“Analysis of Nonlinear Partial Differential Equations” (Mathematics)

15 May 2010

The team is addressing different aspects of the contemporary analysis of nonlinear PDEs - from regularity issues for singular and degenerate equations to free boundary problems, from kinetic equations to ocean and climate modeling.

Nonlinear partial differential equations (PDEs) are central to modern applied mathematics, both in view of the significance of the concrete problems that they model and the novel techniques that their analysis generates. In recent years, the subject has developed immensely and in many unexpected and challenging directions. A new range of applications emerged with the advent of biomathematics. This project is structured into two strongly connected branches - the analysis and the applications. They bridge the gap between theoretical aspects related to the analysis of the PDEs and the production of sound information that may have a strong impact in terms of the applications. The ultimate goal is to solve concrete relevant problems based on solid mathematics and the mastery of up-to-date analytical techniques. The team is addressing different aspects of the contemporary analysis of nonlinear PDEs - from regularity issues for singular and degenerate equations to free boundary problems, from kinetic equations to ocean and climate modeling.

An important trend of the project concerns the mathematical and numerical modeling of the ocean and climate. This is a major issue from an economic, environmental, and public health perspective. Taking into account both the rotation of the Earth and the density stratification of the ocean, the ROMS numerical model obtained simulations of the Madeira island wake. The team performed several sensitivity studies that changed the values of the Reynolds number Re, the Rossby number Ro, and the Burger number Bu. Together, they control the three instabilities that may occur in the wake: centrifugal, barotropic, and baroclinic.

The team compared results with similar simulations around an idealized circular island. It found that the predominance of cyclones, regardless of the value of Re, formed the main difference. Contrary to expectations, the team could not find a regime of anticyclonic eddy dominance for moderate values of Re and increasing lambda equals Ro/Bu or dynamical symmetry for small values of lambda. The particular island contour formed the only explanation. (See image. It represents the normalized relative vorticity at two different times of the eddy shedding cycle corresponding to Re equals 400 and lambda equals 0.077. Cyclones and anticyclones are almost aligned, in contrast to the von Kármán wake theory.) As a step closer to reality, future numerical Madeira island wake simulations should include the real ocean bathymetry, a time-varying incoming flow, the effect of the bottom drag on the sloping nearshore topography, and the interaction between the oceanic and atmospheric island wakes.

The team's activity is also centered on the qualitative study of strongly coupled elliptic systems modeling. For example, a binary mixture of Bose-Einstein condensates; the trend to equilibrium of a chemically reactive mixture modeled by means of the spatially homogeneous Boltzmann equation; or the understanding of the local properties of solutions of singular and degenerate PDEs arising from different applications.

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