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3D Euler, Vortex Dynamics and PDE

Final Report Summary - VORT3DEULER (3D Euler, Vortex Dynamics and PDE)

The project focused on a range of mathematical analysis topics arising in fluid mechanics. A wide range of model equations were considered; representative examples include the three-dimensional Euler equation and the Surface Quasi-geostrophic equation.

For the three-dimensional Euler we focused on vortex dynamics and, more specifically, on the evolution of isolated vortex lines. These can be seen as a mathematical idealisation of a tornado for an incompressible fluid. The key difficulty on making rigorous sense of these objects is that, mathematically, they give rise to a singular velocity that moves the vortex lines.

For the surface-quasi geostrophic equation, sharp fronts – an idealised weather front (as the equation originally arose from weather prediction) – are the main focus. As for 3D Euler they give rise a singular velocity that moves the front.

The main results can de divided in various categories:

o Development of general tools for the study of singularities in PDEs, both for quantitative and qualitative analysis.
o Existence and uniqueness results to various partial differential equations in incompressible fluid mechanics (including viscous counterparts).
o Development of detailed asymptotics and geometric descriptions of families of solutions with arbitrarily large gradient but simple geometry, including vortex lines and sharp fronts, as described above.