove global well-posedness and scattering in the defocusing case for radially symmetrical initial data.1991 Mathematics Subject Classification. 35, 42, 76. Key words and phrases. Energy-critical fourth order Schrödinger equations, local wellposedness, global well-posedness.

We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R × T d . We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 d 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d 2. As a consequence, we obtain global strong solutions (for d 2) with infinitely growing high Sobolev norms H s .

We develop a dynamic model for suspensions of negatively buoyant particles on an incline. Our model includes settling due to gravity and resuspension of particles by shear-induced migration. We consider the case where the particles settle onto the solid substrate and two distinct fronts form: a faster liquid and a slower particle front. The resulting transport equations for the liquid and the particles are of hyperbolic type and we study the dilute limit for which we compute exact solutions. We also carry out systematic laboratory experiments, focusing on the motion of the two fronts. We show that the dynamic model predictions for small to moderate values of the particle volume fraction and the inclination angle of the solid substrate agree well with the experimental data.